Answer: Let Y1, Y2, . . . , Yn denote a random sample of

Chapter 8, Problem 14E

(choose chapter or problem)

Let \(Y_{1}, Y_{2}, \ldots, Y_{n}\) denote a random sample of size n from a population whose density is given by

\(f(y)=\left\{\begin{array}{ll} \alpha y^{\alpha-1} / \theta^{\alpha}, & 0 \leq y \leq \theta \\  0, & \text { elsewhere }  \end{array}\right.\)

where \(a>0\)is a known, fixed value, but \(\theta\) is unknown. (This is the power family distribution

introduced in Exercise 6.17.) Consider the estimator \(\widehat{\theta}=\max \left(Y_{1}, Y_{2}, \ldots, Y_{n}\right)\)

Show that \(\widehat{\theta}\) is a biased estimator for \(\theta\)Find a multiple of \(\widehat{\theta}\) that is an unbiased estimator of \(\theta\)Derive \(\operatorname{MSE}(\widehat{\theta})\)

Equation Transcription:

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Text Transcription:

Y1, Y2,...,Yn

\(f(y)= \alpha y^\alpha-1 / \theta^\alpha, & 0 \leq y \leq \theta \  0, & elsewhere

a>0    

\widehat\theta=\max(Y_1, Y_2, \ldots, Y_n\right)

\(\widehat\theta\)

\theta

\(\widehat\theta\)

\theta

MSE(\widehat\theta)