Page 282 - Applied statistics and probability for engineers

260 Chapter 7/Point Estimation of Parameters and Sampling Distributions

and
n
d ln L( μ) = σ ( ) ∑ x i ( − μ)
−1
2
dμ i=1
Equating this last result to zero and solving for μ yields
n
∑ X i
ˆ μ = = i 1 = X
n

Conclusion: The sample mean is the maximum likelihood estimator of μ. Notice that this is identical to the moment
estimator.

Example 7-12 Exponential Distribution MLE Let X be exponentially distributed with parameter λ. The
likelihood function of a random sample of size n, say, X X 2 ,… , X n , is
1 ,
n
n ∑ x i
L λ ( ) = Π λ e −λ x i = λ n e − λ i=1

i=1
The log likelihood is
ln ( λ) = n ln − λ ∑ x i
n
L
λ
= i 1
Now
d ln L( λ) n n
= − ∑ x i
dλ λ i=1
and upon equating this last result to zero, we obtain

n
ˆ
λ = n / ∑ X i = / X1
= i 1
Conclusion: Thus, the maximum likelihood estimator of λ 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 7-9(a) plots the log of the likelihood function for the exponential parameter from
Example 7-12, using the n = 8 observations on failure time given following Example 7-6.
It is common for the log likelihood function to be negative. We found that the estimate of λ
ˆ
was λ = .0 0462. From Example 7-12, we know that this is a maximum likelihood estimate.
Figure 7-9(a) shows clearly that the log likelihood function is maximized at a value of λ
that is approximately equal to 0.0462. Notice that the log likelihood function is relatively
lat 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. 7-9(b) where we have
plotted the difference in log likelihoods for the maximum value, assuming that the sample
sizes were n = 8 20, , and 40 but that the sample average time to failure remained constant at