To solve this question, we need to assess each statement concerning the properties of the random variables \( Y_1, Y_2, Y_3, \) and \( Y_4 \), which are uncorrelated observations having the given expectations and variances.
- Statement: \(p_1\beta_1 + p_2\beta_2 + p_3\beta_3\) is estimable if and only if \(p_1 + p_2 = 2p_3\).
- The expectation equations given are:
- \(E(Y_1) = \beta_1 + \beta_2 + \beta_3\)
- \(E(Y_2) = \beta_1 + \beta_2 + \beta_3\)
- \(E(Y_3) = \beta_1 - \beta_2\)
- \(E(Y_4) = \beta_1 - \beta_2\)
- To express an estimable function \(p_1\beta_1 + p_2\beta_2 + p_3\beta_3\), it should be a linear combination of the given expectations.
- By comparing coefficients in linear combinations, for the function to be estimable, we require \(p_1 + p_2 = 2p_3\), as this balances the contributions of each parameter's estimation in the linear combinations.
- Thus, this statement is true.
- The expectation equations given are:
- Statement: An unbiased estimator of \(\sigma^2\) is \([(Y_1 - Y_2)^2 + (Y_3 - Y_4)^2]/4\).
- Since the variables are uncorrelated, the variance of \((Y_1 - Y_2)\) is \(2\sigma^2\) and the same for \((Y_3 - Y_4)\).
- Thus, the variance of \(\dfrac{1}{4}[(Y_1 - Y_2)^2 + (Y_3 - Y_4)^2]\) is \(\dfrac{1}{2}(\dfrac{2\sigma^2 + 2\sigma^2}{4}) = \sigma^2\).
- The equation indeed provides the correct divisor for getting an unbiased estimator for \(\sigma^2\).
- Thus, this statement is true.
- Statement: The best linear unbiased estimator of \(\beta_1 - \beta_2\) is \(\frac{1}{2}(Y_3 + Y_4)\).
- The estimator \(\frac{1}{2}(Y_3 + Y_4)\) provides the unbiased estimate of \(E(Y_3) = \beta_1 - \beta_2\) with minimal variance, as it combines two independent estimators.
- This ensures that the variance of the estimate is minimized as they are uncorrelated.
- Thus, this statement is true.
- Statement: The variance of the best linear unbiased estimator of \(\beta_1 + \beta_2 + \beta_3\) is \(\sigma^2\).
- The best linear unbiased estimator of \(\beta_1 + \beta_2 + \beta_3\) using \(Y_1\) or \(Y_2\) has variance \(\sigma^2\) since \(Var(Y_i) = \sigma^2\).
- Thus, this statement is a bit redundant but technically correct.
- \(\sigma^2\) is indeed the variance directly from the given observations.
- This statement is correct considering the provided variance of the observations.
Important Questions from Gauss-Markov Models
- Let $X_1, X_2,..., X_n$ ($n > 5$) be independent random variables such that
$X_t = \alpha + \alpha^2 t + \epsilon_t$, for $t = 1,..., n$,
where, $\epsilon_1, \epsilon_2,..., \epsilon_n$ are independent and identically distributed $N(0, \sigma^2)$ random variables. Here $\alpha \in R$ and $\sigma > 0$ are unknown parameters. Which of the following statements are true? Consider the linear statistical model
$y_{ij} = \mu + \tau_i + \varepsilon_{ij}; \quad i = 1, 2, \cdots, a; \quad j = 1, 2, \cdots, n$
where $\mu$ is unknown, $\tau_i$ are independently and identically distributed as $N(0, \sigma_\tau^2)$, $\varepsilon_{ij}$ are independently and identically distributed as $N(0, \sigma^2)$; $\tau_i$ and $\varepsilon_{ij}$ are independent for all $i$ and $j$. Note that $\tau_i$ is the $i^{th}$ treatment effect. Suppose $SS_{total}, SS_{treatment}, SS_{error}$ are total sum of squares, total treatment sum of squares and error sum of squares, respectively. To test :$H_0 : \sigma_\tau^2 = 0$ vs $H_A : \sigma_\tau^2 > 0$ which of the following statements is not true?
Consider the model $Y = X\beta + \epsilon$, where $Y = \begin{bmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{bmatrix}, X = \left( (x_{ij}) \right)_{n \times p}$ and $\beta = \begin{bmatrix} \beta_1 \\ \vdots \\ \beta_p \end{bmatrix}, \epsilon = \begin{bmatrix} \epsilon_1 \\ \epsilon_2 \\ \vdots \\ \epsilon_n \end{bmatrix}$. $E(\epsilon) = 0$ and $D(\epsilon) = \sigma^2 I_n, p < n$. Let $\hat{\beta}$ be the solution of $X^T X \beta = X^T Y$. Which of the following are true?
Let $X_1, X_2, \ldots$ be a random sample from uniform $(0, 3\theta), \theta > 0$. Define
$T_n = \frac{1}{3} \max\{X_1, X_2, \ldots, X_n\}$. Which of the following is NOT true?
- Consider the following regression problem $y_i = \beta_1 e^i + \beta_2 e^{-i} + \epsilon_i; \ i = 1, \ldots, n$. Here $\epsilon_1, \ldots, \epsilon_n$ are i.i.d. $N(0, \sigma^2)$ random variables. If $\hat{\beta}_1$ and $\hat{\beta}_2$ are the least square estimators of $\beta_1$ and $\beta_2$, respectively, then which of the following statements are correct?
