To solve this problem, we need to analyze the given regression model and evaluate the truthfulness of the provided statements concerning the least squares estimators, \(\hat{\beta}_1\)and \(\hat{\beta}_2\).
The given regression model is: \(y_i = \beta_1 e^i + \beta_2 e^{-i} + \epsilon_i; \quad i = 1, \ldots, n\)where \(\epsilon_i \sim N(0, \sigma^2) \\) are independently and identically distributed (i.i.d.) random variables.
- The least squares estimators \(\hat{\beta}_1\)and \(\hat{\beta}_2\)are unbiased estimators of \(\beta_1\)and \(\beta_2\), respectively. This means that:
- \(E(\hat{\beta}_1) = \beta_1\)is correct because the property of unbiasedness in linear regression ensures that the expectation of the estimator equals the parameter being estimated.
- \(E(\hat{\beta}_2) = \beta_2\)is also correct for the same reason, demonstrating unbiasedness.
- Analyzing the variances and covariance:
- The statement \(\text{Var}(\hat{\beta}_1) > \text{Var}(\hat{\beta}_2)\)would require specific information about the variances which is not provided in the problem. Thus, we cannot confirm this statement without further information such as the design matrix or computing the variance directly.
- The statement \(\text{Cov}(\hat{\beta}_1, \hat{\beta}_2) < 0\)indicates the relationship between the two estimators. Given the form of the model where they relate oppositely through exponential and reciprocal exponential terms, it is plausible that the covariance is negative. In many regression models with opposing terms (one increasing while the other decreasing), a negative covariance is common, supporting the correctness of this statement.
From this analysis, the correct statements based on standard properties of least squares estimators and logical deduction are:
- \(E(\hat{\beta}_1) = \beta_1\)
- \(E(\hat{\beta}_2) = \beta_2\)
- \(\text{Cov}(\hat{\beta}_1, \hat{\beta}_2) < 0\)
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 multiple linear regression model
$Y_i = \beta_1 x_{i1} + \dots + \beta_p x_{ip} + \epsilon_i$, with $E(\epsilon_i) = 0$, $Cov(\epsilon_i, \epsilon_k) = 0$,
if $i \ne k$ and $Var(\epsilon_i) = \sigma^2$, for $i, k = 1, \dots, n$. If $\hat{y}_i$ is the least squares fit of $y_i$ and $\hat{\epsilon}_i$ is the corresponding estimated residual for $i = 1, \dots, n$, then which of the following statements are always correct?
