that n = 2k for some integer k. Consider ReferenceLet Y1,

Chapter 9, Problem 23E

(choose chapter or problem)

Refer to Exercise 9.21. Suppose that \(Y_{1}, Y_{2}, \ldots, Y_{n}\) is a random sample of size \(n\) from a population for which the first four moments are finite. That is, \(m_{1}^{\prime}=E\left(Y_{1}\right)<\infty, m_{2}^{\prime}=E\left(Y_{1}^{2}\right)<\infty, m_{3}^{\prime}=E\left(Y_{1}^{3}\right)<\infty, \text { and } m_{4}^{\prime}=E\left(Y_{1}^{4}\right)<\infty\) (Note: This assumption is valid for the normal and Poisson distributions in Exercises 9.21 and 9.22, respectively.) Again, assume that \(n=2 k\) for some integer \(k\). Consider

                                                                                                            \(\widehat{\sigma}^{2}=\frac{1}{2 k} \sum_{i=1}^{k}\left(Y_{2 i}-Y_{2 i-1}\right)^{2}\)

a Show that  \(\hat{\sigma}^{2}\) is an unbiased estimator for \(\sigma^{2}\).

b Show that  \(\hat{\sigma}^{2}\) is a consistent estimator for \(\sigma^{2}\)

c Why did you need the assumption that \(m_{4}^{\prime}=E\left(Y_{1}^{4}\right)<\infty\)?

Equation Transcription:

 and

Text Transcription:  

Y_1, Y_2, …., Y_n

n

M’_1 = E(Y_1) < infty, m’_2 = E(Y_{1}^{2}) < infty,m’_3 = E(Y_{1}^{3}) < infty, and  m’_4 = E(Y_{1}^{4}) < infty

n = 2 k

k

hat{sigma}^{2} = frac{1}{2 k} sum_{i=1}^{k} (Y_{2 i}-Y_{2 i-1})^2

hat{sigma}^{2}

sigma^{2}

hat{sigma}^{2}

sigma^{2}

m’_4 = E(Y_{1}^{4}) < infty