Page 258 - Probability, Random Variables and Random Processes

250 ESTIMATION THEORY [CHAP 7

and the minimum m.s. error em is (Prob. 7.22)

where a,, = Cov(X, Y) and p,, is the correlation coefficient of X and Y. Note that Eq. (7.21) states
that the optimum linear m.s. estimator p= ax + 6 of Y is such that the estimation error Y - P = Y -
(ax + b) is orthogonal to the observation X. This is known as the orthogonality principle. The line
y = ax + b is often called a regression line.
Next, we consider the estimator ? of Y with a linear combination of the random sample
(XI, . - , Xn) by

Again, we maintain that in order to produce the linear estimator with the minimum m.s. error, the
coefficients ai must be such that the following orthogonality conditions are satisfied (Prob. 7.35):

Solving Eq. (7.25) for ai, we obtain

a = ['I] .=[:I
where

an "=E(YXj) R= Rij = E(XiXj)
and R- ' is the inverse of R.

Solved Problems

PROPERTIES OF POINT ESTIMATORS
7.1. Let (XI, . . . , Xn) be a random sample of X having unknown mean p. Show that the estimator of
p defined by

is an unbiased estimator of p. Note that X is known as the sample mean (Prob. 4.64).
By Eq. (4.1 O8),

Thus, M is an unbiased estimator of p.

7.2. Let (XI, . . . , X,) be a random sample of X having unknown mean p and variance a2. Show that
the estimator of a2 defined by