ReferenceLet Y1, Y2, . . . , Yn denote a random sample of

Chapter 9, Problem 4E

(choose chapter or problem)

Let \(Y_{1}, Y_{2}, \ldots, Y_{n}\) denote a random sample of size \(n\) from a uniform distribution on the interval \((0, \theta)\). If \(Y_{(1)}=\min \left(Y_{1}, Y_{2}, \ldots, Y_{n}\right)\), the result of Exercise 8.18 is that \(\hat{\theta}_{1}=(n+1) Y_{(1)}\) is an unbiased estimator for \(\theta\). If \(Y_{(n)}=\max \left(Y_{1}, Y_{2}, \ldots, Y_{n}\right)\), the results of Example 9.1 imply that \(\hat{\theta}_{2}=[(n+1) / n] Y_{(n)}\) is another unbiased estimator for \(\theta\). Show that the efficiency of \(\hat{\theta}_{1}\) to \(\hat{\theta}_{2}\) is \(1 / n^{2}\). Notice that this implies that \(\hat{\theta}_{2}\)is a markedly superior estimator.

Equation Transcription:

=  min  

  =

  =  max

   =  

 

 

 

Text Transcription:

Y_{1}, Y_{2}, ..., Y_{n}

n

(0, theta)

Y_{(1)} = min (Y_{1}, Y_{2}, ..., Y_{n})

hat{theta}_{1}=(n+1) Y_{(1)}

theta

Y_{(n)} = max (Y_{1}, Y_{2}, ..., Y_{n})

hat{theta}_{2} = [(n+1) / n] Y_(n)

theta

hat{theta}_{1}

hat{theta}_{2}

1/n^2

hat{theta}_{2}