Page 260 - Probability, Random Variables and Random Processes

ESTIMATION THEORY [CHAP 7

is the most efficient linear unbiased estimator of p.
Assume that

is a linear unbiased estimator of p with lower variance than M. Since MI is unbiased, we must have

which implies that xy= '=, ai = 1. By Eq. (4.1 1 Z),

Var(M) = 1 02 and Var(M ,) = o2 ai2
n i= 1
By assumption,

Consider the sum

which, by assumption (7.30), is less than 0. This is impossible unless ai = l/n, implying that M is the most
efficient linear unbiased estimator of p.

7.5. Showthatif
lim E(O,) = 8 and limVar(O,) = 0
n-rm n-+ a)
then the estimator On is consistent.
Using Chebyshev's inequality (2.97), we can write

Thus, if
lim E(On) = 8 and lim Var(G3,) = 0
n- m n-rm
then limP(I0, - 81 ~E)=O
n-r m
that is, On is consistent [see Eq. (7.6)].

7.6. Let (XI, . . . , X,) be a random sample of a uniform r.v. X over (0, a), where a is unknown. Show
that

is a consistent estimator of the parameter a.