Solution Found!
Let X1,X2, . . . ,Xn be a random sample from a
Chapter 6, Problem 4E(choose chapter or problem)
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: