Variance of estimators

8 The independent random variables \(X _ { 1 }\) and \(X _ { 2 }\) have the distributions \(\mathrm { B } \left( n _ { 1 } , \theta \right)\) and \(\mathrm { B } \left( n _ { 2 } , \theta \right)\) respectively. Two possible estimators for \(\theta\) are $$T _ { 1 } = \frac { 1 } { 2 } \left( \frac { X _ { 1 } } { n _ { 1 } } + \frac { X _ { 2 } } { n _ { 2 } } \right) \text { and } T _ { 2 } = \frac { X _ { 1 } + X _ { 2 } } { n _ { 1 } + n _ { 2 } } .$$

1. Show that \(T _ { 1 }\) and \(T _ { 2 }\) are both unbiased estimators, and calculate their variances.
2. Find \(\frac { \operatorname { Var } \left( T _ { 1 } \right) } { \operatorname { Var } \left( T _ { 2 } \right) }\). Given that \(n _ { 1 } \neq n _ { 2 }\), use the inequality \(\left( n _ { 1 } - n _ { 2 } \right) ^ { 2 } > 0\) to find which of \(T _ { 1 }\) and \(T _ { 2 }\) is the more efficient estimator.

The probability density function of the continuous random variable \(X\) is given by $$f(x) = \frac{3x^2}{\alpha^3} \quad \text{for } 0 \leq x \leq \alpha$$ $$f(x) = 0 \quad \text{otherwise.}$$ \(\overline{X}\) is the mean of a random sample of \(n\) observations of \(X\).

1. 1. Show that \(U = \frac{4\overline{X}}{3}\) is an unbiased estimator for \(\alpha\). [5]
   2. If \(\alpha\) is an integer, what is the smallest value of \(n\) that gives a rational value for the standard error of \(U\)? [9]
2. \(\overline{X}_1\) and \(\overline{X}_2\) are the means of independent random samples of \(X\), each of size \(n\). The estimator \(V = 4\overline{X}_1 - \frac{8}{3}\overline{X}_2\) is also an unbiased estimator for \(\alpha\).
   1. Show that \(\frac{\text{Var}(U)}{\text{Var}(V)} = \frac{1}{13}\). [4]
   2. Hence state, with a reason, which of \(U\) or \(V\) is the better estimator. [1]