7 A continuous random variable \(Y\) has cumulative distribution function
$$\mathrm { F } ( y ) = \left\{ \begin{array} { c c }
0 & y < a
1 - \frac { a ^ { 5 } } { y ^ { 5 } } & y \geqslant a
\end{array} \right.$$
where \(a\) is a parameter.
Two independent observations of \(Y\) are denoted by \(Y _ { 1 }\) and \(Y _ { 2 }\). The smaller of them is denoted by S .
- Show that \(P ( S > \mathrm { s } ) = \frac { a ^ { 10 } } { s ^ { 10 } }\) and hence find the probability density function of \(S\).
- Show that \(S\) is not an unbiased estimator of \(a\), and construct an unbiased estimator of \(a , T _ { 1 }\) based on \(S\).
- Construct another unbiased estimator of \(a , T _ { 2 }\), of the form \(k \left( Y _ { 1 } + Y _ { 2 } \right)\), where \(k\) is a constant to be found.
- Without further calculation, explain how you would decide which of \(T _ { 1 }\) and \(T _ { 2 }\) is the more efficient estimator.