Page 259 - Probability, Random Variables and Random Processes

CHAP. 71 ESTIMATION THEORY

where R is the sample mean, is a biased estimator of a2.
By definition, we have

Now

By Eqs. (4.1 12) and (7.27), we have

Thus

which shows that S2 is a biased estimator of c2.

7.3. Let (XI, . . . , X,) be a random sample of a Poisson r.v. X with unknown parameter A.
(a) Show that

Al = Xi and A, = $(x, + X2)
n i=l
are both unbiased estimators of A.
(b) Which estimator is more efficient?
(a) By Eqs. (2.42) and (4.108), we have

Thus, both estimators are unbiased estimators of I.
(b) By Eqs. (2.43) and (4.1 12),
1 1 1 1
Var(Al) = - Z Var(Xi) = - C Var(Xi) = - (ni) = -
n2 i=l n2 i=l n2 n

Thus, if n > 2, A, is a more efficient estimator of A than A,, since 1/n < 112.

7.4. Let (XI, . . . , X,) be a random sample of X with mean p and variance a2. A linear estimator of p
is defined to be a linear function of X,, . . ., X,, l(X,, .. ., X,). Show that the linear estimator
defined by [Eq. (7.27)],