Compare estimator properties

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1. The statistic \(T\) is derived from a random sample taken from a population which has an unknown parameter \(\theta\). \(T\) is an unbiased estimator of \(\theta\). What does the statement ' \(T\) is an unbiased estimator of \(\theta ^ { \prime }\) imply?
2. A random sample of size \(n\) is taken from each of two independent populations. The first population has a non-zero mean \(\mu\) and variance \(\sigma ^ { 2 }\) and \(\bar { X } _ { 1 }\) denotes the sample mean. The second population has mean \(\frac { 1 } { 2 } \mu\) and variance \(b \sigma ^ { 2 }\), where \(b\) is a positive constant, and \(\bar { X } _ { 2 }\) denotes the sample mean. Two unbiased estimators for \(\mu\) are defined by $$T _ { 1 } = 3 \bar { X } _ { 1 } - a \bar { X } _ { 2 } \quad \text { and } \quad T _ { 2 } = \frac { 1 } { 5 } \left( 4 \bar { X } _ { 1 } + 2 \bar { X } _ { 2 } \right) .$$
   1. Determine the value of \(a\).
   2. Show that \(\operatorname { Var } \left( T _ { 1 } \right) = \frac { \sigma ^ { 2 } } { n } ( 9 + 16 b )\) and find a similar expression for \(\operatorname { Var } \left( T _ { 2 } \right)\).
   3. The estimator with the smaller variance is preferred. State which of \(T _ { 1 }\) and \(T _ { 2 }\) is the preferred estimator of \(\mu\).

Faults occur in a roll of material at a rate of \(\lambda\) per m\(^2\). To estimate \(\lambda\), three pieces of material of sizes 3 m\(^2\), 7 m\(^2\) and 10 m\(^2\) are selected and the number of faults \(X_1\), \(X_2\) and \(X_3\) respectively are recorded. The estimator \(\hat{\lambda}\), where $$\hat{\lambda} = k(X_1 + X_2 + X_3)$$ is an unbiased estimator of \(\lambda\).

1. Write down the distributions of \(X_1\), \(X_2\) and \(X_3\) and find the value of \(k\). [4]
2. Find Var(\(\hat{\lambda}\)). [3]

A random sample of \(n\) pieces of this material, each of size 4 m\(^2\), was taken. The number of faults on each piece, \(Y\), was recorded.

1. Show that \(\frac{1}{4}\bar{Y}\) is an unbiased estimator of \(\lambda\). [2]
2. Find Var(\(\frac{1}{4}\bar{Y}\)). [3]
3. Find the minimum value of \(n\) for which \(\frac{1}{4}\bar{Y}\) becomes a better estimator of \(\lambda\) than \(\hat{\lambda}\). [2]

A technician is trying to estimate the area \(\mu^2\) of a metal square. The independent random variables \(X_1\) and \(X_2\) are each distributed \(\text{N}(\mu, \sigma^2)\) and represent two measurements of the sides of the square. Two estimators of the area, \(A_1\) and \(A_2\), are proposed where $$A_1 = X_1X_2 \text{ and } A_2 = \left(\frac{X_1 + X_2}{2}\right)^2.$$ [You may assume that if \(X_1\) and \(X_2\) are independent random variables then $$\text{E}(X_1X_2) = \text{E}(X_1)\text{E}(X_2)$$]

1. Find \(\text{E}(A_1)\) and show that \(\text{E}(A_2) = \mu^2 + \frac{\sigma^2}{2}\). [4]
2. Find the bias of each of these estimators. [2]

The technician is told that \(\text{Var}(A_1) = \sigma^4 + 2\mu^2\sigma^2\) and \(\text{Var}(A_2) = \frac{1}{2}\sigma^4 + 2\mu^2\sigma^2\). The technician decided to use \(A_1\) as the estimator for \(\mu^2\).

1. Suggest a possible reason for this decision. [1]

A statistician suggests taking a random sample of \(n\) measurements of sides of the square and finding the mean \(\overline{X}\). He knows that \(\text{E}(\overline{X}^2) = \mu^2 + \frac{\sigma^2}{n}\) and $$\text{Var}(\overline{X}^2) = \frac{2\sigma^4}{n^2} + \frac{4\sigma^2\mu^2}{n}.$$

1. Explain whether or not \(\overline{X}^2\) is a consistent estimator of \(\mu^2\). [3]

When a tree seed is planted the probability of it germinating is \(p\). A random sample of size \(n\) is taken and the number of tree seeds, \(X\), which germinate is recorded.

1. 1. Show that \(\hat{p}_1 = \frac{X}{n}\) is an unbiased estimator of \(p\).
   2. Find the variance of \(\hat{p}_1\). [4]
   A second sample of size \(m\) is taken and the number of tree seeds, \(Y\), which germinate is recorded. Given that \(\hat{p}_2 = \frac{Y}{m}\) and that \(\hat{p}_3 = a(3\hat{p}_1 + 2\hat{p}_2)\) is an unbiased estimator of \(p\),
2. show that
   1. \(a = \frac{1}{5}\),
   2. \(\text{Var}(\hat{p}_3) = \frac{p(1-p)}{25}\left(\frac{9}{n} + \frac{4}{m}\right)\). [6]
3. Find the range of values of \(\frac{n}{m}\) for which $$\text{Var}(\hat{p}_3) < \text{Var}(\hat{p}_1) \text{ and } \text{Var}(\hat{p}_3) < \text{Var}(\hat{p}_2)$$ [3]
4. Given that \(n = 20\) and \(m = 60\), explain which of \(\hat{p}_1\), \(\hat{p}_2\) or \(\hat{p}_3\) is the best estimator. [3]