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Question 1

Let $X_1, X_2, \dots, X_n$ be a random sample of size $n$ from a distribution with mean $a+b$ and variance $a$ , where $a, b \in \mathbb{R}$ and $a > 0$ .

Let $\bar X = \frac{1}{n} \sum_{i=1}^n X_i$ be the sample mean and $S_n^2 = \frac{1}{n} \sum_{i=1}^n (X_i - \bar X)^2$ be the sample variance.

Consider the estimator $T_1 = \bar X - S_n^2$ for the parameter $b$ .

Show that the estimator $T_1$ is not unbiased for $b$ and compute its bias.

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Question 2

Let $\bar{X}$ be the sample mean and $S^2$ be the sample variance of a random sample. It is given that $E(\bar{X}) = a + b$ and $E(S^2) = a$ , where $a$ and $b$ are parameters.

Find one linear combination $T = \alpha \bar{X} + \beta S^2$ that is an unbiased estimator for $b$ . Specify the values of $\alpha$ and $\beta$ .

[4]

Question 3

Assuming $X_i \sim N(a+b, a)$ so that $\bar{X}$ and $S^2$ are independent, express the variance of the estimator $T = \bar{X} - \frac{(n-1)S^2}{n}$ in terms of $a$ and $n$ .

[5]

Question 4

Using the method of moments for a sample from a distribution with $E(X) = a + b$ and $\text{Var}(X) = a$ , derive the moment estimators $\hat{a}$ and $\hat{b}$ and state whether each is unbiased.

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Question 5

Properties of estimators: unbiasedness of the sample mean.

Show that the sample mean $\displaystyle \bar{X} = \frac{1}{n}\sum_{i=1}^n X_i$ is an unbiased estimator of the parameter $a + b$ , given that $E(X_i)=a+b$ .

[3]

Question 6

In this question, $a$ represents the population variance, commonly denoted by $\sigma^2$ .

Show that the sample variance $S^2 = \displaystyle \frac{1}{n-1}\sum_{i=1}^n (X_i - \bar{X})^2$ is an unbiased estimator for $a$ .

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Question 7

Derive the variance of the sample mean $\bar{X}$ in terms of $a$ for $n$ i.i.d. observations where $\operatorname{Var}(X_i) = a$ .

[3]

Question 8

Determine the bias of the estimator $S_n^2 = \displaystyle \frac{1}{n}\sum_{i=1}^n (X_i - \bar X)^2$ for the parameter $a$ , where $\text{Var}(X_i)=a$ .

[4]

Question 9

Let $X_1, X_2, \dots, X_n$ be a random sample of size $n$ from a distribution with mean $a+b$ and variance $a$ , where $a, b \in \mathbb{R}, a > 0$ and $n > 1$ . Let $\bar X$ be the sample mean and $S_n^2 = \frac{1}{n} \sum_{i=1}^n (X_i - \bar X)^2$ be the biased sample variance.

Find the constant $k$ , in terms of $n$ , such that $T_2 = \bar X - k S_n^2$ is an unbiased estimator of $b$ .

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Question 10

Let $S_n^2$ be an estimator for the parameter $a$ , where $E(S_n^2) = \frac{n-1}{n}a$ .

Find the constant $c$ such that the estimator $c S_n^2$ is unbiased for $a$ .

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Question 11

Use the Cramér–Rao inequality to find a lower bound for the variance of any unbiased estimator of $b$ when $X_i \sim N(a+b, a)$ and $a$ is known.

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Question 12

Let $X_1, X_2, \dots, X_n$ be a random sample of size $n$ from a distribution such that $E(\bar X) = a + b$ and $E\left[\sum_{i=1}^n (X_i - \bar X)^2 \right] = na$ , where $a$ and $b$ are constants and $n > 1$ . Let $S^2$ be the unbiased sample variance given by $S^2 = \frac{1}{n-1} \sum_{i=1}^n (X_i - \bar X)^2$ .

Show that the estimator $T = \bar X - \frac{(n-1)S^2}{n}$ is an unbiased estimator for $b$ .

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Question Type 4: Working with parameters in determining unbiased estimators Exercises