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?
Analyzing Statistic Properties for Uniform Distribution
A random sample $X_1, X_2, \ldots$ is drawn from a uniform distribution $U(0, 3\theta)$, where $\theta > 0$. The statistic is $T_n = \frac{1}{3} \max\{X_1, X_2, \ldots, X_n\}$. We need to find the statement about $T_n$ that is NOT true.
Deriving the Distribution of $T_n$
Let $Y_n = \max\{X_1, X_2, \ldots, X_n\}$. The cumulative distribution function (CDF) of $X_i$ is $F(x; \theta) = \frac{x}{3\theta}$ for $0 < x < 3\theta$. The CDF of $Y_n$ is $F_{Y_n}(y) = P(Y_n \le y) = [F(y; \theta)]^n = \left(\frac{y}{3\theta}\right)^n$ for $0 < y < 3\theta$. Since $T_n = \frac{1}{3} Y_n$, its CDF is $F_{T_n}(t) = P(T_n \le t) = P(Y_n \le 3t) = \left(\frac{3t}{3\theta}\right)^n = \left(\frac{t}{\theta}\right)^n$ for $0 < t < \theta$. The probability density function (PDF) of $T_n$ is $f_{T_n}(t) = \frac{d}{dt} F_{T_n}(t) = \frac{n t^{n-1}}{\theta^n}$ for $0 < t < \theta$.
Evaluating $T_n$ Consistency for $\theta$
Consistency: $T_n$ is consistent for $\theta$ if $T_n \xrightarrow{P} \theta$ as $n \to \infty$. We calculate the expectation and variance:
- $E[T_n] = \int_0^\theta t \cdot \frac{n t^{n-1}}{\theta^n} dt = \int_0^\theta \frac{n t^n}{\theta^n} dt = \frac{n}{\theta^n} \left[ \frac{t^{n+1}}{n+1} \right]_0^\theta = \frac{n}{n+1} \theta$.
- $E[T_n^2] = \int_0^\theta t^2 \cdot \frac{n t^{n-1}}{\theta^n} dt = \int_0^\theta \frac{n t^{n+1}}{\theta^n} dt = \frac{n}{\theta^n} \left[ \frac{t^{n+2}}{n+2} \right]_0^\theta = \frac{n}{n+2} \theta^2$.
- $Var(T_n) = E[T_n^2] - (E[T_n])^2 = \frac{n}{n+2} \theta^2 - \left(\frac{n}{n+1} \theta\right)^2 = \frac{n \theta^2}{(n+2)(n+1)^2}$.
As $n \to \infty$, $E[T_n] \to \theta$ and $Var(T_n) \to 0$. By Chebyshev's inequality, $T_n$ is consistent for $\theta$. Therefore, statement 1 is true.
Checking $T_n$ Unbiasedness for $\theta$
Unbiasedness: $T_n$ is unbiased if $E[T_n] = \theta$. From the calculation above, $E[T_n] = \frac{n}{n+1} \theta$. Since $\frac{n}{n+1} \neq 1$ for finite $n$, $E[T_n] \neq \theta$. Thus, $T_n$ is not unbiased for $\theta$. Statement 2 is NOT true.
Verifying $T_n$ Sufficiency
Sufficiency: The joint PDF of the sample is $f(x_1, \ldots, x_n; \theta) = \prod_{i=1}^n \frac{1}{3\theta} \mathbb{I}(0 < x_i < 3\theta) = \left(\frac{1}{3\theta}\right)^n \mathbb{I}(\max\{x_i\} < 3\theta)$. Let $Y_n = \max\{x_i\}$. Then $f(\ldots; \theta) = \left(\frac{1}{3\theta}\right)^n \mathbb{I}(Y_n < 3\theta)$. Since $T_n = Y_n/3$, we have $Y_n = 3T_n$. The condition becomes $3T_n < 3\theta$, or $T_n < \theta$. So, $f(\ldots; \theta) = \left(\frac{1}{3\theta}\right)^n \mathbb{I}(T_n < \theta)$. This likelihood function depends on the data only through the statistic $T_n$. By the factorization criterion, $T_n$ is sufficient for $\theta$. Statement 3 is true.
Confirming $T_n$ Completeness
Completeness: The PDF of $T_n$ is $f_{T_n}(t) = \frac{n t^{n-1}}{\theta^n}$ for $0 < t < \theta$. Consider the transformation $U_n = T_n / \theta$. The PDF of $U_n$ can be derived using the change of variable method. Substituting $t = u\theta$, we get $dt = \theta du$. The PDF is $f_{U_n}(u) = f_{T_n}(u\theta) \left| \frac{dt}{du} \right| = \frac{n (u\theta)^{n-1}}{\theta^n} \cdot \theta = n u^{n-1}$ for $0 < u < 1$. This is the PDF of a Beta distribution with parameters $n$ and $1$ (denoted Beta(n, 1)). The family of Beta(k, 1) distributions, for $k>0$, is known to be complete. Since $n \ge 1$, the distribution of $U_n = T_n/\theta$ is complete. This property implies that the statistic $T_n$ is complete for $\theta$. Statement 4 is true.
Conclusion
The statement "$T_n$ is unbiased for $\theta$" (Option B) is the only one that is NOT true.
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?
- 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?
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?
