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7-2 GENERAL CONCEPTS OF POINT ESTIMATION 225

In a sense, the MVUE is most likely among all unbiased estimators to produce an estimate ˆ

that is close to the true value of . It has been possible to develop methodology to identify the
MVUE in many practical situations. While this methodology is beyond the scope of this book,
we give one very important result concerning the normal distribution.

Theorem 7-1
If X , X , p , X n is a random sample of size n from a normal distribution with mean
2
1
and variance 2 , the sample mean X is the MVUE for .

In situations in which we do not know whether an MVUE exists, we could still use a minimum
variance principle to choose among competing estimators. Suppose, for example, we wish to es-
timate the mean of a population (not necessarily a normal population). We have a random sample

and we wish to compare two possible estimators for : the sam-
of n observations X 1 , X 2 , p , X n
ple mean X and a single observation from the sample, say, . Note that both XX i and X are unbi-
i
ased estimators of ; for the sample mean, we have V1X 2 n 2 from Equation 5-40b and the
variance of any observation is V1X 2 2 . Since V1X 2 V1X 2 for sample sizes n
2, we
i
i
X
would conclude that the sample mean is a better estimator of than a single observation . i
7-2.4 Standard Error: Reporting a Point Estimate
When the numerical value or point estimate of a parameter is reported, it is usually desirable
to give some idea of the precision of estimation. The measure of precision usually employed
is the standard error of the estimator that has been used.

Definition
ˆ
The standard error of an estimator is its standard deviation, given by
ˆ
ˆ 2V1 2 . If the standard error involves unknown parameters that can be esti-

mated, substitution of those values into ˆ produces an estimated standard error,
denoted by ˆ .
ˆ
ˆ
Sometimes the estimated standard error is denoted by s ˆ or se1 2 .
Suppose we are sampling from a normal distribution with mean and variance 2 . Now
the distribution of X is normal with mean and variance n 2 , so the standard error of X is

X
1n
If we did not know but substituted the sample standard deviation S into the above equation,
the estimated standard error of X would be

S
ˆ
X 1n

When the estimator follows a normal distribution, as in the above situation, we can be rea-
sonably confident that the true value of the parameter lies within two standard errors of the