Chapter 6, Problem 38SE

Each of n specimens is to be weighed twice on the same scale. Let X_i and Y_i denote the two observed weights for the ith specimen. Suppose X_i and Y_i are independent of one another, each normally distributed with mean value μ_i (the true weight of specimen i) and variance σ².

1. a. Show that the maximum likelihood estimator of σ² is $\widehat{{\sigma }^2}\text{ = }{\displaystyle \sum {(X_i\text{ - }{Y}_i)}^2/(4n)}$. [Hint: If $\bar{Z}$ = (z₁ + z₂)/2, then ${\displaystyle \sum {(z_i\text{ - }\bar{z})}^2\text{ = (}{z}_1\text{ - }{z}_2{\text{)}}^2/2}$.]
2. b. Is the mle a² an unbiased estimator of σ²? Find an unbiased estimator of σ². [Hint: For any rv Z, E(Z²) = V(Z) + [E(Z)]². Apply this to Z = X_i – Y_i.]