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Chapter 9, Problem 60E

(choose chapter or problem)

Let \(Y_{1}, Y_{2, \cdots,} Y_{n}\) denote a random sample from the probability density function

                     

                                         \(f(y \mid \theta)=\left\{\begin{array}{ll} \theta y^{\theta-1}, & 0<y<1, \theta>0 \\ 0, & \text {elsewhere } \end{array}\right.\)

a Show that this density function is in the (one-parameter) exponential family and that \(\sum_{i=1}^{n}-\ln \left(Y_{i}\right)\) is sufficient for \(\theta\). (See Exercise 9.45.)

b If \(W_{i}=-\ln \left(Y_{i}\right)\) show that \(W_{i}\) has an exponential distribution with mean \(1 / \theta\).

c Use methods similar to those in Example 9.10 to show that \(2 \theta \sum_{i=1}^{n} W_{i}\) has a \(x^{2}\) distribution with \(2n\) df.

d Show that

                                        \(E\left(\frac{1}{2 \theta \sum_{i-1}^{n} W_{i}}\right)=\frac{1}{2(n-1)}\)

[Hint: Recall Exercise 4.112.]

e What is the MVUE for \(\theta\)?

Equation Transcription:

{

Text Transcription:

Y_1, Y_2, cdots, Y_n

f(y mid theta) = {begin l theta y^theta-1, 0<y<1, theta>0  0,elsewhere }

sum_i = 1^n-lnY_i

theta

W_i=-ln (Y_i

W_i

1 / theta

2 theta sum_i=1^n W_i

x^2

2n

E(1/2 theta sum_i-1^n W_i=(1/2(n-1)

theta