Chapter 9, Problem 108SE

The MLE obtained in Exercise 9.107 is a function of the minimal sufficient statistic for θ, but it is not unbiased. Use the Rao–Blackwell theorem to find the MVUE of e^−t/θ by the following steps.

1. a Let

   $V=\{\begin{array}{ll}1,\hfill & Y_1>t,\hfill \\ 0,\hfill & \text{elsewhere}\text{.}\hfill \end{array}$

   Show that V is an unbiased estimator of e^−t/θ.

2. b Because $U={\displaystyle {\sum }_{i=1}^n{Y}_i}$ is the minimal sufficient statistic for θ, show that the conditional density function for Y₁, given U = u, is

   $f_{Y_1}{}_{|U}(y_1|u)=\{\begin{array}{ll}\left(\frac{n-1}{u^{n-1}}\right)\left(u-y_1\right){,}^{n-2}\hfill & 0<y_1>u,\hfill \\ 0,\hfill & \text{elsewhere}\text{.}\hfill \end{array}$

3. c Show that

   $E\left(V/U\right)=P\left(Y_1>t|U\right)={\left(1-\frac{t}{U}\right)}^{n-1}$

9.107. Suppose that a random sample of length-of-life measurements, Y₁, Y₂, . . . , Y_n, is to be taken of components whose length of life has an exponential distribution with mean θ. It is frequently of interest to estimate

F(t) = 1 − F(t) = e^−t/θ ,

the reliability at time t of such a component. For any fixed value of t, find the MLE of $\bar{F}\left(t\right).$