Applied Statistics and Probability for Engineers

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

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.



ˆ

If is a random sample of size nfrom a normal distribution with mean

and variance ^2 , the sample mean Xis the MVUE for .

X 1 , X 2 ,p, Xn

Theorem 7-1

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 normalpopulation). We have a random sample

of nobservations and we wish to compare two possible estimators for : the sam-

ple mean and a single observation from the sample, say,. Note that both and Xiare unbi-

ased estimators of ; for the sample mean, we have from Equation 5-40b and the

variance of any observation is. Since for sample sizes we

would conclude that the sample mean is a better estimator of than a single observation.

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.

Xi

V 1 Xi 2 ^2 V 1 X 2 V 1 Xi 2 n 2,

 V 1 X 2 ^2 n

X Xi X

X 1 , X 2 ,p, Xn 

The standard errorof an estimator is its standard deviation, given by

. If the standard error involves unknown parameters that can be esti-
mated, substitution of those values into produces an estimated standard error,
denoted by ˆˆ.

ˆ

ˆ 2 V 1 ˆ 2

ˆ

Definition

Sometimes the estimated standard error is denoted by or.

Suppose we are sampling from a normal distribution with mean and variance. Now

the distribution of is normal with mean and variance , so the standard error of is

If we did not know but substituted the sample standard deviation Sinto the above equation,

the estimated standard error of would be

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

ˆX

S

1 n

X

X



1 n

X  ^2 n X

 ^2

sˆ se 1 ˆ 2

7-2 GENERAL CONCEPTS OF POINT ESTIMATION 225

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