Chapter 9.3, Problem 23E

Refer to Exercise 9.21. Suppose that Y₁, Y₂,…, Y_n is a random sample of size n from a population for which the first four moments are finite. That is, ${m^{\prime }}_1=E(Y_1)<\infty , {m^{\prime }}_2=E(Y_1^2)<\infty ,{m^{\prime }}_3=E(Y_1^3)<\infty$, and ${m^{\prime }}_4=E(Y_1^4)<\infty$. (Note: This assumption is valid for the normal and Poisson distributions in Exercises 9.21 and 9.22, respectively.) Again, assume that n = 2k for some integer k. Consider

${\hat{\sigma }}^2=\frac{1}{2k}{\displaystyle \underset{i=1}{\overset{k}{\sum }}{(Y_{2i}-Y_{2i-1})}^2.}$

1. a Show that ${\hat{\sigma }}^2$ is an unbiased estimator for σ².
2. b Show that ${\hat{\sigma }}^2$ is a consistent estimator for σ².
3. c Why did you need the assumption that ${m^{\prime }}_4=E(Y_1^4)<\infty$?

9.21 Let Y₁, Y₂,…, Y_n be a random sample of size n from a normal population with mean μ and variance σ². Assuming that n = 2k for some integer k, one possible estimator for σ² is given by

${\hat{\sigma }}^2=\frac{1}{2k}{\displaystyle \underset{i=1}{\overset{k}{\sum }}{(Y_{2i}-Y_{2i-1})}^2.}$

1. a Show that ${\hat{\sigma }}^2$ is an unbiased estimator for σ².
2. b Show that ${\hat{\sigma }}^2$ is a consistent estimator for σ².