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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
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