CSIR NET Maths previous year question papers

Consider the multiple linear regression model $y_{i} = \beta_{0} + \beta_{1}x_{1i} + \dots + \beta_{8}x_{8i} + \varepsilon_{i}, i = 1, 2, \dots, 29$, where $\varepsilon_{1}, \varepsilon_{2}, \dots, \varepsilon_{29}$ are independent and identically normal distributed with mean 0 and variance $\sigma^{2}$. Suppose the model is fitted using the method of least squares. If the calculated value of the F-statistic for testing the significance of regression is 2.50, then the possible values of $R^{2}$ and Adjusted $R^{2}$ are respectively

Suppose $(X_{1}, X_{2}, X_{3})^{T}$ ~ $N_{3}(0, I_{3})$, where 0 is the zero mean vector and $I_{3}$ is the 3 × 3 identity matrix, and $(X, Y, Z)^{T} = A(X_{1}, X_{2}, X_{3})^{T}$ where A has rows (3, 0, 0), (2, 2, 0) and (4, 0, 4). Then the partial correlation coefficient $\rho_{YZ.X}$ is

Let $X = (X_{1},X_{2},X_{3})^{T}$ be a 3×1 random vector with E(X) = (3,2,1)ᵀ and $Cov(X) = \sum$, with rows (3,−2,0), (−2,3,−2), (0,−2,3). Suppose that $Y = (Y_{1},Y_{2},Y_{3})^{T} = (1/\sqrt{3})X$. Then the value of the multiple correlation coefficient between $Y_{1}$ and $(Y_{2},Y_{3})$ equals

Consider the linear model $Y_{1} = \beta_{1}+\beta_{2}+\varepsilon_{1}, Y_{2} = \beta_{1}+2\beta_{2}+\varepsilon_{2}, Y_{3} = \beta_{1}+c\beta_{2}+\varepsilon_{3}$, where $\beta_{1},\beta_{2} \in \mathbb{R}$ are unknown parameters, and the uncorrelated errors $\varepsilon_{i}, i=1,2,3$ have zero mean and finite variance $\sigma^{2}(>0)$. The constant c is such that $\beta$̂$_{1}$ and $\beta$̂$_{2}$ are uncorrelated, where $\beta$̂$_{1}$ and $\beta$̂$_{2}$ are the best linear unbiased estimators of $\beta_{1}$ and $\beta_{2}$, respectively. Which of the following statements is the correct option for $(Var(\beta$̂$_{1}), Var(\beta$̂$_{2}))$?

Consider a paired data $(x_{i}, y_{i}); i = 1, 2, 3, 4, 5$, where $(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}) = (-2, -1, 0, 1, 2)$ and $y_{i} = x_{i}^{2}$ for all i = 1, 2, 3, 4, 5. On this data, a simple linear regression model with an intercept term and a simple linear regression model without an intercept term are fitted using the method of least squares. Which of the following statements are true?

An analyst fits a multiple linear regression model $Y_{i} = \beta_{0} + \beta_{1}X_{1i} + \dots + \beta_{4}X_{4i} + \varepsilon_{i}, i = 1, 2, \dots, n$, using the method of least squares. However, his coordinator insists that the intercept and two regressors $Z_{1} = X_{1} + X_{3}$ and $Z_{2} = X_{2} - X_{4}$ are enough to represent the model. Suppose the coordinator's claim can be tested in the form of a general linear hypothesis, viz., $H_{0} : L^{T}\beta = 0$ against $H_A : H_{0}$ is not true, where $\beta^{T} = (\beta_{0}, \beta_{1}, \dots, \beta_{4})$. Suppose we have n observations on the response Y and each regressor. Further, assume that the errors with or without restrictions are independent $N(0, \sigma^{2})$ variables. Then, which of the following statements are true?

Let $(X_{1}, Y_{1}), (X_{2}, Y_{2}), \dots, (X_{10}, Y_{10})$ be a random sample from a bivariate normal distribution BVN$(\mu_{1}, \mu_{2}, \sigma_{1}^{2}, \sigma_{2}^{2}, \rho)$ with $\mu_{1} = 5, \mu_{2} = 6, \sigma_{1}^{2} = 4, \sigma_{2}^{2} = 9$ and $\rho = 1/2$. Then, which of the following statements are true?

Let the random vector $(X_{1}, X_{2}, X_{3})^{T}$ have the positive definite dispersion matrix with rows $(1, \rho, \rho), (\rho, 1, \rho)$ and $(\rho, \rho, 1)$. Then, which of the following statements are true?

Let $X_{1}, X_{2}, \dots, X_{20}$ be a random sample from $N_{12}(\mu,\sum)$, where $\mu\in\mathbb{R}^{12}$ and $\sum$ is positive definite. Suppose ${\bar{X}} = (1/20)\sum_{i}$₌$_{1}^{20}X_{i}$ and $S = (1/19)\sum_{i}$₌$_{1}^{20}(X_{i}-{\bar{X}})(X_{i}-{\bar{X}})^{T}$. Then which of the following statements are true?

Let $Y=X\beta+\varepsilon$ be a multiple linear regression model with p regressors and an intercept, where $\beta=(\beta_{0},\beta_{1},\dots,\beta_{p})^{T}$ and the random error $\varepsilon$~$N_{n}(0,\sigma^{2}I), \sigma>0$ and n>p+1. The least squares method provides a unique estimator $\beta$̂. Let the total sum of squares (corrected), sum of squares due to regression, and sum of squares due to error, based on $\beta$̂, be denoted YᵀAY, YᵀBY and YᵀCY respectively, so YᵀAY=YᵀBY+YᵀCY. Which of the following statements are always true?

Consider the simple linear regression model $Y_{i}=\beta{}x_{i}+\varepsilon_{i}, i=1,2,\dots,n$, where $\beta>0, \sum{}x_{i}^{2}>0$, and the uncorrelated errors $\varepsilon_{i}$ have zero mean and finite variance $\sigma^{2}(>0)$. Let $\beta$̃$_{1}=\sum{}a_{i}$$Y_{i}$, where $a_{i}$'s minimize $E(\sum{}a_{i} Y_{i}-\beta)^{2}$ with respect to scalars $a_{1},a_{2},\dots,a_{n}$. Let $\beta$̃$_{2}$ be the ordinary least squares estimator of $\beta$. Which of the following statements are true?

Consider a multiple linear regression model $Y_{i} = \beta_{0} + \beta_{1}x_{i1} + \cdots + \beta_p x_ip + \varepsilon_{i}, 1 \le i \le n, n > (p + 1)$, where errors $\varepsilon_{i}'s$ are uncorrelated with zero mean and finite variance $\sigma^{2} > 0$. Here, $Y_{i}$ is the i-th response. Let Ŷ$_{i}$ be the i-th predicted response by the least squares estimation method, and let $\varepsilon$̂$_{i} = Y_{i} -$ Ŷ$_{i}, 1 \le i \le n$. Then, which of the following statements is true?

Let X, Y and Z be independent and identically distributed (i.i.d.) random variables with distribution N(0, 1). Define U = 2X, V = 3X + Y, W = X + 4Z. Then the partial correlation coefficient of V and W, given U is

Consider a multiple linear regression model $Y_{i} = \beta_{0} + \beta_{1}x_{i1} + \cdots + \beta_p x_ip + \varepsilon_{i}, 1 \le i \le n, n > (p + 1)$, where the errors $\varepsilon_{i}'s$ are uncorrelated with zero mean and finite variance $\sigma^{2} > 0$. Here, $Y_{i}$ is the i-th response. Let Ŷ$_{i}$ be the i-th predicted response by the least squares estimation method, and let $\varepsilon$̂$_{i} = Y_{i} -$ Ŷ$_{i}, 1 \le i \le n$. Then, which of the following statements is true?

Let X, Y and Z be random variables such that S = [[X, Y], [Y, Z]] ~ $W_{2}(10, \sum)$, where $W_{2}$ denotes the Wishart distribution and $\sum = [[1, 1/2], [1/2, 1]]$. Define $T = Z - Y^{2}/X$. Then, Var(T) equals

An analyst considers standardized values of observations on three variables, consumption (C), saving (S) and total income (TI) so that they have zero means and unit variances. She further considers disposable income (DI) where DI = C + S. In the simple linear regressions of DI on TI, DI on C and S on TI, the regression coefficients are 0.8, 0.5 and 0.4, respectively. There are 21 sample observations. Sample covariances and variances are calculated with divisor 20. Then, the value of sum of squared residuals in the regression of DI on S is

Let $X_{0}, X_{1}, \dots, X_p (p \ge 2)$ be independent and identically distributed random variables with mean 0 and variance 1. Suppose $Y_{i} = X_{0} + X_{i}, i = 1, \dots, p$. The first principal component based on the covariance matrix of $Y = (Y_{1}, \dots, Y_p)^{T}$ is

Consider a linear model $Y_{i} = \beta_{1} + \beta_{2} + \cdots + \beta_{i} + \varepsilon_{i}, 1 \le i \le n$, where the errors $\varepsilon_{i}'s$ are uncorrelated with zero mean and finite variance $\sigma^{2} > 0$. Then, which of the following statements are true?

Consider a linear regression model $Y = X\beta + \varepsilon$, with r regressors and an intercept. Random error $\varepsilon$ ~ $N_{n}(0, \sigma^{2}I_{n})$ and X has full column rank. Here $I_{n}$ denotes the identity matrix of order n. Regression coefficients are estimated by the least squares estimation method. Let $\sigma$̂$^{2}$ and $\sigma$̂$^{2}($MLE), respectively, be the mean squares residuals and the maximum likelihood estimator of $\sigma^{2}$. Then, which of the following statements are true?

If $(X_{1}, X_{2})$ᵗ ~ $N_{2}((0, 0)$ᵗ, [[3, 1], [1, 3]]), then which of the following statements are true?

Consider a linear model $Y_{i} = \beta_{1} + \beta_{2} + \cdots + \beta_{i} + \varepsilon_{i}, 1 \le i \le n$, where errors $\varepsilon_{i}'s$ are uncorrelated with zero mean and finite variance $\sigma^{2} > 0$. Let $\beta$̂$_{i}$ be the best linear unbiased estimator (BLUE) of $\beta_{i}, i = 1, 2, \dots, n$. Then, which of the following statements are true?

Consider a linear regression model $Y = X\beta + \varepsilon$, with r regressors and an intercept. Random error $\varepsilon$ ~ $N_{n}(0, \sigma^{2}I_{n})$ and X has full column rank. Here $I_{n}$ denotes the identity matrix of order n. Regression coefficients are estimated by the least squares estimation method. Let $\sigma$̂$^{2}$ and $\sigma$̂$^{2}($MLE), respectively, be the mean squares residuals and the maximum likelihood estimator of $\sigma^{2}$. Then, which of the following statements are true?

Let $(X_{1}, X_{2})$ be a bivariate normal random vector with $E(X_{1}) = 1, E(X_{2}) = 0, Var(X_{1}) = 1, Var(X_{2}) = 1$, and correlation coefficient 1/2. Let U be a U(0, 1) random variable, which is independent of $(X_{1}, X_{2})$. If Z = (UX$_{1} + X_{2} - U)/\sqrt{U^{2} + U + 1}$, then which of the following statements are true?

In a standard linear regression model, let $R^{2}$ and ${\bar{R}}^{2}$, respectively, denote the coefficient of determination and adjusted coefficient of determination. Which of the following statements are true?

Consider the two-way ANOVA model $Y_{ij} = \mu + \alpha_{i} + \beta_{j} + \varepsilon_{ij}, i = 1, 2; j = 1, 2$, where $\mu$ is the overall mean effect, $\alpha_{i}$ is the effect of the i-th level of factor $A, \beta_{j}$ is the effect of j-th level of factor $B, Y_{ij}$ is the response of the (i, j)-th experimental unit and $\varepsilon_{ij}$ is the corresponding error with $E(\varepsilon_{ij}) = 0$ for i = 1, 2; j = 1, 2. Which of the following are estimable linear parametric functions?