Let $\{X_n\}$ be a sequence of independent random variables where the distribution of $X_n$ is normal with mean $\mu$ and variance $n$ for $n = 1, 2, \ldots$ Define $\bar{X}_n = \frac{\sum_{i=1}^n X_i}{n}$ and $S_n = \frac{\sum_{i=1}^n \frac{X_i}{i}}{\sum_{i=1}^n \frac{1}{i}}$. Which of the following are true?
The problem asks us to evaluate the truthfulness of four statements concerning two statistics, $\bar{X}_n$ and $S_n$, derived from a sequence of independent normal random variables $\{X_n\}$ where $X_n \sim N(\mu, n)$.
Analyzing Expectation: $E(\bar{X}_n) = E(S_n)$
We calculate the expectation for both statistics:
- For $\bar{X}_n = \frac{1}{n} \sum_{i=1}^n X_i$: $E(\bar{X}_n) = E\left(\frac{1}{n} \sum_{i=1}^n X_i\right) = \frac{1}{n} \sum_{i=1}^n E(X_i)$ Since $E(X_i) = \mu$, we have: $E(\bar{X}_n) = \frac{1}{n} \sum_{i=1}^n \mu = \frac{1}{n} (n\mu) = \mu$.
- For $S_n = \frac{\sum_{i=1}^n \frac{X_i}{i}}{\sum_{i=1}^n \frac{1}{i}}$: Let $C_n = \sum_{i=1}^n \frac{1}{i}$. $E(S_n) = E\left(\frac{1}{C_n} \sum_{i=1}^n \frac{X_i}{i}\right) = \frac{1}{C_n} \sum_{i=1}^n E\left(\frac{X_i}{i}\right)$ $E(S_n) = \frac{1}{C_n} \sum_{i=1}^n \frac{E(X_i)}{i} = \frac{1}{C_n} \sum_{i=1}^n \frac{\mu}{i}$ $E(S_n) = \frac{\mu}{C_n} \sum_{i=1}^n \frac{1}{i} = \frac{\mu}{C_n} C_n = \mu$.
Since both $E(\bar{X}_n) = \mu$ and $E(S_n) = \mu$, the statement $E(\bar{X}_n) = E(S_n)$ is true for all $n$, including sufficiently large $n$. Thus, option A is true.
Analyzing Variance: $\text{Var}(S_n) < \text{Var}(\bar{X}_n)$
We calculate the variance for both statistics:
- For $\bar{X}_n$: $\text{Var}(\bar{X}_n) = \text{Var}\left(\frac{1}{n} \sum_{i=1}^n X_i\right) = \frac{1}{n^2} \sum_{i=1}^n \text{Var}(X_i)$ (due to independence) Since $\text{Var}(X_i) = i$: $\text{Var}(\bar{X}_n) = \frac{1}{n^2} \sum_{i=1}^n i = \frac{1}{n^2} \frac{n(n+1)}{2} = \frac{n+1}{2n}$.
- For $S_n$: $\text{Var}(S_n) = \text{Var}\left(\frac{1}{C_n} \sum_{i=1}^n \frac{X_i}{i}\right) = \frac{1}{C_n^2} \sum_{i=1}^n \text{Var}\left(\frac{X_i}{i}\right)$ (due to independence) $\text{Var}(S_n) = \frac{1}{C_n^2} \sum_{i=1}^n \frac{\text{Var}(X_i)}{i^2} = \frac{1}{C_n^2} \sum_{i=1}^n \frac{i}{i^2} = \frac{1}{C_n^2} \sum_{i=1}^n \frac{1}{i}$ $\text{Var}(S_n) = \frac{1}{C_n^2} C_n = \frac{1}{C_n}$.
Now, consider the behavior for large $n$. We know $C_n = \sum_{i=1}^n \frac{1}{i} \approx \ln(n) + \gamma$, which grows indefinitely as $n \to \infty$.
- As $n \to \infty$, $\text{Var}(\bar{X}_n) = \frac{n+1}{2n} \to \frac{1}{2}$.
- As $n \to \infty$, $\text{Var}(S_n) = \frac{1}{C_n} \to 0$.
Since $0 < \frac{1}{2}$, for sufficiently large $n$, we have $\text{Var}(S_n) < \text{Var}(\bar{X}_n)$. Thus, option B is true.
Evaluating Consistency: $\bar{X}_n$ for $\mu$
An estimator $T_n$ is consistent for $\theta$ if $T_n \to \theta$ in probability.
We check if $\bar{X}_n \to \mu$ in probability using Chebyshev's inequality:
$P(|\bar{X}_n - E(\bar{X}_n)| \ge \epsilon) \le \frac{\text{Var}(\bar{X}_n)}{\epsilon^2}$Substituting $E(\bar{X}_n) = \mu$ and $\text{Var}(\bar{X}_n) = \frac{n+1}{2n}$:
$P(|\bar{X}_n - \mu| \ge \epsilon) \le \frac{n+1}{2n\epsilon^2}$As $n \to \infty$, the right side converges to $\frac{1}{2\epsilon^2}$, which is not 0.
Since the probability does not converge to 0, $\bar{X}_n$ is not consistent for $\mu$. Thus, option C is false.
Assessing Sufficiency: $\bar{X}_n$ for $\mu$
A statistic $T$ is sufficient for $\theta$ if the conditional distribution of the sample $X = (X_1, \dots, X_n)$ given $T=t$ does not depend on $\theta$.
The joint probability density function is:
$f(x_1, \dots, x_n; \mu) = \prod_{i=1}^n \frac{1}{\sqrt{2\pi i}} \exp\left(-\frac{(x_i - \mu)^2}{2i}\right)$The exponent term involves $\sum_{i=1}^n \frac{(x_i - \mu)^2}{i}$. Expanding this shows dependence on terms like $\sum \frac{x_i}{i}$ and $\sum \frac{x_i^2}{i}$. These terms involve more information than just $\bar{X}_n = \frac{1}{n}\sum x_i$. Therefore, $\bar{X}_n$ alone does not contain all the information about $\mu$. Thus, option D is false.
Conclusion
Based on the analysis, statements A and B are true.
Important Questions from Best Linear Unbiased Estimators
Consider the simple linear regression model $Y_i = \beta x_i + \epsilon_i, i = 1,2,..., n$, where $\beta > 0$, $\sum_{i=1}^n x_i^2 > 0$, and the uncorrelated errors $\epsilon_i, i = 1,2,..., n$, have zero mean and finite variance $\sigma^2 (> 0)$. Let $\tilde{\beta}_1 = \sum_{i=1}^n a_i^*Y_i$, where $a_i$'s minimize $E(\sum_{i=1}^n a_i Y_i - \beta)^2$ with respect to scalars $a_1, a_2,..., a_n$. Let $\tilde{\beta}_2$ be the ordinary least squares estimator of $\beta$. Which of the following statements are true?
Consider the model $y_i = i\beta + \varepsilon_i, i = 1, 2, 3$ where $\varepsilon_1, \varepsilon_2, \varepsilon_3$ are independent with mean $0$ and variance $\sigma^2, 2\sigma^2, 3\sigma^2$ respectively. Which of the following is the best linear unbiased estimate of $\beta$?
Consider the linear model $Y \sim N_n(X\beta, \sigma^2 I)$, where $X$ is a $n \times (k + 1)$ matrix of rank $k + 1 < n$. Let $\hat{\beta}$ and $\hat{\sigma}^2$ be the maximum likelihood estimators of $\beta$ and $\sigma^2$ respectively. Then which of the following statements are true?
- Let $X_1, X_2, \ldots, X_n$ be a random sample from $f_\theta(x)$, a probability density function or a probability mass function. Define $s_n^2 = \frac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X}_n)^2$, where $\bar{X}_n = \frac{1}{n} \sum_{i=1}^n X_i$. Then $s_n^2$ is unbiased for $\theta$ if
- Let $Y_1, Y_2, Y_3$ be uncorrelated observations with common variance $\sigma^2$ and expectations given by $\mathbb{E}(Y_1) = \beta_1, \mathbb{E}(Y_2) = \beta_2$ and $\mathbb{E}(Y_3) = \beta_1 + \beta_2,$ where $\beta_1, \beta_2$ are unknown parameters. The best linear unbiased estimator of $\beta_1 + \beta_2$ is
