is the MLE estimate of any parameter,
, then the (
)% two-sided confidence bounds on the
parameter are:In general, the MLE estimates of the parameters are asymptotically normal.
This means that for large sample sizes the distribution of the estimates
from the same population would be very close to the normal distribution
[12].
If is the MLE estimate of any parameter,
, then the ()% two-sided confidence bounds on the
parameter are:
(18)
where 
represents the variance of 
and 
is the critical value corresponding
to a significance level of 
on the standard normal distribution.
[Note]
The variance of the parameter, 
, is obtained using the Fisher
information matrix. For 
parameters, the Fisher information
matrix is obtained from the log-likelihood function 
as follows:
(19)
The variance-covariance matrix is obtained by inverting the Fisher matrix
:
Once the variance-covariance matrix is known the variance of any parameter
can be obtained from the diagonal elements of the matrix. Note that if
a parameter, 
, can take only positive values, it
is assumed that the 
follows the normal distribution [12].
The bounds on the parameter in this case are:
Using 
we get 
. Substituting this value we have:
(20)
Knowing 
from the variance-covariance matrix,
the confidence bounds on 
can then be determined.
Example 11.2
Continuing with Example 11.1,
the confidence bounds on the MLE estimates of the parameters 
, 
, 
and 
can now be obtained. The Fisher information
matrix for the example is:
The variance-covariance matrix can be obtained by taking the inverse
of the Fisher matrix 
:
Inverting 
returns the following matrix:
Therefore, the variance of the parameter estimates are:
Knowing the variance, the confidence bounds on the parameters can be
calculated. For example, the 90% bounds (
) on 
can be calculated as shown next:
The 90% bounds on 
are (considering that 
can only take positive values):
The standard error for the parameters can be obtained by taking the
positive square root of the variance. For example, the standard error
for 
is:
The 
statistic for 
is:
The 
value corresponding to this statistic
based on the standard normal distribution is:
The previous calculation results are displayed as MLE Information in
the results obtained from DOE++
as shown in Figure 11.4. In the figure, the Effect
corresponding to each factor is simply twice the MLE estimate of the coefficient
for that factor. Generally, the 
value corresponding to any coefficient
in the MLE Information table should match the value obtained from the
likelihood ratio test (displayed in the Likelihood Ratio Test table of
Figure 11.3). If the sample size is not large enough,
as in the case of the present example, a difference may be seen in the
two values. In such cases, the 
value from the likelihood ratio test
should be given preference. For the present example, the 
value of 0.8318 for 
, obtained from the likelihood ratio
test, would be preferred to the 
value of 0.8313 displayed under MLE
information. For details see [12].
|
Figure 11.4: MLE information from DOE++ for Example 11.2. |
