Use the method described in Exercise 9.26 to show that, if

Chapter 9, Problem 27E

(choose chapter or problem)

Use the method described in Exercise 9.26 to show that, if \(Y_{(1)}=\min \left(Y_{1}, Y_{2}, \ldots, Y_{n}\right)\) when \(Y_{1}, Y_{2}, \ldots, Y_{n}\) are independent uniform random variables on the interval \((0, \theta)\), then \(Y_{(1)}\) is not a consistent estimator for \(\theta\). [Hint: Based on the methods of Section 6.7, \(Y_{(1)}\) has the distribution function

                                                                              \(F_{(1)}(y)=\left\{\begin{array}{ll}0, & y<0 \\1-(y / \theta)^{n}, & 0 \leq y \leq 8, \\1, & y>\theta .]\end{array}\right.\)

Equation Transcription:

 = min

 {

Text Transcription:  

Y_(1) = min (Y_1, Y_2, …., Y_n)

Y_1, Y_2, …., Y_n

(0, theta)

Y_(1)

theta

Y_(1)

F_{(1)}(y) = {0, & y < 0 \\1- (y / theta)^{n}, & 0 leq y leq 8, \\1, & y > theta .]

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