Thus, we see that all these acceptable estimators are asymptotically unbiased in the sense that their limit distribution as n [right arrow] [infinity] is the uniform distribution-the same as for the maximum likelihood estimators
, which are the unique unbiased estimators for the parameter p.
Therefore the maximum likelihood estimators
for [mathematical expression not reproducible] are the solutions to equations:
Setting the partial derivatives ([partial derivative]/[partial derivative][alpha])l([phi], [mu], [alpha]; t), ([partial derivative]/ [partial derivative][mu])l([phi], [mu], [alpha]; t), and ([partial derivative]/[partial derivative][phi])l([phi], [mu], [alpha]; t) equal to 0,we obtain the following maximum likelihood estimators
The effectiveness of maximum likelihood estimator
is also verified through the simulations.
Given a random sample of size n, the maximum likelihood estimators
of these parameters can be explicitly computed as:
Performance of maximum likelihood estimator
for [lambda] developed in previous section is measured using MATLAB.
The "Penalized Maximum Likelihood Estimator
and Risk Measure Estimators" section then presents the details about the fitting procedure.
In the follwing we report the definitions of the probability density function (PDF), the distribution function (DF), the survival function (S) and the maximum likelihood estimator
(MLE) for the two distributions analyzed here.
where [??] is the maximum likelihood estimator
. This is an approximate (1 - [alpha])-prediction interval only for large sample sizes, because it does not include any information about the uncertainty of the estimation.
The use of the Maximum Likelihood Estimator
enables us to estimate the population parameters that are shown in Table 4.
The proposed estimator is based on moment generating function and is compared with maximum likelihood estimator
using the mean squared error as a performance criterion.
Maximum likelihood estimator
(MLE) is very popular both in the literature and in practice.