Applied Statistics and Probability for Engineers

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 1 11.96, x 2 5.03, x 3 67.40, x 4

16.07, x 5 31.50, x 6 7.73, x 7 11.10, and x 8 22.38. Because , the moment

estimate of is

EXAMPLE 7-4 Suppose that X 1 , X 2 ,, Xnis a random sample from a normal distribution with parameters 

and ^2. For the normal distribution E(X) and E(X^2 ) ^2  ^2. EquatingE(X) to and

E(X^2 ) to gives

Solving these equations gives the moment estimators

Notice that the moment estimator of ^2 is not an unbiased estimator.

EXAMPLE 7-5 Suppose that X 1 ,X 2 ,, Xnis a random sample from a gamma distribution with parametersr

and . For the gamma distribution and The moment esti-

mators are found by solving

The resulting estimators are

To illustrate, consider the time to failure data introduced following Example 7-3. For this data,

and , so the moment estimates are

When r1, the gamma reduces to the exponential distribution. Because slightly exceeds

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.

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