From Algorithms to Z-Scores - matloff - University of California, Davis

266 CHAPTER 12. GENERAL STATISTICAL ESTIMATION AND INFERENCE
i = 0,...,n/2-1, with the n/2 pairs being independent. Find the Method of Moments estimators of
µ and ρ.
12. Suppose we have a random sample X1, ..., Xn from some population in which EX = µ and
V ar(X) = σ 2 . Let X = (X1 + ... + Xn)/n be the sample mean. Suppose the data points Xi are
collected by a machine, and that due to a defect, the machine always records the last number as
0, i.e. Xn = 0. Each of the other Xi is distributed as the population, i.e. each has mean µ and
variance σ 2 . Find the mean squared error of X as an estimator of µ, separating the MSE into
variance and squared bias components as in Section 12.2.
13. Suppose we have a random sample X1, ..., Xn from a population in which X is uniformly
distributed on the region (0, 1) ∪ (2, c) for some unknown c > 2. Find closed-form expressions
for the Method of Moments and Maximum Likelihood Estimators, to be denoted by T1 and T2,
respectively.