Let X1,X2, . . . ,Xn be a random sample from a

Chapter 6, Problem 4E

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QUESTION:

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from a distribution with pdf \(f(x ; \theta)=\theta x^{\theta-1}, 0<x<1\), where \(0<\theta\).

(a) Find a sufficient statistic \(Y\) for \(\theta\).

(b) Show that the maximum likelihood estimator \(\widehat{\theta}\) is a function of \(Y\).

(c) Argue that \(\widehat{\theta}\) is also sufficient for \(\theta\).

Equation Transcription:

 


 

Text Transcription:

X_1,X_2,…,X_n  

f(x;theta)=theta x^theta-1, 0<x<1  

0<theta  

Y

Theta

Widehat theta

Questions & Answers

QUESTION:

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from a distribution with pdf \(f(x ; \theta)=\theta x^{\theta-1}, 0<x<1\), where \(0<\theta\).

(a) Find a sufficient statistic \(Y\) for \(\theta\).

(b) Show that the maximum likelihood estimator \(\widehat{\theta}\) is a function of \(Y\).

(c) Argue that \(\widehat{\theta}\) is also sufficient for \(\theta\).

Equation Transcription:

 


 

Text Transcription:

X_1,X_2,…,X_n  

f(x;theta)=theta x^theta-1, 0<x<1  

0<theta  

Y

Theta

Widehat theta

ANSWER:

Step 1 of 4

Given:

An i.i.d. sample of random variables   have the following density function:

                                               

 

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