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230 CHAPTER 7 POINT ESTIMATION OF PARAMETERS

The time to failure is exponentially distributed. Eight units are randomly selected and
tested, resulting in the following failure time (in hours): x 11.96, x 5.03, x 67.40, x 4
1
2
3

16.07, x 31.50, x 7.73, x 11.10, and x 22.38. Because x 21.65 , the moment
6
5
7
8
estimate of is 1 x 1 21.65 0.0462.
EXAMPLE 7-4 Suppose that X 1 , X 2 , p , X n is a random sample from a normal distribution with parameters
2
2
2
2
and . For the normal distribution E(X) and E(X ) . Equating E(X) to X and
2 1 n 2
n
E(X ) to g i 1 X i gives
1 n 2
2
2
n a
X, X i
i 1
Solving these equations gives the moment estimators
n 1 n 2 n
2
2
a X i a X i b a 1X i X 2 2
n a
2
ˆ X, ˆ i 1 i 1 i 1
n n
2
Notice that the moment estimator of is not an unbiased estimator.
EXAMPLE 7-5 Suppose that X , X , p , X is a random sample from a gamma distribution with parameters r
2
1
n
2
2
and . For the gamma distribution E1X 2 r and E1X 2 r 1r 12 . The moment esti-
mators are found by solving
1 n 2
2
r X, r1r 12 X i
n a
i 1
The resulting estimators are
X 2 X
ˆ
r ˆ
n n
2
2
11 n2 a X i X i 2 11 n2 a X i X 2
i 1 i 1
To illustrate, consider the time to failure data introduced following Example 7-3. For this data,
8 2
x 21.65 and g i 1 x i 6639.40 , so the moment estimates are
121.652 2 21.65
ˆ
r ˆ 2 1.30, 2 0.0599
11 82 6639.40 121.652 11 82 6639.40 121.652
When r 1, the gamma reduces to the exponential distribution. Because slightly exceeds
r ˆ
unity, it is quite possible that either the gamma or the exponential distribution would provide
a reasonable model for the data.
7-3.2 Method of Maximum Likelihood
One of the best methods of obtaining a point estimator of a parameter is the method of maxi-
mum likelihood. This technique was developed in the 1920s by a famous British statistician,
Sir R. A. Fisher. As the name implies, the estimator will be the value of the parameter that
maximizes the likelihood function.