Question

3. (25 pts) Suppose \{X_1, X_2, X_3, ..., X_n\} is a random sample from $N(\mu, \sigma^2)$. Let $\overline{X}$ be the sample mean, \\ and $S^2 = \frac{1}{n-1}\sum_{i=1}^{n}(X_i - \overline{X})^2$ be the sample variance. Let \\ $\hat{\theta} = \left(\frac{n+2}{n-3}\right)S^2$ \\ (a) (10 pts) Show that $\hat{\theta}$ is a biased estimator for $\sigma^2$. Calculate the bias of $\hat{\theta}$. \\ (b) (8 pts) Calculate the variance of $\hat{\theta}$. \\ (c) (7 pts) Calculate the mean squared error of $\hat{\theta}$. \\ Show all your work, including calculations of $E(S^2)$ and $V(S^2)$. Hint: you can use the fact that $\frac{(n-1)S^2}{\sigma^2}$ \\ has the Chi-Squared distribution with df = (n - 1). Assume $n \ge 4.$

          3. (25 pts) Suppose \{X_1, X_2, X_3, ..., X_n\} is a random sample from $N(\mu, \sigma^2)$. Let $\overline{X}$ be the sample mean, \\
and $S^2 = \frac{1}{n-1}\sum_{i=1}^{n}(X_i - \overline{X})^2$ be the sample variance. Let \\
$\hat{\theta} = \left(\frac{n+2}{n-3}\right)S^2$ \\
(a) (10 pts) Show that $\hat{\theta}$ is a biased estimator for $\sigma^2$. Calculate the bias of $\hat{\theta}$. \\
(b) (8 pts) Calculate the variance of $\hat{\theta}$. \\
(c) (7 pts) Calculate the mean squared error of $\hat{\theta}$. \\
Show all your work, including calculations of $E(S^2)$ and $V(S^2)$. Hint: you can use the fact that $\frac{(n-1)S^2}{\sigma^2}$ \\
has the Chi-Squared distribution with df = (n - 1). Assume $n \ge 4.$

$3. (25 pts) Suppose {X1, X2, X3, ..., Xn} is a random sample from N(μ, σ^2). Let X be the sample mean, and S^2 = (1)/(n-1)∑i=1^n(Xi - X)^2 be the sample variance. Let θ̂ = ((n+2)/(n-3))S^2 (a) (10 pts) Show that θ̂ is a biased estimator for σ^2. Calculate the bias of θ̂. (b) (8 pts) Calculate the variance of θ̂. (c) (7 pts) Calculate the mean squared error of θ̂. Show all your work, including calculations of E(S^2) and V(S^2). Hint: you can use the fact that ((n-1)S^2)/(σ^2) has the Chi-Squared distribution with df = (n - 1). Assume n ≥ 4.$

Added by Mary R.

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