Teaching CSIR NET Mock Test Series Mathematical Science Statistics & Exploratory Data Analysis Independent Random Variables
Consider the multiple linear regression model \(\underline{Y}=X \underline{\beta}+\underline{ϵ}\), where \(\underline{Y}=\left(Y_1, \ldots, Y_n\right)^T\), \(\underline{ϵ}=\left(ϵ_1, \ldots, ϵ_n\right)^T\), \(\underline{\beta}=\left(\beta_0, \beta_1, \ldots, \beta_p\right)^T, \) X is a fixed n × (p + 1) matrix (n > p + 1) of rank (p + 1), and ϵ1, ..., ϵn are independent and identically distributed (i.i.d.) N(0, σ2), (σ > 0) variables. If \(\underline{\hat{\beta}}\) is the OLS estimator of \(\underline{\beta}\), then which of the following statements are true?
1
\(\frac{1}{\sigma^2} \underline{Y}^T X \underline{\hat{\beta}}\) has a central \(\chi_{p+1}^2\) distribution
2
\(\frac{1}{\sigma^2}(\underline{Y}-X \underline{\hat{\beta}})^T(\underline{Y}-X \underline{\hat{\beta}})\) has a central \(\chi_{n-p-1}^2\) distribution
3
\(X \underline{\hat{\beta}}\) and \((\underline{Y}-X \underline{\hat{\beta}})^T(\underline{Y}-X \underline{\hat{\beta}})\) are independently distributed
4
\(\frac{1}{\sigma^2} \sum_{i=1}^n\left(Y_i-\bar{Y}\right)^2\) has a central \(\chi_{n-1}^2\) distribution, where \(\bar{Y}=\frac{1}{n} \sum_{i=1}^n Y_i\)