Applied Statistics and Probability for Engineers

7-3 METHODS OF POINT ESTIMATION 233

Now

and upon equating this last result to zero we obtain

Thus the maximum likelihood estimator of is the reciprocal of the sample mean. Notice that

this is the same as the moment estimator.

It is easy to illustrate graphically just how the method of maximum likelihood works.

Figure 7-3(a) plots the log of the likelihood function for the exponential parameter from

Example 7-8, using the n8 observations on failure time given following Example 7-3. We

found that the estimate of was. From Example 7-8, we know that this is a

maximum likelihood estimate. Figure 7-3(a) shows clearly that the log likelihood function is

maximized at a value of that is approximately equal to 0.0462. Notice that the log likelihood

function is relatively flat in the region of the maximum. This implies that the parameter is not

estimated very precisely. If the parameter were estimated precisely, the log likelihood function

would be very peaked at the maximum value. The sample size here is relatively small, and this

has led to the imprecision in estimation. This is illustrated in Fig. 7-3(b) where we have plot-

ted the difference in log likelihoods for the maximum value, assuming that the sample sizes

were n8, 20, and 40 but that the sample average time to failure remained constant at

. Notice how much steeper the log likelihood is for n20 in comparsion to n8,
and for n40 in comparison to both smaller sample sizes.
The method of maximum likelihood can be used in situations where there are several un-
known parameters, say,  1 ,  2 ,, kto estimate. In such cases, the likelihood function is a func-
tion of the kunknown parameters  1 ,  2 ,, k, and the maximum likelihood estimators
would be found by equating the kpartial derivatives to
zero and solving the resulting system of equations.

L 1  1 ,  2 ,p, k (^2) i, i1, 2,p, k
p 5 ˆi 6
p
x21.65
ˆ0.0462
ˆna
n
i 1
Xi (^1)  X
d ln L 1  2
d

n

 a
n
i 1
xi
(a)
–32.69
–32.67
–32.65
–32.63
–32.61
–32.59
.040 .042 .044 .046 .048 .050 .052
Log likelihood
λ
(b)
λ

• 0.4

0.038 0.040 0.042 0.044 0.046 0.048 0.050 0.052 0.054

• 0.3


• 0.2


• 0.1

0.0

Difference in log likelihood

n = 8

n = 20

n = 40

Figure 7-3 Log likelihood for the exponential distribution, using the failure time data. (a) Log likelihood with n8 (original

data). (b) Log likelihood if n8, 20, and 40.

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