Let Y be the largest order statistic of a random sample of

QUESTION:

Let \(Y\) be the largest order statistic of a random sample of size \(n\) from a distribution with pdf \(f(x \mid \theta)=\) \(1 / \theta, 0<x<\theta\). Say \(\theta\) has the prior pdf

\(h(\theta)=\beta \alpha^{\beta} / \theta^{\beta+1}, \quad \alpha<\theta<\infty,\)

where \(\alpha>0, \beta>0\).

(a) If \(w(Y)\) is the Bayes estimator of \(\theta\) and \([\theta-w(Y)]^{2}\) is the loss function, find \(w(Y)\).

(b) If \(n=4, \alpha=1\), and \(\beta=2\), find the Bayesian estimator \(w(Y)\) if the loss function is \(|\theta-w(Y)|\).

Equation Transcription:

,

Text Transcription:

Y  

n  

f(x∣)= 1/,0<x<theta

Theta

h(theta)=beta alpha^beta/theta^ beta +1, alpha  < theta<infinity

Alpha,  > 0,beta>0  

w(Y)  

[-w(Y)]^2  

n=4,alpha=1

beta=2

|theta-w(Y)|