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

Question 1

Skill question

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 $a, b \in R$ and $a > 0$ .

Let $\overset{ˉ}{X} = \frac{1}{n} \sum_{i = 1}^{n} X_{i}$ be the sample mean and $S_{n}^{2} = \frac{1}{n} \sum_{i = 1}^{n} (X_{i} - \overset{ˉ}{X})^{2}$ be the sample variance.

Consider the estimator $T_{1} = \overset{ˉ}{X} - S_{n}^{2}$ 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

Skill question

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

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

[4]

Question 3

Skill question

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

[5]

Question 4

Skill question

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

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Question 5

Skill question

Properties of estimators: unbiasedness of the sample mean.

Show that the sample mean $\overset{ˉ}{X} = \frac{1}{n} i = 1 \sum n X_{i}$ is an unbiased estimator of the parameter $a + b$ , given that $E (X_{i}) = a + b$ .

[3]

Question 6

Skill question

In this question, $a$ represents the population variance, commonly denoted by $σ^{2}$ .

Show that the sample variance $S^{2} = \frac{1}{n - 1} i = 1 \sum n (X_{i} - \overset{ˉ}{X})^{2}$ is an unbiased estimator for $a$ .

[3]

Question 7

Skill question

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

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Question 8

Skill question

Determine the bias of the estimator $S_{n}^{2} = \frac{1}{n} i = 1 \sum n (X_{i} - \overset{ˉ}{X})^{2}$ for the parameter $a$ , where $Var (X_{i}) = a$ .

[4]

Question 9

Skill question

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 $a, b \in R, a > 0$ and $n > 1$ . Let $\overset{ˉ}{X}$ be the sample mean and $S_{n}^{2} = \frac{1}{n} \sum_{i = 1}^{n} (X_{i} - \overset{ˉ}{X})^{2}$ be the biased sample variance.

Find the constant $k$ , in terms of $n$ , such that $T_{2} = \overset{ˉ}{X} - k S_{n}^{2}$ is an unbiased estimator of $b$ .

[5]

Question 10

Skill question

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

Skill question

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

Skill question

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

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

[3]

IB Mathematics Applications & Interpretation Question Type 4: Working with Parameters in Determining Unbiased Estimators Exercises