Linear Inference and Multivariate Analysis – Statistics Optional PYQs (1995–2026)

Here are all 218 Linear Inference and Multivariate Analysis questions asked in UPSC Statistics Optional Paper I from 1995 to 2026, arranged under the 7 topics of the official syllabus. Within each topic the latest questions come first.

Each question ends with a code such as [2024/15m/3b] – the year, the marks, the word limit (where UPSC set one) and the question number in the paper.

3.1] Linear models and least squares

48 questions

1. Let $Y_1,Y_2,\cdots ,Y_n$ be $n$ independent observations from a population with mean $\mu$ and variance ${\sigma }^2$. Obtain the best linear unbiased estimator of $\mu$ and an unbiased quadratic estimator for ${\sigma }^2$. [2026/10m/5a]

2. Discussing Gauss-Markov linear model $\underset{n\times 1}{𝐘}=\underset{n\times p}{X}\underset{p\times 1}{𝛃}+\underset{n\times 1}{𝐮}$ with $E\left(𝐘\right)=X𝛃$ and $D\left(𝐘\right)$ = variance and covariance matrix of $𝐘={\sigma }^{2}I$, show that $E[𝐘\prime \{I-X{\left(X\prime X\right)}^{-1}X\prime \}𝐘]={\sigma }^2(n-p)$ [2026/10m/5c]

3. Let $𝐥\prime 𝛃$ be any estimable linear function of parameters ${\beta }_1,{\beta }_2,\cdots ,{\beta }_p$. Then prove that– (i) there exists a unique linear function $𝐜\prime 𝐘$ of random variables $Y_1,Y_2,\cdots ,Y_n$ such that $𝐜\in V\left(A\prime \right)$ and $E\left(𝐜\prime 𝐘\right)=𝐥\prime 𝛃$, where $V\left(A\prime \right)$ is the vector space generated by row vectors of $A\prime$ and $E\left(𝐘\right)=A𝛃$; (ii) variance of $𝐜\prime 𝐘$ is less than the variance of any other linear unbiased estimator. [2026/20m/6a]

4. If $X_1,X_2,\dots ,X_n$ is a random sample from a standard normal population, then using quadratic forms show that the sample mean $\bar{X}=\frac{1}{n}{\sum }_{j=1}^n{X}_j$ and sample variance $S^2=\frac{1}{n-1}{\sum }_{j=1}^n{(X_j-\bar{X})}^2$ are stochastically independent. [2025/10m/5c]

5. Consider the model: $\underset{n\times 1}{Y}=\underset{n\times k}{X}\underset{k\times 1}{\theta }+\underset{n\times 1}{\epsilon },$ where $\epsilon$ is an $n\times 1$ vector of unobservable random variables such that $E\left(\epsilon \right)=0$ and $D\left(\epsilon \right)={\sigma }^2\Omega$, $\sigma >0$ unknown, $\Omega$ is a positive definite matrix of known constants and $rank\left(X\right)=k<n$. Then (i) Derive least square estimator of $\theta$ and (ii) Derive an unbiased estimator of ${\sigma }^2$ [2025/15m/6c]

6. How will you justify the usage of the principle of least squares in estimating the parameters of a linear regression model? With usual notations, for the regression model $\underset{\sim }{y}=X\underset{\sim }{\beta }+\underset{\sim }{\epsilon }$, show that the least square estimator of $\underset{\sim }{\beta }$ is $\widehat{\underset{\sim }{\beta }}={\left(X\prime X\right)}^{-1}X\prime \underset{\sim }{y}$ [2024/10m/5a]

7. Let three mutually independent variables $Y_1,Y_2$ and $Y_3$ having common variance ${\sigma }^2$ and $E\left(Y_1\right)={\beta }_1+{\beta }_2,E\left(Y_2\right)={\beta }_1+{\beta }_3,E\left(Y_3\right)={\beta }_1+{\beta }_2$ be given. Show that the linear parametric function $p_1{\beta }_1+p_2{\beta }_2+p_3{\beta }_3$ is estimable if and only if $p_1=p_2+p_3$, clearly stating the assumptions used, if any. [2023/5m/6b(ii)]

8. Define general linear model with usual assumptions. If $y_1={\beta }_1+u_1,y_2=-{\beta }_1+{\beta }_2+u_2,y_3=-{\beta }_2+u_3,$ where $u_1,u_2,u_3$ are mutually independent random variables with mean zero and variance ${\sigma }^2$, then find the least square estimators of ${\beta }_1$ and ${\beta }_2$. [2022/10m/5a]

9. (i) What are orthogonal polynomials? How do you fit an orthogonal polynomial of degree ‘$p$’? (ii) For the model $\underset{n\times 1}{𝐘}=\underset{n\times k}{X}\underset{k\times 1}{𝛃}+\underset{n\times 1}{𝐮},E\left(𝐮𝐮\prime \right)={\sigma }^2{I}_n$, where $\underset{n\times k}{X}$ is a matrix of rank $k(k<n)$, find out the value of $E\left[𝐘\prime (I_n-X{\left(X\prime X\right)}^{-1}X\prime )𝐘\right].$ [2022/20m/8a]

10. Define Gauss-Markov linear model. State and prove necessary and sufficient condition for the linear function $l\prime \beta$ of the parameters to be linearly estimable. [2020/10m/5d]

11. Let $Y_1,Y_2,Y_3$ be uncorrelated variables with common variance ${\sigma }^2$ such that $E\left(Y_1\right)={\beta }_1+{\beta }_{2}E\left(Y_2\right)=2{\beta }_{1}E\left(Y_3\right)={\beta }_1-{\beta }_2$ Compute residual mean square. [2020/10m/6b]

12. In order to estimate two parameters $\theta$ and $\varphi$, a number of independent measurements are taken, each having errors with mean zero and variance ${\sigma }^2$. Suppose there are (i) $n$ observations with mean $\theta$, (ii) $m$ observations with mean $(\theta -\varphi )$, and (iii) $m$ observations with mean $(\varphi -\theta )$. Set up this in the form of a standard linear model. What are the least square estimates of $\theta$ and $\varphi$? Find $var\left(\hat{\varphi }\right)$, $cov(\hat{\theta },\hat{\varphi })$ and $var(\hat{\varphi }-\hat{\theta })$. [2019/10m/5a]

13. Consider a standard linear model with design matrix $X=\left(\begin{array}{cc}\hfill -2\hfill & \hfill -2\hfill \\ \hfill -1\hfill & \hfill -1\hfill \\ \hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill 2\hfill \\ \hfill -1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill -1\hfill \end{array}\right)$ and unknown coefficients ${\beta }_1$ and ${\beta }_2$. Which linear combinations $\hat{\psi }=a_1{\hat{\beta }}_1+a_2{\hat{\beta }}_2$ have largest and smallest variance subject to $a_1^2+a_2^2=1$? [2019/10m/5b]

14. The period of oscillation, $t$ of a pendulum is $2\pi \sqrt{l/g}$ where $l$ is the length of the pendulum and $g$ is the gravitational constant. The periods observed are $t_{ij}$ ($j=1,2,\dots ,n_i$) and lengths $l_i$ ($i=1,\dots ,k$) of the pendulum, in an experiment. Assuming the errors of observations to be uncorrelated with zero means and variances ${\sigma }^2$, show that the best linear estimator of $\beta =2\pi /\sqrt{g}$ is $\hat{\beta }=\frac{\sum {\left(l_i\right)}^{\frac{1}{2}}{\sum }_j^{}{t}_{ij}}{{\sum }_i^{}{n}_i{l}_i}\mathrm{\text{and}}V\left(\hat{\beta }\right)=\frac{{\sigma }^2}{{\sum }_i^{}{n}_i{l}_i}$ [2018/10m/5a]

15. For a model $Y_i={\beta }_0+{\beta }_1{X}_i+{\epsilon }_i,i=1,2,3$ where $X_1=-1$, $X_2=0$, $X_3=1$, find BLUE’s of ${\beta }_1$, ${\beta }_0$. Suppose the model is not correct and the true model is $Y_i={\beta }_0+{\beta }_1{X}_i+{\beta }_2{X}_i^2+{\epsilon }_i.$ Find the bias of the BLUE’s obtained. [2018/15m/8c]

16. $Y_1,Y_2,Y_3,Y_4$ are four independent variables with $E\left(Y_1\right)=E\left(Y_3\right)={\theta }_1+{\theta }_3+{\theta }_4$, $E\left(Y_2\right)=E\left(Y_4\right)={\theta }_1-{\theta }_2$ and $V\left(Y_i\right)={\sigma }^2$ (for $i=1,2,3,4$). Verify whether ${\theta }_1+{\theta }_3$ and ${\theta }_2+{\theta }_3+{\theta }_4$ are estimable. If so, obtain their BLUEs. Also obtain the variances of the BLUEs. [2017/10m/5a]

17. Show that in the model $y=X\beta +\epsilon$ subject to $C\beta =0$, the estimator of $\beta$ is ${\hat{\beta }}_C=\hat{\beta }-{\left(X\prime X\right)}^{-1}C\prime {\left[C{\left(X\prime X\right)}^{-1}C\prime \right]}^{-1}C\hat{\beta }$ where $\hat{\beta }={\left(X\prime X\right)}^{-1}X\prime y$. Note here that $y$ is $n\times 1$, $X$ is $n\times k$, $\beta$ is $k\times 1$ and $C$ is $q\times k$ of rank $q\le k$. [2017/15m/6b]

18. Let $y_1={\beta }_1-{\beta }_2+e_1$, $y_2={\beta }_3+e_2$ and $y_3={\beta }_1+{\beta }_2+e_3$, where $e_1$, $e_2$ and $e_3$ are iid $N(0,{\sigma }^2)$. Obtain best linear unbiased estimator (BLUE) of $(2{\beta }_1-{\beta }_2)$. Is this linear parametric function testable? [2017/15m/7c]

19. If $l\prime y$ and $m\prime y$ be two estimates of the estimable function $\lambda \prime \beta$, and $l_s$ and $m_s$ are the projections of $l$ and $m$ respectively on $V_s$ (the column space of coefficient matrix $A$), then show that $l_s=m_s$. [2016/10m/5b]

20. Consider four random variables $Y_1,Y_2,Y_3$ and $Y_4$ with means and variances as follows: $E\left(Y_1\right)=E\left(Y_2\right)={\theta }_1+{\theta }_2,E\left(Y_3\right)=E\left(Y_4\right)={\theta }_1+{\theta }_3\mathrm{\text{and}}V\left(Y_i\right)={\sigma }^2,i=1,2,3,4$ Determine the condition of estimability of the parametric function $𝐥\prime 𝛉=l_1{\theta }_1+l_2{\theta }_2+l_3{\theta }_3$ [2016/15m/6c]

21. Consider three independent random variables $Y_1$, $Y_2$ and $Y_3$ having common variance ${\sigma }^2$ and $E\left(Y_1\right)={\beta }_1+{\beta }_3$, $E\left(Y_2\right)={\beta }_1+{\beta }_2$, $E\left(Y_3\right)={\beta }_1+{\beta }_3$. Determine the condition of estimability of the linear parametric function $l\prime \beta$. Obtain a solution of the normal equation and the S.S. due to error. [2015/10m/5c]

22. For the Gauss Markov Model $(\underset{\sim }{Y},X\underset{\sim }{\beta },{\sigma }^{2}I)$, the estimator $\underset{\sim }{t}\prime \underset{\sim }{Y}$ is the best linear unbiased estimator (BLUE) for $E\left(\underset{\sim }{t}\prime \underset{\sim }{Y}\right)$ iff $\underset{\sim }{t}\prime \underset{\sim }{Y}$ is uncorrelated with all unbiased estimators of zero. [2015/15m/6c]

23. Let $Y_1,Y_2,Y_3$ be three independent observations having expectations $E\left(Y_1\right)={\beta }_0-{\beta }_1+{\beta }_2,E\left(Y_2\right)={\beta }_0-2{\beta }_2,E\left(Y_3\right)={\beta }_0+{\beta }_1+{\beta }_2\mathrm{\text{and}}V\left(Y_i\right)={\sigma }^2\mathrm{\text{for}}i=1,2,3.$ Obtain least square estimates of ${\beta }_0$, ${\beta }_1$ and ${\beta }_2$. Can you obtain unbiased estimate of ${\sigma }^2$? Justify your claim. [2014/10m/5a]

24. In the Gauss-Markov set up $(Y,X\beta ,{\sigma }^{2}I)$ a least square estimate is given by a solution of the system: $X\prime \times \hat{\beta }=X\prime Y$ (i) Justify the above statement. (ii) Establish that the above system of equations is always consistent. [2014/15m/8c]

25. Consider the following linear model: $y_1={\theta }_1+{\theta }_2+{\theta }_3+{\theta }_4+{\epsilon }_1,y_2={\theta }_1+{\theta }_3-{\theta }_2-{\theta }_4+{\epsilon }_2,y_3={\theta }_1+{\theta }_2-{\theta }_3-{\theta }_4+{\epsilon }_3,y_4={\theta }_1+{\theta }_4-{\theta }_2-{\theta }_3+{\epsilon }_4$ where ${\theta }_1,{\theta }_2,{\theta }_3$ and ${\theta }_4$ are parameters, ${\epsilon }_j\sim (0,{\sigma }^2),j=1\dots 4$. ${\epsilon }_j$’s are all independently distributed. Derive normal equations and obtain the BLUE of ${\theta }_1+{\theta }_2+{\theta }_3+{\theta }_4$. Also find the variance of the BLUE. [2013/10m/5a]

26. Consider a model $y_{ij}=\mu +{\alpha }_i+{\epsilon }_{ij},i=1,\dots k,j=1,\dots n$, ${\epsilon }_{ij}$’s are i.i.d. $N(0,{\sigma }^2)$. Obtain least squares solutions of $\mu$ and ${\alpha }_i$. When is a linear parametric function of ${\alpha }_i$’s (only) estimable? Derive a test statistic for testing $H_0:{\sum }_{i=1}^n{C}_i{\alpha }_i=0$. [2013/10m/5b]

27. $Y_1,Y_2,Y_3$ and $Y_4$ are independent stochastic variables with common variance ${\sigma }^2$ and $E\left(Y_1\right)=E\left(Y_2\right)=\mu +{\alpha }_1$ and $E\left(Y_3\right)=E\left(Y_4\right)=\mu +{\alpha }_2$ (i) Examine whether ${\alpha }_1$ is estimable. (ii) Find the best unbiased estimator of $({\alpha }_1-{\alpha }_2)$. (iii) Obtain the variance of the best unbiased estimator of $({\alpha }_1-{\alpha }_2)$. [2012/12m/5b]

28. Derive the normal equations for estimating the parameters of a linear regression model by the method of least squares. [2012/12m/5e]

29. Assuming the model $\begin{array}{cc}\hfill x_i\hfill & \hfill =m_1+e_{1i}\hfill \\ \hfill y_i\hfill & \hfill =m_2+e_{2i}\hfill \\ \hfill z_i\hfill & \hfill =m_1+m_2+e_{3i},i=1\left(1\right)n\hfill \end{array}$ where $e_{1i}$, $e_{2i}$ and $e_{3i}$ are the errors, obtain least square estimates of $m_1$ and $m_2$. [2011/12m/5b]

30. Let $y^{n\times 1}$ be a random vector distributed as $N_n(0,{\sigma }^2{I}_n)$ Derive necessary and sufficient conditions for the quadratic form $Y\prime AY$ to be distributed as ${\chi }^2$. [2007/20m/6a]

31. In a Gauss-Markoff set-up, define an estimable parametric function. Show that if $r\left(X\right)=p$, then every parametric function is estimable where we consider the set-up $(y^{n\times 1},X^{n\times p}{\beta }^{p\times 1},{\sigma }^2{I}_n)$ [2007/20m/6c]

32. Discuss least squares method of estimation and the relation between least squares estimators and best linear unbiased estimators. [2006/20m/4a]

33. Consider the model $E\left(y_1\right)=2{\beta }_1+{\beta }_2,E\left(y_2\right)={\beta }_1-{\beta }_2,E\left(y_3\right)={\beta }_1+\alpha {\beta }_2$ with the usual assumptions. Determine $\alpha$ so that the BLUE’s of ${\beta }_1$ and ${\beta }_2$ are uncorrelated. [2005/20m/6a]

34. Show that a linear function of observations belongs to the error space if and only if its coefficient vector is orthogonal to the columns of the matrix X in the usual linear model $Y=X\beta +\epsilon$. [2004/12m/5b]

35. For the model $y_1={\beta }_1+{\beta }_2+{\epsilon }_1,y_2={\beta }_1+{\beta }_3+{\epsilon }_2,y_3={\beta }_1+{\beta }_2+{\epsilon }_3$ show that ${\lambda }_1{\beta }_1+{\lambda }_2{\beta }_2+{\lambda }_3{\beta }_3$ is estimable if and only if ${\lambda }_1={\lambda }_2+{\lambda }_3$. [2004/20m/6a]

36. If $Y=X\beta +e$ is a general linear hypothesis model of full rank. Find the estimate of $\beta$ the unknown parameter using appropriate method when the distribution of e is unspecified. [2003/12m/5a]

37. Fill up the blanks in parts (i), (ii) and (iii) and answer (iv). (i) The arithmetic mean of a positive valued statistical variable is than the reciprocal of the arithmetic mean of the reciprocal of the variable. (ii) If one observation is missing in a Randomized Block Design with v treatments and b blocks, in the Analysis of Variance Table, the number of degrees of freedom for treatments will be and that for error will be (iii) The Gauss-Markov Theorem requires the error components to have mean values, variance and all pair-wise covariance’s to be (iv) Write down the structure of the Analysis of Variance Table for a Latin Square Design of order m. [2001/12m/5c]

38. In connection with the Gauss-Markov linear models, what do you mean by Design matrix and Normal equations? Discuss the least squares principle and use it to derive the normal equations. [2000/12m/5a]

39. Identify, giving reasons, which of the following models are linear? Wherever possible, suggest a transformation $Z_i=g\left(Y_i\right)$ that will result in a linear model for $Z_i$: (i) $Y_i={\beta }_1+{\beta }_2{x}_i^2+e_i$ (ii) $Y_i={\beta }_1+{\beta }_2{x}_i+e_i^2$ [2000/12m/5b]

40. Consider the following linear model $\begin{array}{cc}\hfill y_1\hfill & \hfill ={\theta }_1+{\theta }_4+{\theta }_5+e_1\hfill \\ \hfill y_2\hfill & \hfill ={\theta }_2-{\theta }_4+e_2\hfill \\ \hfill y_3\hfill & \hfill ={\theta }_3+2{\theta }_5+e_3\hfill \\ \hfill y_4\hfill & \hfill ={\theta }_1+{\theta }_2+2{\theta }_5+e_4\hfill \\ \hfill y_5\hfill & \hfill ={\theta }_1+{\theta }_3+{\theta }_4-e_5\hfill \end{array}$ Where $e_i$’s are i.i.d. $N(0,{\sigma }^2)$ variables (i) Show that ${\sum }_{i=1}^5{l}_i{\theta }_i$ is estimable if and only if $l_4=l_1-l_2$ and $l_5=l_1+l_2-2l_3$. (ii) Obtain the BLUe of ${\theta }_1+{\theta }_2+{\theta }_3$ and ${\theta }_3-2{\theta }_5$ [2000/10m/6a]

41. State and prove Gauss-Markoff theorem. [1999/30m/9a]

42. $y_1,y_2$ and $y_3$ are uncorrelated observations with a common variance ${\sigma }^2$ and expectations $E\left(y_1\right)={\theta }_1+{\theta }_2,E\left(y_2\right)={\theta }_1+{\theta }_3\mathrm{\text{and}}E\left(y_3\right)={\theta }_1+{\theta }_2$ (i) Show $l_1{\theta }_1+l_2{\theta }_2+l_3{\theta }_3$ that is estimable if and only if $l_1=l_2+l_3$. (ii) Find an unbiased estimate of ${\sigma }^2$. [1999/30m/9b]

43. Discuss least square estimate and show that least-square estimates are the same as the best linear estimates in linear estimation. [1998/20m/9a]

44. For the model $y=X\beta +e$, we have $\hat{\beta }={\left(X\prime X\right)}^{-1}X\prime y$, $E(\hat{\beta }-\beta )(\hat{\beta }-\beta )\prime ={\sigma }^2{\left(X\prime X\right)}^{-1}$, then test statistic $F=\frac{Q_2\left({\beta }_0\right)}{p}/\frac{Q_1}{n-p}$ for $H_0:\beta ={\beta }_0$, where $Q_1=(y-X\hat{\beta })\prime (y-X\hat{\beta })$, $Q_2=\left(X\prime e\right)\prime {\left(X\prime X\right)}^{-1}\left(X\prime e\right)$, $e=(y-X{\beta }_0)$, find the distribution of F and state its properties. [1998/25m/10b]

45. In connection with the problem of linear estimation, explain the terms ‘estimable parametric function’, ‘error function’ and ‘best linear unbiased estimator (BLUE)’. [1997/24m/9a]

46. Consider three independent random variables $Y_1$, $Y_2$ and $Y_3$ having common variance ${\sigma }^2$ and $E\left(Y_1\right)=2{\mu }_1+3{\mu }_2,E\left(Y_1\right)=3{\mu }_1+4{\mu }_2\mathrm{\text{and}}E\left(Y_3\right)=4{\mu }_1+5{\mu }_2$ Show that ${\mu }_1$ and ${\mu }_2$ are estimable. Find their best linear unbiased estimates. [1997/36m/9b]

47. With the usual Gauss-Markov linear model, define the estimation space and the error space. Obtain expressions for the SS due to error and the SS due to all possible BLUEs of parametric functions. Find the expectation of the former. [1996/30m/9a]

48. Let $l_1\prime \hat{\beta }$ and $l_2\prime \hat{\beta }$ be the LS estimators of two estimable functions $l_1\prime \beta$ and $l_2\prime \beta$ respectively. Obtain the variances and covariances of the estimators. [1996/30m/9b]

3.2] Analysis of variance

14 questions

1. Write down the model used in the analysis of a two-way classification with interactions, stating the assumptions. What are the hypotheses tested in this scenario? Obtain the expression for the sum of squares and complete the ANOVA. [2024/15m/8c]

2. Describe the complete analysis of two-way classified data with multiple (but equal) observations per cell, clearly stating the assumptions used. Also state two examples where such type of analysis is used. [2023/15m/6b(i)]

3. In a set of two-way classified data according to $k$ levels of factor A and $r$ levels of factor B, there is one observation in each cell. Show that the total number of error contrasts is $(r-1)(k-1)$. [2022/15m/6a]

4. Three sides of an equilateral triangle were measured by 5 persons with the following results:

Sides \ Persons	A	B	C	D	E
$a$	5.44	5.41	5.43	5.42	5.43
$b$	5.43	5.41	5.42	5.43	5.44
$c$	5.45	5.42	5.43	5.43	5.44

Is there any significant difference between (i) measurements by the persons and (ii) the sides of the triangle? [2017/20m/8a]

5. Write down the model used for one way analysis of variance. The lifetimes (in hours) of samples taken from three different brands of batteries gave the following: $n_1=5$, $n_2=4$, $n_3=6$, $\bar{x}=40$, $\bar{y}=55$, $\bar{z}=60$, ${\sum }_{i=1}^5{(x_i-\bar{x})}^2=400$, ${\sum }_{i=1}^4{(y_i-\bar{y})}^2=350$ and ${\sum }_{i=1}^6{(z_i-\bar{z})}^2=850$. Test the hypothesis that there is no difference in lifetimes for the three brands. Also write down the ANOVA $(F_{2,12,.05}=3.89)$. [2012/20m/6b]

6. Discuss the two-way analysis of variance model giving its analysis and ANOVA table. [2006/20m/8a]

7. Describe Tukey’s test for non-additivity for a two-way layout with one observation per cell. [2005/12m/5b]

8. Explain the concept of one-way classified data. Write the linear model for it Obtain the least square estimate of the parameters involved in it. [2003/20m/6c]

9. Describe an appropriate test for equality of means of several populations on the basis of independent samples drawn from them. [2001/15m/6iii]

10. In the case of two-way classified data with unequal number of observations per cell derive the expectation of the MS components of the ANOVA table. [1998/20m/9c]

11. Define contrasts, orthogonal contrasts and show that SS due to n observations is equal to the total SS due to the orthogonal contrasts. [1998/20m/10a]

12. Write down the model of a complete three-way classified data with equal number of observations per cell. Write down the corresponding ANOVA table. [1996/15m/10c]

13. Describe the model and the analysis of variance of a complete two-way classified data with equal ($>1$) number of observations per cell. State clearly what you will do if the interaction effects (i) are not significant (ii) are significant. [1995/30m/9a]

14. In the case of two-way classified data with one observation per cell derive the expectations of the MS components of the ANOVA table. [1995/30m/9b]

3.3] Regression analysis

31 questions

1. Discuss orthogonal polynomial regression on a single variable along with its principal advantages. [2026/15m/6c]

2. For a two variable linear regression model $Y_i=a+b{X}_i+e_i$, where $E\left(e_i\right)=0$, $Var\left(e_i\right)={\sigma }_e^2$, $Cov(e_i,e_j)=0$ for $i\ne j,(i,j)\in \{1,2,\dots ,n\}$, if $\hat{a}$ and $\hat{b}$ are least square estimators of $a$ and $b$ respectively, derive expressions for $Var\left(\hat{a}\right)$, $Var\left(\hat{b}\right)$ and $Cov(\hat{a},\hat{b})$. [2025/10m/5a]

3. For the simple linear regression model $y={\beta }_0+{\beta }_{1}x+\epsilon$, where ${\beta }_0$ and ${\beta }_1$ are parameters and $\epsilon$ has zero mean and an unknown variance ${\sigma }^2$, find the estimates of ${\beta }_0$ and ${\beta }_1$ by the principle of least squares as well as the method of maximum likelihood. Examine whether they are identical. [2024/15m/6c]

4. Let $x_1,x_2,\dots ,x_n$ be $n$ given observations, and suppose that $Y_i={\beta }_0+{\beta }_1{x}_i+e_i;i=1,2,\dots ,n,$ where ${\beta }_0,{\beta }_1$ are unknown parameters and $e_i$ are mutually independent normal random variables with $E\left(e_i\right)=0$ and $V\left(e_i\right)={\sigma }^2,i=1,2,\dots ,n$. Also, ${\sigma }^2$ is assumed to be unknown. Test the null hypothesis $H_0:{\beta }_0={\beta }_1=0$. [2023/10m/5c]

5. (i) What are orthogonal polynomials? How do you fit an orthogonal polynomial of degree ‘$p$’? (ii) For the model $\underset{n\times 1}{𝐘}=\underset{n\times k}{X}\underset{k\times 1}{𝛃}+\underset{n\times 1}{𝐮},E\left(𝐮𝐮\prime \right)={\sigma }^2{I}_n$, where $\underset{n\times k}{X}$ is a matrix of rank $k(k<n)$, find out the value of $E\left[𝐘\prime (I_n-X{\left(X\prime X\right)}^{-1}X\prime )𝐘\right].$ [2022/20m/8a]

6. For a simple linear regression model $Y={\beta }_0+{\beta }_1{X}_i+{\epsilon }_i,i=1,\dots ,n$ (i) Derive the least square estimators of ${\beta }_0$ and ${\beta }_1$, clearly stating the conditions assumed. (ii) For $e_i=Y_i-{\hat{Y}}_i$ where ${\hat{Y}}_i$ is the fitted value, show that 1. ${\sum }_{i=1}^n{e}_i=0$ 2. ${\sum }_{i=1}^n{Y}_i={\sum }_{i=1}^n{\hat{Y}}_i$ 3. ${\sum }_{i=1}^n{X}_i{e}_i=0$ 4. ${\sum }_{i=1}^n{\hat{Y}}_i{e}_i=0$ 5. The regression line passes through $(\bar{X},\bar{Y})$. [2021/10m/5a]

7. For the multiple linear regression model with two predictor variables $X_1$ and $X_2$, show that the estimate of regression coefficient of $X_1$ is unchanged when $X_2$ is added to the regression model, whenever $X_1$ and $X_2$ are uncorrelated. [2021/10m/5c]

8. (i) For a second order polynomial model with one predictor variable, derive the least squares normal equations clearly stating the conditions assumed. How will you interpret the parameters in this model? (ii) Describe why it is recommended to work with predictor variables centred around the mean. Comment on fitted values of the response variable in this case. Prove your claim. [2021/15m/8b]

9. A student fits the following linear regression model to a given data: $y_i=\alpha +\beta x_i+u_i,i=1,2,\dots ,n$ $u_i$’s are iid normal values ($N(0,{\sigma }^2)$), where ${\sigma }^2$ is unknown. The following information is available: $n=7,\sum {(x_i-\bar{x})}^2=280000,\sum (x_i-\bar{x})(y_i-\bar{y})=16500$ 95% confidence interval for $\beta$ is: $(0.030,0.088)$. Calculate what proportion of the total variability of the responses is explained by the model. Given that $t_{0.025,5}=2.57$. [2020/10m/5a]

10. Consider a simple regression model $y={\beta }_0+{\beta }_{1}x+e$ based on $n$ observations. Show that the regression sum of squares is ${\hat{\beta }}_1^2{\sum }_{i=1}^n{(x_i-\bar{x})}^2$. Suppose the summarized data on 10 observations yield ${\sum }_i^{}{(y_i-{\hat{y}}_i)}^2=3.267,{\sum }_i^{}{y}_i=63.3,{\sum }_i^{}{y}_i^2=423.49,{\sum }_i^{}{x}_i=139,{\sum }_i^{}{x}_i^2=2239$ Estimate ${\beta }_1$ assuming positive correlation between $X$ and $Y$. Also obtain coefficient of determination. [2017/15m/8b]

11. Consider the simple linear regression model $y_i={\beta }_0+{\beta }_1{x}_i+e_i$, $i=1,2,\dots ,n$. Show that if the $x_i$s’ are equally spaced (i.e. $x_i=u+iv$ for fixed values of $u$ and $v$), then $y_i={\gamma }_0+{\gamma }_{1}i+e_i$ is an equivalent reparametrization (in the sense that both the design matrices have the same column space). [2015/15m/7c]

12. Give the outline of the procedure, using ANOVA technique, for testing if the response variable $y$ is dependent on a fixed set of variables $x_1,x_2,\dots ,x_k$, based on an observed set of data on these variables. [2011/20m/6a]

13. Consider the mean vector for $X=\left[\begin{array}{c}\hfill X_1\hfill \\ \hfill X_2\hfill \end{array}\right]$ to be ${\mu }_x=\left[\begin{array}{c}\hfill 3\hfill \\ \hfill -2\hfill \end{array}\right]$ and for $Y$ to be ${\mu }_y=4$. The covariance matrices for $X_1$, $X_2$ and $Y$ are ${\Sigma }_{xx}=\left[\begin{array}{cc}\hfill 2\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill \end{array}\right],{\sigma }_{YY}=9\mathrm{\text{and}}{\Sigma }_{XY}=\left[\begin{array}{c}\hfill 3\hfill \\ \hfill 1\hfill \end{array}\right]$ Fit the equation $Y=b_0+b_1{X}_1+b_2{X}_2$ and calculate multiple correlation coefficient. [2010/12m/5d]

14. Obtain an expression for estimated value of $Y$ for given $X=x_0$ by fitting a curve $e^{X}Y=a+b{e}^{2X}$ to $n$ given points $(x_i,y_i)$, $i=1,2,\dots ,n$ using least-square method. [2010/20m/8b]

15. Based on a set of data $(x_i,y_{ij})$, $j=1\left(1\right)n_i$, $i=1\left(1\right)p$, discuss how you would use the technique of Analysis of Variance for testing the hypothesis that the regression equation of $y$ on $x$ is linear. [2009/20m/6b]

16. What are outliers in the context of regression analysis? How can you use Cook’s distance to detect outliers? [2007/12m/5a]

17. Describe curvilinear regression. Given that ${\sum }_1^{8}X=158,{\sum }_1^8{X}^2=5530,{\sum }_1^{8}logy=13.0312\mathrm{\text{and}}{\sum }_1^{8}Xlogy=212.1224,$ fit the exponential curve $y=a{b}^X$ [?] applying the method of least squares. [2006/20m/6b]

18. Derive the Shapiro-Wilk statistic for testing normality of errors in a regression model. [2005/12m/5c]

19. Define regression analysis. Write the form of simple linear regression model. State the various assumptions of the model. [2003/12m/5b]

20. Explain the linear regression model. Calculate the least squares estimate of $\beta =\left[\begin{array}{c}\hfill {\beta }_0\hfill \\ \hfill {\beta }_1\hfill \end{array}\right]$ for a straight line model $Y_j={\beta }_0+{\beta }_1{Z}_{j1}+{\epsilon }_j$ from the data.

$Z_j$	0	1	2	3	4
y	1	4	3	8	9

[2002/20m/6a]

21. Let $y_i=m+b{x}_i+e_i(i=1,2,\dots ,n)$, find estimate of m and b and express $Var\left(\hat{b}\right)$. [1998/20m/9b]

22. In the multiple linear regression model with normality assumption obtain $100(1-\alpha )\%$ confidence interval to find out whether the regression on the regressor variables is significant or not. [1998/15m/10c]

23. Using the linear regression model $E(Y\mid x_1,x_2,x_3)={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_3$ and n independent sets of observations, derive the tests for the following hypotheses (i) $H_0:{\beta }_1={\beta }_2={\beta }_3$ (ii) $H_0:{\beta }_1+{\beta }_2+{\beta }_3=1$ You may make the usual assumptions. In each case state clearly the alternative hypothesis you consider. [1997/30m/10a]

24. Explain how orthogonal polynomials are used in fitting regression curves. Develop an appropriate test for choosing the appropriate degree of orthogonal polynomial using ANOVA table. [1997/30m/10b]

25. Given two regression lines $Y_{ki}={\beta }_k{X}_i+e_{ki}(k=1,2;i=1,2,\dots ,n)$ show that the F-statistic for testing $H:{\beta }_1={\beta }_2$ can be put in the form $F=\frac{{({\hat{\beta }}_1-{\hat{\beta }}_2)}^2}{2S^2{\left({\sum }_1^n{x}_i^2\right)}^{-1}}$ Write the expression of $S^2$. [1996/20m/10a]

26. In the multiple linear regression model with $(p-1)$ independent variables, derive a test to find out whether the regression on the regressor variables is significant or not. Show that the test statistic can be expressed in terms of the sample multiple correlation coefficient. [1996/25m/10b]

27. Show that under usual normal population regression model the joint distribution of the partial regression coefficients is multivariate normal. [1996/30m/11b]

28. In the multiple linear regression model with k independent variables, ${\beta }_k$ is the estimated regression coefficient for the $k^{\mathrm{\text{th}}}$ independent variable ${\beta }_k$. Show that the statistic for testing the hypothesis ${\beta }_k=0$ is provided by $t=\frac{{\hat{\beta }}_k}{\sqrt{\widehat{Var}\left({\hat{\beta }}_k\right)}},$ where $\widehat{Var}\left({\hat{\beta }}_k\right)$ is the estimated variance of ${\hat{\beta }}_k$. What is the critical region for the alternatives ${\beta }_k\ne 0$? [1995/30m/10a]

29. In the multiple linear regression model with usual normality assumptions obtain the $100(1-\alpha )\%$ confidence interval for a regression coefficient ${\beta }_i$ which is symmetric with respect to ${\hat{\beta }}_i$. [1995/15m/10b]

30. Indicate, with the help of the ANOVA table, how you will test for the linearity of regression. [1995/15m/10c]

31. What are orthogonal polynomials? State how these are used in fitting regression curves and obtain the LS estimates of the coefficients in the regression equation. Indicate how the technique of ANOVA is used in deciding the appropriate degree of the polynomial to be fitted. [1995/30m/12b]

3.4] Multiple and partial correlation

23 questions

1. If $X_1$, $X_2$ and $X_3$ are three variables, obtain the expression for the partial correlation coefficient between $X_1$ and $X_2$ eliminating the effect of $X_3$, ${\rho }_{12\cdot 3}$, in terms of simple correlation coefficients. [2024/5m/5b(ii)]

2. Let $𝐗=\left(X_1{X}_2{X}_3\right)\prime$ be distributed as $N_3(𝛍,\Sigma )$, where $𝛍=(2-13)\prime$ and $\Sigma =\left(\begin{array}{ccc}\hfill 4\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 2\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 3\hfill \end{array}\right).$ Find (i) the conditional distribution of $\left(X_1{X}_2\right)\prime$ given $X_3=2$. (ii) partial correlation coefficient ${\rho }_{12.3}$ and multiple correlation coefficient $R_{1.23}$. [2023/15m/6a]

3. (i) If $X_1=Y_1+Y_2,X_2=Y_2+Y_3,X_3=Y_3+Y_1$, where $Y_1,Y_2$ and $Y_3$ are uncorrelated random variables and each of which has zero mean and unit standard deviation, find the multiple correlation coefficient between $X_3$ and $X_1,X_2$. (ii) Let $𝐗$ be a 3-dimensional random vector with dispersion matrix $\Sigma =\left(\begin{array}{ccc}\hfill 9\hfill & \hfill 3\hfill & \hfill 3\hfill \\ \hfill 3\hfill & \hfill 9\hfill & \hfill 3\hfill \\ \hfill 3\hfill & \hfill 3\hfill & \hfill 9\hfill \end{array}\right).$ Determine the first principal component and the proportion of the total variability that it explains. [2022/15m/6c]

4. For a multiple linear regression model with three covariates $X_1,X_2$ and $X_3$, let $r_{ij}$ denote the correlation coefficient between $X_i$ and $X_j$. For a data, it was found $r_{12}=0.77,r_{23}=0.52,r_{13}=0.72$. (i) Check the consistency of the above data. (ii) If $r_{13}$ is unknown, obtain the limits within which $r_{13}$ lies given the above values for $r_{12}$ and $r_{23}$. [2021/20m/6a]

5. Let $𝐗\sim N_3(\left(\begin{array}{c}\hfill 1\hfill \\ \hfill 2\hfill \\ \hfill 3\hfill \end{array}\right),\left(\begin{array}{ccc}\hfill 11\hfill & \hfill -6\hfill & \hfill 2\hfill \\ \hfill -6\hfill & \hfill 10\hfill & \hfill -4\hfill \\ \hfill 2\hfill & \hfill -4\hfill & \hfill 6\hfill \end{array}\right))$ $Z_1=X_2-X_3{Z}_2=X_2+X_3{Z}_3\mid Z_1,Z_2\sim N(Z_1+Z_2,10)$ Derive: (i) The joint distribution of $Z_1$, $Z_2$ and $Z_3$ (ii) The conditional mean of $X_3$ given $X_1$ and $X_2$ (iii) ${\rho }_{3\cdot 12}^2$ [2019/20m/7a]

6. For a random sample of size $n$ from $N_4(\mu ,\Sigma )$, we obtain $\hat{\mu }=(18,15,18,14)$ and $\hat{\Sigma }=\left[\begin{array}{cccc}\hfill 9.5\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ \hfill 5.2\hfill & \hfill 5.4\hfill & \hfill \hfill & \hfill \hfill \\ \hfill 6.9\hfill & \hfill 5.1\hfill & \hfill 10.0\hfill & \hfill \hfill \\ \hfill 4.6\hfill & \hfill 3.5\hfill & \hfill 5.6\hfill & \hfill 4.5\hfill \end{array}\right].$ (i) Find the estimates of the parameters of the conditional distribution of $(X_3,X_4)$ given $(X_1,X_2)$. (ii) Find the partial correlation $r_{34\cdot 12}$. [2018/20m/7a]

7. If $X_1=Y_1+Y_2$, $X_2=Y_2+Y_3$, $X_3=Y_3+Y_1$, where $Y_1$, $Y_2$ and $Y_3$ are uncorrelated random variables and each of which has zero mean and unit standard deviation, find the multiple correlation coefficient between $X_1$ and $X_2$, $X_3$. [2016/10m/5a]

8. Let $X=(X_1,X_2\dots X_p)\prime$ be a p-dimensional random vector with $E\left(X\right)=0$, $V\left(X\right)=\Sigma$. Define partial correlation coefficient ${\rho }_{12.\left(3\dots p\right)}$ between $X_1$, $X_2$. Show that ${\rho }_{12.\left(3\dots p\right)}^2=\frac{{\left({\sigma }^{12}\right)}^2}{{\sigma }^{11}{\sigma }^{22}}$ where ${\sigma }^{ij}$ is the ${(i,j)}^{\mathrm{\text{th}}}$ element of ${\Sigma }^{-1}$. [2015/10m/5b]

9. Define multiple correlation coefficient ${\rho }_{1.23\dots p}$ between $X_1$ and $X_2,X_3,\dots X_p$ and show that $1-{\rho }_{1.2\dots p}^2={\left({\sigma }_{11}{\sigma }^{11}\right)}^{-1}$ where ${\sigma }_{11}$ and ${\sigma }^{11}$ are ${(1,1)}^{\mathrm{\text{th}}}$ elements of the dispersion matrix $\Sigma$ and ${\Sigma }^{-1}$ respectively. [2013/10m/5e]

10. If $r_{12}$, $r_{13}$ and $r_{23}$ are the simple correlation coefficients between $X_1$ and $X_2$; $X_1$ and $X_3$; $X_2$ and $X_3$ respectively then show that $r_{12.3}=\frac{r_{12}-r_{13}{r}_{23}}{\sqrt{(1-r_{13}^2)(1-r_{23}^2)}}.$ Also examine the validity of the statement “the range of the partial regression coefficient is $-\infty$ to $+\infty$.” [2012/12m/5c]

11. Suppose that $\underset{\sim }{X}$ follow the $p$ variate normal distribution with co-variance matrix $\Sigma =\left[\begin{array}{ccccc}\hfill {\sigma }^2\hfill & \hfill \rho {\sigma }^2\hfill & \hfill \rho {\sigma }^2\hfill & \hfill \dots \hfill & \hfill \rho {\sigma }^2\hfill \\ \hfill \rho {\sigma }^2\hfill & \hfill {\sigma }^2\hfill & \hfill \rho {\sigma }^2\hfill & \hfill \dots \hfill & \hfill \rho {\sigma }^2\hfill \\ \hfill \rho {\sigma }^2\hfill & \hfill \rho {\sigma }^2\hfill & \hfill {\sigma }^2\hfill & \hfill \dots \hfill & \hfill \rho {\sigma }^2\hfill \\ \hfill \dots \hfill & \hfill \dots \hfill & \hfill \dots \hfill & \hfill \dots \hfill & \hfill \dots \hfill \\ \hfill \rho {\sigma }^2\hfill & \hfill \rho {\sigma }^2\hfill & \hfill \rho {\sigma }^2\hfill & \hfill \dots \hfill & \hfill {\sigma }^2\hfill \end{array}\right]$ Obtain the expression for the multiple correlation coefficient $R_{1.23\dots p}$. [2012/20m/6a]

12. Suppose, with usual notation $r_{12}=0\cdot 36$, $r_{13}=0\cdot 29$, $r_{14}=-0\cdot 62$, $r_{23}=0\cdot 41$, $r_{24}=-0\cdot 24$, $r_{34}=-0\cdot 52$. For predicting $x_1$, the variable $x_4$ has already been included in the regression equation. Which one of $x_2$ and $x_3$ is worth including in the regression equation in addition to $x_4$? [2011/12m/5e]

13. Does the observed correlation between two variables $X_1$ and $X_2$ always imply causal relationship between them? Give an example. Define partial correlation coefficient $r_{12\cdot 3}$ between $X_1$ and $X_2$ eliminating from each the effect of $X_3$. Why is $r_{12\cdot 3}$ more suitable for throwing light on the net correlation between $X_1$ and $X_2$? Based on the observed data set $x_{ij}$, $i=1,2,3$ and $j=1\left(1\right)n$, derive the expression for $r_{12\cdot 3}$. [2011/20m/8a]

14. Consider the mean vector for $X=\left[\begin{array}{c}\hfill X_1\hfill \\ \hfill X_2\hfill \end{array}\right]$ to be ${\mu }_x=\left[\begin{array}{c}\hfill 3\hfill \\ \hfill -2\hfill \end{array}\right]$ and for $Y$ to be ${\mu }_y=4$. The covariance matrices for $X_1$, $X_2$ and $Y$ are ${\Sigma }_{xx}=\left[\begin{array}{cc}\hfill 2\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill \end{array}\right],{\sigma }_{YY}=9\mathrm{\text{and}}{\Sigma }_{XY}=\left[\begin{array}{c}\hfill 3\hfill \\ \hfill 1\hfill \end{array}\right]$ Fit the equation $Y=b_0+b_1{X}_1+b_2{X}_2$ and calculate multiple correlation coefficient. [2010/12m/5d]

15. Examine, after deducing necessary result, if the following set of correlation coefficients are internally consistent: $r_{12}=0\cdot 62$, $r_{13}=0\cdot 55$ and $r_{23}=-0\cdot 42$. [2009/12m/5b]

16. Describing briefly the concept of multiple regression, define multiple and partial correlations and obtain their formulae. [2006/12m/5a]

17. Establish the range of a multiple correlation coefficient. [2002/12m/5b]

18. Show that the correlation matrix is at least positive semi-definite. [2001/15m/7iv]

19. If $x=4y+5$ and $y=\left(1/16\right)x+4$ are the regression lines of x on y and y on x respectively, find the value of the 4 correlation coefficient between x and y. [2001/15m/8iii]

20. In standard notations, given that $r_{12}=0.5,r_{13}=0.4$ and $r_{23}=0.2$. calculate $R_{1.23}$ [2001/15m/8iv]

21. If all correlations of order zero in a set of p variables are equal to $\rho$, show that (i) every partial correlation of order s is $\frac{\rho }{1+s\rho }$ and (ii) $1-R_{1.23}^2=\frac{(1-\rho )(1+2\rho )}{1+\rho }$, symbols having usual meanings. [1999/30m/10a]

22. Define partial correlation coefficients in multivariate analysis. Show that ${\hat{\Sigma }}_{11.2}={\hat{\Sigma }}_{11}-{\hat{\Sigma }}_{12}{\hat{\Sigma }}_{22}^{-1}{\hat{\Sigma }}_{21}$ [1998/20m/11c]

23. Let the covariance matrix of a four-dimensional normal random vector $X=(X_1,X_2,X_3,X_4)\prime$ be given by $\Sigma ={\sigma }^2\left(\begin{array}{cccc}\hfill 1\hfill & \hfill \rho \hfill & \hfill {\rho }^2\hfill & \hfill {\rho }^2\hfill \\ \hfill \rho \hfill & \hfill 1\hfill & \hfill \rho \hfill & \hfill {\rho }^2\hfill \\ \hfill {\rho }^2\hfill & \hfill \rho \hfill & \hfill 1\hfill & \hfill {\rho }^2\hfill \\ \hfill {\rho }^3\hfill & \hfill {\rho }^2\hfill & \hfill \rho \hfill & \hfill 1\hfill \end{array}\right)$ Find the partial correlation coefficient between ${(i-1)}^{\mathrm{\text{th}}}$ and ${(i+i)}^{\mathrm{\text{th}}}$ components of X when the $i^{\mathrm{\text{th}}}$ component is held fixed. [1995/15m/11c]

3.5] Multivariate normal distribution

47 questions

1. Define multivariate normal distribution and write down its important properties. Let ${𝐗}_1$, ${𝐗}_2$, ${𝐗}_3$ and ${𝐗}_4$ are mutually independent random vectors following $N_p(𝛍,\Sigma )$. Then find the marginal distributions of (i) ${𝐖}_1=\frac{1}{4}{𝐗}_1-\frac{1}{4}{𝐗}_2+\frac{1}{4}{𝐗}_3-\frac{1}{4}{𝐗}_4$ (ii) ${𝐖}_2=\frac{1}{4}{𝐗}_1+\frac{1}{4}{𝐗}_2-\frac{1}{4}{𝐗}_3-\frac{1}{4}{𝐗}_4$ Also obtain the joint distribution of random vectors ${𝐖}_1$ and ${𝐖}_2$. [2026/20m/8a]

2. Let $X=\left(X_1{X}_2{X}_3\right)\prime \sim N_3(\mu ,\Sigma )$, where $\mu =\left(\begin{array}{c}\hfill 1\hfill \\ \hfill 2\hfill \\ \hfill 1\hfill \end{array}\right)\mathrm{\text{and}}\Sigma =\left(\begin{array}{ccc}\hfill 9\hfill & \hfill 2\hfill & \hfill 2\hfill \\ \hfill 2\hfill & \hfill 3\hfill & \hfill 0\hfill \\ \hfill 2\hfill & \hfill 0\hfill & \hfill 2\hfill \end{array}\right).$ Find the joint distribution of $Y_1=X_1+X_2+X_3$ and $Y_2=X_2-X_3$. [2025/10m/5b]

3. Let $X=(X_1,X_2,X_3)\prime \sim N_3(\mu ,\Sigma )$, in which $\mu =\left(\begin{array}{c}\hfill 2\hfill \\ \hfill 1\hfill \\ \hfill 3\hfill \end{array}\right)\mathrm{\text{and}}\Sigma =\left(\begin{array}{ccc}\hfill 9\hfill & \hfill 2\hfill & \hfill -2\hfill \\ \hfill 2\hfill & \hfill 2\hfill & \hfill -3\hfill \\ \hfill -2\hfill & \hfill -3\hfill & \hfill 9\hfill \end{array}\right)$ Obtain (i) $E\{X_1\mid X_2=x_2,X_3=x_3\}$ and (ii) $Var\{X_1\mid X_2=x_2,X_3=x_3\}$. [2025/15m/6b]

4. If $\underset{\sim }{X}$ is distributed as $N_3(\underset{\sim }{\mu },\Sigma )$, find the distribution of $\left[\begin{array}{c}\hfill X_1-X_2\hfill \\ \hfill X_2-X_3\hfill \end{array}\right]$. [2024/5m/5b(i)]

5. State the multivariate extension of (i) when $\underset{\sim }{X}$ follows $N_p(\underset{\sim }{\mu },\Sigma )$. [2024/4m/6a(iv)]

6. (i) If $𝐗=\left(X_1{X}_2{X}_3\right)\prime$ is distributed as $N_3(𝛍,\Sigma )$, find the distribution of $\left[(X_1-X_2)(X_2-X_3)\right]\prime$. (ii) Suppose that $𝐗=\left(X_1{X}_2{X}_3\right)\prime \sim N_3(0,\Sigma )$, where $\Sigma =\left(\begin{array}{ccc}\hfill 1\hfill & \hfill \rho \hfill & \hfill 0\hfill \\ \hfill \rho \hfill & \hfill 1\hfill & \hfill \rho \hfill \\ \hfill 0\hfill & \hfill \rho \hfill & \hfill 1\hfill \end{array}\right)$. Is there a value of $\rho$ for which $(X_1+X_2+X_3)$ and $(X_1-X_2-X_3)$ are independent? [2023/10m/5a]

7. Show that $𝐗=(X_1,X_2,\dots ,X_p)\prime$ has p-variate normal distribution if and only if every linear combination $(l_1{X}_1+l_2{X}_2+\dots +l_p{X}_p)$ of $𝐗$ follows a univariate normal distribution. [2023/10m/5b]

8. Let $𝐗=\left(X_1{X}_2{X}_3\right)\prime$ be distributed as $N_3(𝛍,\Sigma )$, where $𝛍=(2-13)\prime$ and $\Sigma =\left(\begin{array}{ccc}\hfill 4\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 2\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 3\hfill \end{array}\right).$ Find (i) the conditional distribution of $\left(X_1{X}_2\right)\prime$ given $X_3=2$. (ii) partial correlation coefficient ${\rho }_{12.3}$ and multiple correlation coefficient $R_{1.23}$. [2023/15m/6a]

9. Given $𝐗\sim N_3(𝛍,\Sigma )$, where $𝛍=(2,4,3)\prime$ and $\Sigma =\left(\begin{array}{ccc}\hfill 8\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill 2\hfill & \hfill 4\hfill & \hfill 1\hfill \\ \hfill 3\hfill & \hfill 1\hfill & \hfill 3\hfill \end{array}\right),$ (i) find the regression function of $X_1$ on $X_2$ and $X_3$, and (ii) compute the conditional variance of $X_1$ given $X_2$ and $X_3$. [2022/10m/5b]

10. Let $Z_{3\times 1}=\left(\begin{array}{c}\hfill X_{1\times 1}\hfill \\ \hfill Y_{2\times 1}\hfill \end{array}\right)\sim N_3(\left(\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 1\hfill \end{array}\right),\left(\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill 5\hfill & \hfill 2\hfill \\ \hfill 1\hfill & \hfill 2\hfill & \hfill 2\hfill \end{array}\right))$. Show that conditional on $X_{1\times 1}$, the two components of $Y_{2\times 1}$ are independent but marginally they are not. [2021/15m/6c]

11. Let $X\sim N_2(\mu ,\Sigma )$ with probability distribution function (pdf), $f\left(x\right)=C{e}^{-Q/2},$ where $Q=3x^2+2y^2-2xy-32x+4y+92$. Obtain $\mu$ and $\Sigma$. [2020/10m/5b]

12. Let $𝐗\sim N_3(\left(\begin{array}{c}\hfill 1\hfill \\ \hfill 2\hfill \\ \hfill 3\hfill \end{array}\right),\left(\begin{array}{ccc}\hfill 11\hfill & \hfill -6\hfill & \hfill 2\hfill \\ \hfill -6\hfill & \hfill 10\hfill & \hfill -4\hfill \\ \hfill 2\hfill & \hfill -4\hfill & \hfill 6\hfill \end{array}\right))$ $Z_1=X_2-X_3{Z}_2=X_2+X_3{Z}_3\mid Z_1,Z_2\sim N(Z_1+Z_2,10)$ Derive: (i) The joint distribution of $Z_1$, $Z_2$ and $Z_3$ (ii) The conditional mean of $X_3$ given $X_1$ and $X_2$ (iii) ${\rho }_{3\cdot 12}^2$ [2019/20m/7a]

13. If $X$ is a $p$-component vector of random variables distributed as $N_p(0,\Sigma )$, then show that $X\prime {\Sigma }^{-1}X$ is distributed as chi-square with $p$ degrees of freedom. [2018/15m/6b]

14. For a random sample of size $n$ from $N_4(\mu ,\Sigma )$, we obtain $\hat{\mu }=(18,15,18,14)$ and $\hat{\Sigma }=\left[\begin{array}{cccc}\hfill 9.5\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ \hfill 5.2\hfill & \hfill 5.4\hfill & \hfill \hfill & \hfill \hfill \\ \hfill 6.9\hfill & \hfill 5.1\hfill & \hfill 10.0\hfill & \hfill \hfill \\ \hfill 4.6\hfill & \hfill 3.5\hfill & \hfill 5.6\hfill & \hfill 4.5\hfill \end{array}\right].$ (i) Find the estimates of the parameters of the conditional distribution of $(X_3,X_4)$ given $(X_1,X_2)$. (ii) Find the partial correlation $r_{34\cdot 12}$. [2018/20m/7a]

15. Let $X=[X_1,X_2,X_3]\prime$ have trivariate normal distribution $N_3(\mu ,\Sigma )$, where $\mu \prime =[1,1,1]$ and $\Sigma =\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 2\hfill & \hfill 2\hfill \\ \hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill \end{array}\right]$ Show that ${(X_2-X_1)}^2+{(X_3-X_2)}^2$ has a chi-squared distribution. State the degrees of freedom. [2017/10m/5b]

16. If $𝐗$ follows $N_p(𝛍,\Sigma )$ and $𝐂$ is an $m\times p$ matrix with rank $m$, then show that $𝐂𝐗\sim N_m(𝐂𝛍,𝐂\Sigma 𝐂\prime )$ [2016/20m/7a]

17. Let $\underset{\sim }{X}=(X_1,X_2,X_3)\prime$ have the joint moment generating function (mgf) $M_X\left(\underset{\sim }{t}\right)=exp[t_1-t_2+2t_3+t_1^2+\frac{t_2^2}{2}+2t_3^2-\frac{t_1{t}_2}{2}-t_1{t}_3]$ where $\underset{\sim }{t}=(t_1,t_2,t_3)\prime$. Then: (i) Obtain the covariance matrix and the mean vector of $\underset{\sim }{X}$. (ii) Find a constant $C$ such that $P(2X_1-3X_2+X_3>C)=0.95$. (iii) Derive the conditional distribution of $X_1$ given $X_2=x_2$ and $X_3=x_3$. (If necessary, you can use $P(\tau >1.645)=0.05$ and $P(\tau >1.96)=0.01$ where $\tau$ is a standard normal variate) [2015/15m/6b]

18. Consider a random sample ${\underset{\sim }{x}}_1,{\underset{\sim }{x}}_2,\dots {\underset{\sim }{x}}_n$ from a p-dimensional normal population with mean vector $\underset{\sim }{\mu }$ and dispersion matrix $\Sigma$. The purpose of the problem is to construct the likelihood ratio test (LRT) for testing $H_0:\Sigma ={\sigma }^2{I}_p$. Find the maximum likelihood estimate (MLE) of $\mu$ and ${\sigma }^2$. Write down the expression for $-2{log}_e\wedge$ in terms of $p$, $n$, $\underset{\sim }{\overline{X}}$ and the elements of the sample covariance matrix $S$. [2015/20m/8a]

19. Let $X_1,X_2,X_3,X_4$ and $X_5$ be independent and identically distributed random vectors with mean vector $\mu$ and covariance matrix $\Sigma$. Find the distribution of $Y=X_1-X_2+X_3-X_4+X_5$. [2014/10m/5e]

20. Let $X\sim N_p(\mu ,\Sigma )$, where $X=\left(\begin{array}{c}\hfill X_{q\times 1}^{\left(1\right)}\hfill \\ \hfill X_{p-q\times 1}^{\left(2\right)}\hfill \end{array}\right),\mu =\left(\begin{array}{c}\hfill {\mu }_{q\times 1}^{\left(1\right)}\hfill \\ \hfill {\mu }_{p-q\times 1}^{\left(2\right)}\hfill \end{array}\right),\Sigma =\left(\begin{array}{cc}\hfill {\Sigma }_{11}\hfill & \hfill {\Sigma }_{12}\hfill \\ \hfill {\Sigma }_{21}\hfill & \hfill {\Sigma }_{22}\hfill \end{array}\right)$ Obtain the conditional distribution of $X^{\left(1\right)}$ given $X^{\left(2\right)}=x^{\left(2\right)}$. [2014/20m/6a]

21. Let ${𝐗}_{2p\times 1}=\left[\begin{array}{c}\hfill X_1\hfill \\ \hfill X_2\hfill \end{array}\right]$ be distributed as $N_{2p}(\mu ,\Sigma )$ where $\mu =\left[\begin{array}{c}\hfill {\mu }_1\hfill \\ \hfill {\mu }_2\hfill \end{array}\right]$, $\Sigma =\left(\begin{array}{cc}\hfill {\Sigma }_1\hfill & \hfill {\Sigma }_2\hfill \\ \hfill {\Sigma }_2\hfill & \hfill {\Sigma }_1\hfill \end{array}\right)$, $X_1,X_2,{\mu }_1$ and ${\mu }_2$ are $p$-vectors and ${\Sigma }_1,{\Sigma }_2$ are $p\times p$ matrices. Obtain the characteristic function of $𝐗$. Derive the distribution of $\left[\begin{array}{c}\hfill X_1+X_2\hfill \\ \hfill X_1-X_2\hfill \end{array}\right]$. [2013/20m/8a]

22. $X$ follow $N_3(0,\Sigma )$ where $\Sigma =\left[\begin{array}{ccc}\hfill 1\hfill & \hfill \rho \hfill & \hfill 0\hfill \\ \hfill \rho \hfill & \hfill 1\hfill & \hfill \rho \hfill \\ \hfill 0\hfill & \hfill \rho \hfill & \hfill 1\hfill \end{array}\right]$. Find $\rho$ such that $(X_1+X_2+X_3)$ and $(X_1-X_2-X_3)$ are independent. [2012/12m/5a]

23. Show that a necessary and sufficient condition for ${\underset{\sim }{X}}_p$ to be distributed as $N_p(\mu ,\Sigma )$ is that every linear function $Y=l\prime \underset{\sim }{X}$ is distributed as univariate normal $N(l\prime \underset{\sim }{\mu },l\prime \Sigma l)$. [2011/20m/7a]

24. $X=[X_1,X_2,X_3]\prime$ has 3-variate normal distribution with mean vector $\mu$ and covariance matrix $\Sigma$, where $\mu =\left[\begin{array}{c}\hfill -3\hfill \\ \hfill 1\hfill \\ \hfill 4\hfill \end{array}\right]\mathrm{\text{and}}\Sigma =\left[\begin{array}{ccc}\hfill 1\hfill & \hfill -2\hfill & \hfill 0\hfill \\ \hfill -2\hfill & \hfill 5\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 2\hfill \end{array}\right]$ Show that $X_1$ and $X_2$ are not independent but $[X_1,X_2]\prime$ is independent of $X_3$. Further show that $\frac{1}{2}(X_1+X_2)$ and $X_3$ are also independent. [2010/12m/5b]

25. $X=[X_1,X_2,X_3]\prime$ has 3-variate normal distribution with mean vector $\mu$ and dispersion matrix $\Sigma$, where $\mu =\left[\begin{array}{c}\hfill 2\hfill \\ \hfill -1\hfill \\ \hfill 3\hfill \end{array}\right]\mathrm{\text{and}}\Sigma =\left[\begin{array}{ccc}\hfill 4\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 2\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 3\hfill \end{array}\right]$ Derive the distribution of $Y=4X_1-6X_2+X_3\mathrm{\text{and}}Z=\left[\begin{array}{c}\hfill X_1-X_2+X_3\hfill \\ \hfill 2X_1+X_2-X_3\hfill \end{array}\right]$ [2010/12m/5c]

26. Let $X=[X_1,X_2,\dots ,X_p]\prime$ denote a column vector of variables $X_i$’s. Prove that $Y$ follows a $p$-variate normal distribution $N_p(\mu ,\Sigma )$ iff there exists a random $X=[X_1,X_2,\dots ,X_p]\prime$ of independent standard normal variables $X_i$’s such that $Y=\mu +BX$ with probability one for some matrix $B$ of full rank and $BB\prime =\Sigma$. [2010/20m/6a]

27. Let $X$ be distributed as $N_p(\mu ,\Sigma )$ (a $p$-variate normal distribution). Prove that $Y=CX$, where $C$ is $r\times p$ matrix of rank $r$ $(1\le r\le p)$, is distributed as $N_r(C\mu ,C\Sigma C\prime )$. [2010/20m/7a]

28. Let $X_1,X_2,\dots ,X_n$ be a random sample from $p$-variate normal distribution $N_p(\mu 1,\Sigma )$, where $X_{\alpha }=[X_{1\alpha },X_{2\alpha },\dots ,X_{p\alpha }]\prime$, $\alpha =1,2,\dots ,n$; $1_{p\times 1}=[1,1,\dots ,1]\prime$ and ${\Sigma }_{p\times p}$ is known. Define $T_1=\frac{1\prime \bar{X}}{1\prime 1}\mathrm{\text{and}}{T}_2=\frac{1\prime {\Sigma }^{-1}\bar{X}}{1\prime {\Sigma }^{-1}1}$ where $\bar{X}=[{\bar{X}}_1,{\bar{X}}_2,\dots ,{\bar{X}}_p]\prime$ and ${\bar{X}}_i=\frac{1}{n}{\sum }_{\alpha =1}^n{X}_{i\alpha },i=1,2,\dots ,p$ Prove that $T_1$ and $T_2$ are unbiased estimators of $\mu$ and $V\left(T_2\right)<V\left(T_1\right)$. [2010/20m/8c]

29. The vector variable $\underset{\sim }{X}$ follows $p$-variate normal distribution with mean vector $\underset{\sim }{\mu }$ and dispersion matrix $\Sigma$. Obtain the distribution of $Q=(\underset{\sim }{X}-\underset{\sim }{\mu })\prime {\Sigma }^{-1}(\underset{\sim }{X}-\underset{\sim }{\mu })$. [2009/20m/6a]

30. Let $Y_1,Y_2,\dots ,Y_n$, $(n\ge p)$ be i.i.d. random vectors, each having $N_p(0,\Sigma )$ distribution. Define $A={\sum }_{\alpha }^{}{y}_{\alpha }y{\prime }_{\alpha }$ Show that for $n>p+1$ $E\left(A\right)=\frac{1}{n-p-1}\Sigma -1$ assuming E to be positive definite. [2007/12m/5d]

31. If $\underset{\_}{X}$ is distributed as $N_5(\mu ,\Sigma )$, find the distribution of $\left(\begin{array}{c}\hfill X_2\hfill \\ \hfill X_4\hfill \end{array}\right)$ [2006/12m/5b]

32. If $\underset{\_}{X}\sim N_p(\mu ,\Sigma )$, prove that every linear function of components of $\underset{\_}{X}$ is univariate normal. [2006/12m/5d]

33. If $X_1,X_2,\dots ,X_n$ have a joint normal distribution, show that a necessary and sufficient condition for one subset of random variables to be independent of the subset consisting of the remaining variables is that each covariance of a variable from one set and a variable from the other set be zero. [2006/20m/6c]

34. If X (with p components) is distributed according to $N_p(\mu ,\Sigma )$, then prove that $Z=DX$ is distributed according to $N_q(D\mu ,D\Sigma D\prime )$ where D is a $q\times p$ matrix of rank $q\le p$. [2004/12m/5a]

35. Let X be distributed $N(\mu ,\Sigma )$, find the characteristic function of X. [2003/20m/6a]

36. Let $X\prime =[X_1,X_2,X_3]$ be $N_3(\mu ,\Sigma )$ with $\Sigma =\left[\begin{array}{ccc}\hfill 4\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 3\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 2\hfill \end{array}\right]$ Are $X_1$ and $X_2$ independent? What about independence of $(X_1,X_2)$ and $X_3$? [2002/12m/5a]

37. If X and Y are two independent p-dimensional random variables and if X + Y has a p-variate normal distribution, then prove that both X and Y have p-variate normal distributions. [2002/20m/6b]

38. Let $X\sim N_p(\mu ,\Sigma )$. Let $G(q\times p)$ be a matrix of rank q. show that $GX\sim N_p(G{\mu }_p,G\Sigma G\prime )$ [2000/15m/6c]

39. $X\sim N_p(\mu ,\Sigma )$ and let C be a $q\times p$ matrix of rank q. Prove that $Y=CX$ follows $N_q(C\mu ,C\Sigma C\prime )$ distribution. Explain how this can be used to find the marginal distribution of a q-component vector of X. [1999/30m/10b]

40. If $X\sim N(\zeta ,{\sigma }^2)$ then $a\prime X\sim N(a\prime \zeta ,a\prime Ba)$ find $B=\tilde{T}\tilde{T}\prime$ when $\tilde{T}y=(X-\zeta )$. [1998/5m/11a(i)]

41. If the probability density function of a random vector X is $kexp\{-\frac{1}{2}(X-\zeta )\prime {\Sigma }^{-1}(X-\zeta )\}$ determine $E\left(X\right)$ and $V\left(X\right)$. [1998/10m/11a(ii)]

42. Consider the sets of random variable $X_1,X_2,\dots ,X_q$ and $X_{q+1},X_{q+2},\dots ,X_p$ with vectors $X^{\left(1\right)}=\left(\begin{array}{c}\hfill X_1\hfill \\ \hfill X_2\hfill \\ \hfill ⋮\hfill \\ \hfill X_q\hfill \end{array}\right),X^{\left(2\right)}=\left(\begin{array}{c}\hfill X_{q+1}\hfill \\ \hfill X_{q+2}\hfill \\ \hfill ⋮\hfill \\ \hfill X_p\hfill \end{array}\right)\mathrm{\text{and}}\mu =\left(\begin{array}{c}\hfill {\mu }^{\left(1\right)}\hfill \\ \hfill {\mu }^{\left(2\right)}\hfill \end{array}\right),\Sigma =\left(\begin{array}{cc}\hfill {\Sigma }_{11}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\Sigma }_{22}\hfill \end{array}\right)$ The marginal distribution of the set is multivariate normal with means, variances and covariances obtained by taking the proper components of $\mu$ and $\Sigma$ respectively. [1998/25m/11b]

43. Let $X\sim N_p(\mu ,\Sigma )$ and let $X=\left(\begin{array}{c}\hfill X^{\left(1\right)}\hfill \\ \hfill X^{\left(2\right)}\hfill \end{array}\right),\mu =\left(\begin{array}{c}\hfill {\mu }^{\left(1\right)}\hfill \\ \hfill {\mu }^{\left(2\right)}\hfill \end{array}\right),\Sigma =\left(\begin{array}{cc}\hfill {\Sigma }_{11}\hfill & \hfill {\Sigma }_{12}\hfill \\ \hfill {\Sigma }_{21}\hfill & \hfill {\Sigma }_{22}\hfill \end{array}\right)$ Show that $X^{\left(1\right)}\sim N({\mu }^{\left(1\right)},{\Sigma }_{11})$ even if $X^{\left(1\right)}$ and $X^{\left(2\right)}$ are not independent. [1997/30m/11a]

44. Let X be a random vector with mean vector $\mu$ and dispersion matrix $\Sigma$. Prove that X is distributed as $N_p(\mu ,\Sigma )$ if, and only if, for every coefficient vector $l$, $l\prime X$ is distributed as $N_1(l\prime \mu ,l\prime \Sigma l)$. [1997/30m/11b]

45. Let $Y=(Y_1,Y_2,\dots ,Y_n)\prime$ be a vector of random variables with pdf $K^{-1}exp[-\frac{1}{2}{(y-\theta )}^t{\Sigma }^{-1}(y-\theta )]$. (i) Show that $K={\left(2\pi \right)}^{n/2}{\left|\Sigma \right|}^{1/2}$ and (ii) Determine $E\left[Y\right]$ and $D\left[Y\right]$. [1996/30m/11a]

46. If $X\sim N_p(\Theta ,\Sigma )$ then show that the marginal distribution of each $X_i\sim N({\theta }_i,{\sigma }_{ii})$, where $X=(X_1,X_2,\dots ,X_p)\prime$, $\Theta =({\Theta }_1,\dots ,{\Theta }_p)\prime$ and $\Sigma =\left({\sigma }_{ij}\right)$. Is the converse true? [1995/25m/11a]

47. If $X=(X_1,X_2)\prime \sim N_p(\Theta ,\Sigma )$ and $X_1,X_2$ are r- and s-vectors respectively with $r+s=0$, then show that $X_1-{\Sigma }_{12}{\Sigma }_{22}^{-1}{X}_2$ and $X_2$ are independently normally distributed. The partition of $\Sigma$ is $\Sigma =\left(\begin{array}{cc}\hfill {\Sigma }_{11}\hfill & \hfill {\Sigma }_{12}\hfill \\ \hfill {\Sigma }_{12}\hfill & \hfill {\Sigma }_{22}\hfill \end{array}\right)$ [1995/20m/11b]

3.6] Mahalanobis’s D² and Hotelling’s T²

24 questions

1. Let ${𝐗}_1,{𝐗}_2,\cdots ,{𝐗}_N$ be a random sample from $N_p(𝛍,\Sigma )$. Show that Hotelling’s $T^2$-statistic for testing $H_0:𝛍={𝛍}_0$ against $H_1:𝛍\ne {𝛍}_0$ is a generalization of $t$-statistic for testing $H_0:\mu ={\mu }_0$ against $H_1:\mu \ne {\mu }_0$ in the univariate case. Calculate $T^2$-statistic for testing $H_0:𝛍=\left(\begin{array}{c}\hfill 9\hfill \\ \hfill 5\hfill \end{array}\right)$ on the basis of the data taken from a bivariate normal distribution given below: $X=\left(\begin{array}{cc}\hfill 6\hfill & \hfill 9\hfill \\ \hfill 10\hfill & \hfill 6\hfill \\ \hfill 8\hfill & \hfill 3\hfill \end{array}\right)$ [2026/15m/6b]

2. $X_1$ and $X_2$ are independent data sets of order $(n_1\times p)$ and $(n_2\times p)$ respectively from $N_p(\underset{\sim }{\mu },\Sigma )$. Show that $\frac{n_1{n}_2{D}^2}{n}$ is distributed as $T^2(p,n-2)$, where $n=n_1+n_2$, and $T^2$ and $D^2$ represent the Hotelling’s $T^2$ and Mahalanobis $D^2$ respectively. [2024/10m/5c]

3. (i) The data matrix of a random sample of size $n=3$ from a bivariate normal population $BVN({\mu }_1,{\mu }_2,{\sigma }_1^2,{\sigma }_2^2,\rho )$ is $X=\left[\begin{array}{cc}\hfill 6\hfill & \hfill 10\hfill \\ \hfill 10\hfill & \hfill 6\hfill \\ \hfill 8\hfill & \hfill 2\hfill \end{array}\right].$ Test the null hypothesis $H_0:𝛍={𝛍}_0$ against $H_1:𝛍\ne {𝛍}_0$, where ${𝛍}_0\prime =(8,5)$, at 10% level of significance. [You are given: $F_{0.10;2,1}=49.5,F_{0.10;1,2}=8.53$] (ii) Suppose $n_1=11$ and $n_2=12$, observations are made on two random vectors ${𝐗}_1$ and ${𝐗}_2$ which are assumed to have bivariate normal distribution with a common covariance matrix $\Sigma$, but possibly different mean vectors ${𝛍}_1$ and ${𝛍}_2$. The sample mean vectors and pooled covariance matrix are ${\overline{𝐗}}_1=\left(\begin{array}{c}\hfill -1\hfill \\ \hfill -1\hfill \end{array}\right),{\overline{𝐗}}_2=\left(\begin{array}{c}\hfill 2\hfill \\ \hfill 1\hfill \end{array}\right),S_{\mathrm{\text{pooled}}}=\left(\begin{array}{cc}\hfill 7\hfill & \hfill -1\hfill \\ \hfill -1\hfill & \hfill 5\hfill \end{array}\right).$ Obtain Mahalanobis sample distance $D^2$ and Fisher’s linear discriminant function. Assign the observation ${𝐗}_0=\left(\begin{array}{c}\hfill 0\hfill \\ \hfill 1\hfill \end{array}\right)$ to either population ${\Pi }_1$ or ${\Pi }_2$. [2022/20m/7b]

4. Let $n=4$, $p=2$, $X=\left(\begin{array}{c}\hfill X_1\hfill \\ \hfill X_2\hfill \end{array}\right)$, where $X_1$: weight (in kg), $X_2$: height (in cm) of infant. Sample data matrix $X=\left[\begin{array}{cccc}\hfill 3.0\hfill & \hfill 4.0\hfill & \hfill 3.0\hfill & \hfill 3.0\hfill \\ \hfill 40\hfill & \hfill 40\hfill & \hfill 30\hfill & \hfill 35\hfill \end{array}\right]$ Test at 5% level of significance the null hypothesis $H_0:\mu ={\mu }_0=\left(\begin{array}{c}\hfill 3.0\hfill \\ \hfill 35.0\hfill \end{array}\right)$. [Given that: $F_{((2,2),0.05)}=19.00$]. [2020/20m/6c]

5. Define Hotelling’s-$T^2$. Show that $T^2$ statistic is a function of likelihood ratio criteria. [2020/20m/7a]

6. Consider two independent i.i.d samples, each of size 10, from two bivariate normal populations. The results are summarised below. ${\bar{x}}_1=\left(\begin{array}{c}\hfill 3\hfill \\ \hfill 1\hfill \end{array}\right),{\bar{x}}_2=\left(\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \end{array}\right),S_1=\left(\begin{array}{cc}\hfill 4\hfill & \hfill -1\hfill \\ \hfill -1\hfill & \hfill 2\hfill \end{array}\right),S_2=\left(\begin{array}{cc}\hfill 2\hfill & \hfill -2\hfill \\ \hfill -2\hfill & \hfill 4\hfill \end{array}\right),$ where $S_j$, $j=1,2$ are sample covariance matrices and ${\bar{x}}_j$, $j=1,2$ are sample means. Provide suitable tests for the following hypotheses at level $\alpha =0.05$ and draw your conclusion. (i) $H_0:{𝛍}_1=\left(\begin{array}{c}\hfill 2.5\hfill \\ \hfill 1\hfill \end{array}\right)$ (ii) $H_0:{𝛍}_1={𝛍}_2$ (iii) $H_0:{\mu }_{11}+2{\mu }_{12}=4$, where ${𝛍}_1=({\mu }_{11},{\mu }_{12})\prime$ (You can use the following values: $F_{0.95,2,17}=3.5915$, $F_{0.95,2,18}=3.55$, $F_{0.95,1,19}=4.38$) [2019/20m/6c]

7. Show that Hotelling’s $T^2$ is a generalization of the Student’s $t$. Prove the invariance property of $T^2$ under the non-singular linear transformations of the variables. Briefly mention different uses of $T^2$. [2018/15m/8b]

8. Suppose $n_1=11$ and $n_2=12$ observations are made on two random vectors $X_1$ and $X_2$ which are assumed to have bivariate normal distribution with a common covariance matrix $\Sigma$, but possibly different mean vectors ${\mu }_1$ and ${\mu }_2$. The sample mean vectors and pooled covariance matrix are ${\bar{X}}_1=\left[\begin{array}{c}\hfill -1\hfill \\ \hfill -1\hfill \end{array}\right],{\bar{X}}_2=\left[\begin{array}{c}\hfill 2\hfill \\ \hfill 1\hfill \end{array}\right],S_{\mathrm{\text{pooled}}}=\left[\begin{array}{cc}\hfill 7\hfill & \hfill -1\hfill \\ \hfill -1\hfill & \hfill 5\hfill \end{array}\right]$ Obtain Mahalanobis sample distance $D^2$ and Fisher’s linear discriminant function. Assign the observation $X_0\prime =(0,1)$ to either population ${\pi }_1$ or ${\pi }_2$. [2017/15m/7b]

9. The data matrix for a random sample of size $n=3$ from a bivariate normal population $BVN({\mu }_1,{\mu }_2,{\sigma }_1^2,{\sigma }_2^2,\rho )$ is $𝐗=\left[\begin{array}{cc}\hfill 6\hfill & \hfill 10\hfill \\ \hfill 10\hfill & \hfill 6\hfill \\ \hfill 8\hfill & \hfill 3\hfill \end{array}\right]$ Test the null hypothesis $H_0:𝛍={𝛍}_0$ against $H_1:𝛍\ne {𝛍}_0$, where ${𝛍}_0\prime =(8,5)$, at 10% level of significance. [You are given: $F_{0.10;2,1}=49.5$, $F_{0.10;1,2}=8.52632$]. [2016/10m/5e]

10. If $X_1$ and $X_2$ be independent data matrices of order $(n_1\times p)$ and $(n_2\times p)$ respectively and if the $n_i$ rows of $X_i(i=1,2)$ be identically and independently distributed as $N_p({\mu }_i,{\Sigma }_i)$, then for ${\mu }_1={\mu }_2$ and ${\Sigma }_1={\Sigma }_2$, show that $\frac{n_1{n}_2}{n}{D}^2\simeq T^2(p,n-2)\mathrm{\text{variable, where}}n=n_1+n_2$ $D^2$ denotes sample Mahalanobis distance statistic. [2016/15m/8b]

11. A random sample of size 4 from a bivariate normal population provides the following statistics $\overline{X}=\left(\begin{array}{c}\hfill 4\hfill \\ \hfill 6\hfill \end{array}\right),S=\left(\begin{array}{cc}\hfill 6\hfill & \hfill -2\hfill \\ \hfill -2\hfill & \hfill 3\hfill \end{array}\right)$ where $\overline{X}$ is the sample mean and $S$ is an unbiased estimator of the population dispersion matrix. Test the hypothesis $H_0:\underset{\sim }{\mu }=(5,5)\prime$ where $\mu$ is the population mean vector. $(F_{.05,1,3}=10.13,F_{.05,2,2}=19.00,F_{.05,3,1}=215.7)$ [2015/10m/5a]

12. If ${\pi }_1$ and ${\pi }_2$ denote two bivariate normal populations where ${\pi }_1\sim N_2({\mu }_1,\Sigma )$ and ${\pi }_2\sim N_2({\mu }_2,\Sigma )$, $\mu {\prime }_1=[10,15]$, ${\mu }_2=[10,25]$ and $\Sigma =\left(\begin{array}{cc}\hfill 18\hfill & \hfill 12\hfill \\ \hfill 12\hfill & \hfill 32\hfill \end{array}\right)$. Compute the Mahalonobis distance between ${\pi }_1$ and ${\pi }_2$. [2014/10m/5b]

13. Define Hotelling $T^2$. State any one of its applications. A random sample of size 3 from a bivariate normal $N_2(\underset{\_}{\mu },\Sigma )$ distribution gave following unbiased estimates of $\underset{\_}{\mu }$ and $\Sigma$ $\bar{x}=\left(\begin{array}{c}\hfill 8\hfill \\ \hfill 6\hfill \end{array}\right),S=\left(\begin{array}{cc}\hfill 4\hfill & \hfill -3\hfill \\ \hfill -3\hfill & \hfill 9\hfill \end{array}\right)$ Compute $T^2$-statistic to test the hypothesis $H_o:\underset{\_}{\mu }={\underset{\_}{\mu }}_o=\left(\begin{array}{c}\hfill 7\hfill \\ \hfill 7\hfill \end{array}\right)$. [2014/15m/7b]

14. Let $X_1\dots X_N(N>p)$ be a random sample from $N_p(\mu ,\Sigma ),\Sigma >0$ where $\mu =\left({\mu }_1\dots {\mu }_p\right)\prime$ and $\Sigma$ are unknown. Show that for testing $H_0:{\mu }_1={\mu }_2=\dots ={\mu }_p$ against $H_1$: not all ${\mu }_i$’s are equal, the test statistic can be expressed as $T=N[\bar{X}\prime \bar{S}\prime \bar{X}-\frac{{\left(\bar{X}\prime \bar{S}\prime \epsilon \right)}^2}{\epsilon \prime \bar{S}\prime \epsilon }]$ where $\bar{X}$ and $S$ are the sample mean vector and the unbiased estimator of $\Sigma$ respectively and $\epsilon =(1,1,\dots 1)\prime$. What would be the non null distribution of $T$? [2013/15m/6b]

15. Define Mahalanobis measure of distance squared between two populations with a common positive definite dispersion matrix and show that it is a non decreasing function of the number of characteristics. [2013/15m/7b]

16. Bring out the relation between Hotelling’s $T^2$ and Mahalanobis $D^2$. Explain how $T^2$ can be interpreted as an extension of the student’s $t$ statistic. Describe the utility of Fisher’s discriminent function in classification problems. [2012/20m/7b]

17. In order to test the equality of mean vectors of $N_p({\mu }_i,\Sigma )i=1,2$ define Hotelling’s $T^2$-statistic and sketch the derivation of its null distribution. [2005/20m/6c]

18. Define Mahalanobis $D^2$. Show that it is invariant under any non-singular linear transformation. [2002/12m/5f]

19. Let $X\sim N_p(\mu ,\Sigma )$ unknown and $\mu =({\mu }_1,{\mu }_2,\dots ,{\mu }_p)\prime$. Suggest a test for $H_0:{\mu }_1={\mu }_2=\dots {\mu }_p$ against $H_1$: all $\mu$ s are not equal. [2000/15m/6b(ii)]

20. Define Hotelling’s $T^2$. Show that it is invariant under non-singular linear transformations. Mention some uses of the $T^2$ statistic. [1999/25m/12b]

21. $T^2$ statistics is the generalization of t-statistic. Explain its relationship with $D^2$-statistic. Explain the uses of $T^2$ statistic in test of hypothesis. [1998/30m/12a]

22. Let $X_{\alpha }(\alpha =1,2,\dots ,N)$ be a random sample of size N from $N_p(\mu ,\Sigma )$, where $\Sigma$ is known. Develop a suitable test for testing the null hypothesis $H_0:\mu ={\mu }_0$, where ${\mu }_0$ is a given vector. [1997/30m/12a]

23. Define Mahalanobis’ distance between two populations as estimated from the sample on the basis of p characters. State how the $D^2$-statistic is used to test the hypothesis of no difference in mean values of the p characters. [1996/30m/12a]

24. Define Hotelling’s $T^2$ statistic and Mahalanobis’ $D^2$ statistic and state the relationship between the two. Describe the various uses of these statistics in test of hypotheses. [1995/30m/12a]

3.7] Multivariate methods

31 questions

1. Explain the problem of classification in discriminant analysis. In case of two known multivariate normal populations, define Fisher’s discriminant function and calculate it for the data given below: ${𝛍}^{\left(1\right)}=\left(\begin{array}{c}\hfill 24\hfill \\ \hfill 12\hfill \end{array}\right),{𝛍}^{\left(2\right)}=\left(\begin{array}{c}\hfill 18\hfill \\ \hfill 10\hfill \end{array}\right)\mathrm{\text{and}}\Sigma =\left(\begin{array}{cc}\hfill 9\hfill & \hfill -5\hfill \\ \hfill -5\hfill & \hfill 8\hfill \end{array}\right)$ [2026/10m/5b]

2. (i) What are principal components? Show that the principal components are uncorrelated. (ii) Obtain the principal components and the amount of variation explained by each principal component associated with the following dispersion matrix: $\Sigma =\left(\begin{array}{ccc}\hfill 4\hfill & \hfill 2\hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill 3\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 2\hfill \end{array}\right)$ Comment on the results. [2025/20m/8a]

3. Define principal components and canonical correlation. How can one attain data reduction using principal components? If $(X_1,X_2)$ has covariance matrix $\Sigma =\left[\begin{array}{cc}\hfill 1\hfill & \hfill \rho \hfill \\ \hfill \rho \hfill & \hfill 1\hfill \end{array}\right]$ then find the principal components. [2024/15m/6b]

4. State briefly three reasons why an analyst may wish to perform a principal component analysis. [2023/6m/6c(i)]

5. Define canonical correlations and give two examples of their application. Describe the procedure of working out canonical correlations and canonical variates. [2023/9m/6c(ii)]

6. (i) If $X_1=Y_1+Y_2,X_2=Y_2+Y_3,X_3=Y_3+Y_1$, where $Y_1,Y_2$ and $Y_3$ are uncorrelated random variables and each of which has zero mean and unit standard deviation, find the multiple correlation coefficient between $X_3$ and $X_1,X_2$. (ii) Let $𝐗$ be a 3-dimensional random vector with dispersion matrix $\Sigma =\left(\begin{array}{ccc}\hfill 9\hfill & \hfill 3\hfill & \hfill 3\hfill \\ \hfill 3\hfill & \hfill 9\hfill & \hfill 3\hfill \\ \hfill 3\hfill & \hfill 3\hfill & \hfill 9\hfill \end{array}\right).$ Determine the first principal component and the proportion of the total variability that it explains. [2022/15m/6c]

7. (i) The data matrix of a random sample of size $n=3$ from a bivariate normal population $BVN({\mu }_1,{\mu }_2,{\sigma }_1^2,{\sigma }_2^2,\rho )$ is $X=\left[\begin{array}{cc}\hfill 6\hfill & \hfill 10\hfill \\ \hfill 10\hfill & \hfill 6\hfill \\ \hfill 8\hfill & \hfill 2\hfill \end{array}\right].$ Test the null hypothesis $H_0:𝛍={𝛍}_0$ against $H_1:𝛍\ne {𝛍}_0$, where ${𝛍}_0\prime =(8,5)$, at 10% level of significance. [You are given: $F_{0.10;2,1}=49.5,F_{0.10;1,2}=8.53$] (ii) Suppose $n_1=11$ and $n_2=12$, observations are made on two random vectors ${𝐗}_1$ and ${𝐗}_2$ which are assumed to have bivariate normal distribution with a common covariance matrix $\Sigma$, but possibly different mean vectors ${𝛍}_1$ and ${𝛍}_2$. The sample mean vectors and pooled covariance matrix are ${\overline{𝐗}}_1=\left(\begin{array}{c}\hfill -1\hfill \\ \hfill -1\hfill \end{array}\right),{\overline{𝐗}}_2=\left(\begin{array}{c}\hfill 2\hfill \\ \hfill 1\hfill \end{array}\right),S_{\mathrm{\text{pooled}}}=\left(\begin{array}{cc}\hfill 7\hfill & \hfill -1\hfill \\ \hfill -1\hfill & \hfill 5\hfill \end{array}\right).$ Obtain Mahalanobis sample distance $D^2$ and Fisher’s linear discriminant function. Assign the observation ${𝐗}_0=\left(\begin{array}{c}\hfill 0\hfill \\ \hfill 1\hfill \end{array}\right)$ to either population ${\Pi }_1$ or ${\Pi }_2$. [2022/20m/7b]

8. What are principal components? Describe how to compute the principal components of the vectors $X_1=\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 0\hfill \\ \hfill -1\hfill \end{array}\right]$ and $X_2=\left[\begin{array}{c}\hfill -1\hfill \\ \hfill 1\hfill \\ \hfill 0\hfill \end{array}\right]$. Give $X_1$ and $X_2$ in terms of the principal components. [2021/15m/7b]

9. Two types of tests viz: written and oral were conducted for seeking admission to MBA course. The result of the test given by the students are as follows: $x=\left(\begin{array}{c}\hfill x_1\hfill \\ \hfill x_2\hfill \end{array}\right)=\left(\begin{array}{c}\hfill \mathrm{\text{marks obtained in written test}}\hfill \\ \hfill \mathrm{\text{marks obtained in oral test}}\hfill \end{array}\right)$ Let ${\pi }_1$: students eligible for admission to MBA course; ${\pi }_2$: students not eligible for admission to MBA course ${\mu }_1=\left(\begin{array}{c}\hfill 30.6\hfill \\ \hfill 14.4\hfill \end{array}\right),{\Sigma }_1=\left(\begin{array}{cc}\hfill 5.60\hfill & \hfill -0.62\hfill \\ \hfill -0.62\hfill & \hfill 2.08\hfill \end{array}\right){\mu }_2=\left(\begin{array}{c}\hfill 24.8\hfill \\ \hfill 11.2\hfill \end{array}\right),{\Sigma }_2=\left(\begin{array}{cc}\hfill 5.60\hfill & \hfill -0.62\hfill \\ \hfill -0.62\hfill & \hfill 2.08\hfill \end{array}\right)$ If one student gets 29.0 marks in written test and 12.0 in oral test, classify the student as eligible or not based on Fisher’s linear discriminant function. Also find the probability of misclassification. [Given that $P(0<Z<1.8277)=0.4664$]. [2020/15m/8c]

10. Consider two sons born first in $n$ families. Let $X_1$: head length of first son $X_2$: head breadth of first son $X_3$: head length of second son $X_4$: head breadth of second son The measurements provide the sample variance covariance matrix $S=\left(\begin{array}{cccc}\hfill 91.481\hfill & \hfill 50.753\hfill & \hfill 66.875\hfill & \hfill 44.267\hfill \\ \hfill 50.753\hfill & \hfill 52.186\hfill & \hfill 49.259\hfill & \hfill 33.651\hfill \\ \hfill 66.875\hfill & \hfill 49.259\hfill & \hfill 96.775\hfill & \hfill 54.278\hfill \\ \hfill 44.267\hfill & \hfill 33.651\hfill & \hfill 54.278\hfill & \hfill 43.222\hfill \end{array}\right)$ that leads to the correlation matrix $R=\left(\begin{array}{cc}\hfill R_{11}\hfill & \hfill R_{12}\hfill \\ \hfill R_{21}\hfill & \hfill R_{22}\hfill \end{array}\right),$ where $R_{11}=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 0.7346\hfill \\ \hfill 0.7346\hfill & \hfill 1\hfill \end{array}\right),R_{22}=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 0.8392\hfill \\ \hfill 0.8392\hfill & \hfill 1\hfill \end{array}\right)$ $R_{12}=R{\prime }_{21}=\left(\begin{array}{cc}\hfill 0.7107\hfill & \hfill 0.7040\hfill \\ \hfill 0.6931\hfill & \hfill 0.7085\hfill \end{array}\right).$ Obtain the canonical correlations and hence the first canonical variables and interpret your result. [2019/15m/8b]

11. Define principal components. Is the principal component affected by standardization? Establish your claim by illustrating the same with the covariance matrix $\Sigma =\left[\begin{array}{cc}\hfill 1\hfill & \hfill 4\hfill \\ \hfill 4\hfill & \hfill 100\hfill \end{array}\right].$ [2018/10m/5b]

12. Obtain first canonical correlation and its associated canonical variable pair for the following correlation matrix: $\rho =\left[\begin{array}{cccc}\hfill 1\hfill & \hfill \frac{1}{2}\hfill & \hfill \frac{2}{3}\hfill & \hfill \frac{2}{3}\hfill \\ \hfill \frac{1}{2}\hfill & \hfill 1\hfill & \hfill \frac{2}{3}\hfill & \hfill \frac{2}{3}\hfill \\ \hfill \frac{2}{3}\hfill & \hfill \frac{2}{3}\hfill & \hfill 1\hfill & \hfill \frac{1}{2}\hfill \\ \hfill \frac{2}{3}\hfill & \hfill \frac{2}{3}\hfill & \hfill \frac{1}{2}\hfill & \hfill 1\hfill \end{array}\right]$ [2017/20m/6a]

13. Suppose $n_1=11$ and $n_2=12$ observations are made on two random vectors $X_1$ and $X_2$ which are assumed to have bivariate normal distribution with a common covariance matrix $\Sigma$, but possibly different mean vectors ${\mu }_1$ and ${\mu }_2$. The sample mean vectors and pooled covariance matrix are ${\bar{X}}_1=\left[\begin{array}{c}\hfill -1\hfill \\ \hfill -1\hfill \end{array}\right],{\bar{X}}_2=\left[\begin{array}{c}\hfill 2\hfill \\ \hfill 1\hfill \end{array}\right],S_{\mathrm{\text{pooled}}}=\left[\begin{array}{cc}\hfill 7\hfill & \hfill -1\hfill \\ \hfill -1\hfill & \hfill 5\hfill \end{array}\right]$ Obtain Mahalanobis sample distance $D^2$ and Fisher’s linear discriminant function. Assign the observation $X_0\prime =(0,1)$ to either population ${\pi }_1$ or ${\pi }_2$. [2017/15m/7b]

14. Let $𝐗=(X_1,X_2,X_3)\prime$ has the following covariance matrix: $\Sigma =\left[\begin{array}{ccc}\hfill 1\hfill & \hfill -2\hfill & \hfill 0\hfill \\ \hfill -2\hfill & \hfill 5\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 2\hfill \end{array}\right]$ Obtain the first three principal components using these values and the proportion of variations explained by the first two principal components. [2016/20m/8a]

15. Suppose a random vector $\underset{\sim }{X}$ has the covariance matrix $\Sigma =\left(\begin{array}{ccc}\hfill 3\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 3\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 5\hfill \end{array}\right)$. Find the principal components of $\underset{\sim }{X}$ and obtain the proportion of the total variance accounted for by the first two principal components. [2015/15m/7b]

16. Let $𝐗=(X_1,X_2,X_3)\prime$ has the correlation matrix $R$ given by $R=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill \frac{1}{4\sqrt{2}}\hfill & \hfill 0\hfill \\ \hfill \frac{1}{4\sqrt{2}}\hfill & \hfill 1\hfill & \hfill \frac{1}{4\sqrt{2}}\hfill \\ \hfill 0\hfill & \hfill \frac{1}{4\sqrt{2}}\hfill & \hfill 1\hfill \end{array}\right]$ Obtain the first two principal components and the population variance explained by the first two principal components. [2013/15m/8b]

17. Suppose $\underset{\sim }{X}={(X_1,X_2,X_3)}^1$ has co-variance matrix $\Sigma =\left[\begin{array}{ccc}\hfill 1\hfill & \hfill -2\hfill & \hfill 0\hfill \\ \hfill -2\hfill & \hfill 5\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 2\hfill \end{array}\right].$ Obtain the first two principal components of $X$. [2012/20m/6c]

18. Bring out the relation between Hotelling’s $T^2$ and Mahalanobis $D^2$. Explain how $T^2$ can be interpreted as an extension of the student’s $t$ statistic. Describe the utility of Fisher’s discriminent function in classification problems. [2012/20m/7b]

19. Explain briefly the method of factor analysis and its uses. [2007/12m/5b]

20. Briefly explain the development and use of linear discriminant functions in the context of two multivariate populations. [2007/20m/6b]

21. Discuss the procedure of classification of an observation into one of two known multivariate normal populations. [2006/20m/6a]

22. What is the purpose of MANOVA? Explain the one-way MANOVA model. [2004/12m/5f]

23. Explain the purpose of discriminant analysis. What is Fisher’s discriminant function for two populations? Give the allocation rule based on it. [2004/20m/6b]

24. Outlining the purpose of factor analysis, describe the orthogonal factor model along with its covariance structure. [2004/20m/6c]

25. What are the situations where discriminant analysis is used? Give some examples in support of your answer. Write the objectives of two group discriminant analysis. Explain how the discriminant criterion is calculated. [2003/20m/6b]

26. What are principal components? Discuss the method of finding them. [2002/20m/6c]

27. Define principal components and explain their significance in statistical practice. [2000/10m/6b(i)]

28. What are principal components? Explain the differences between principal components analysis and factor analysis. Briefly indicate how you can obtain maximum likelihood estimates of the principal components, given a random sample from $N_p(0,\Sigma )$ where $\Sigma$ has all its characteristic roots distinct. [1999/35m/12a]

29. Let $N_p({\mu }_1,\Sigma )$ and $N_p({\mu }_2,\Sigma )$ and linear discriminant function $L\left(U\right)=({\mu }_2-{\mu }_1)\prime {\Sigma }^{-1}U$ is sufficient for the class of normal densities $N_p(\alpha {\mu }_1+\beta {\mu }_2,\Sigma )$, $\alpha +\beta =1$, prove it. Discuss the methods used in it. [1998/30m/12b]

30. Define fishers discriminant function and obtain a test of the adequacy of a hypothetical discriminant function. [1997/30m/12b]

31. Define Fisher’s discriminant function between two p-variate populations. Indicate the connection between the discriminant function coefficients and the $D^2$-statistic. Describe a test to judge the adequacy of a preassigned discriminant function. [1996/30m/12b]

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