The Likelihood Function and Maximum-Likelihood Estimation
The log-likelihood function
can be expressed in terms of the mean and the dispersion parameter :
- Inverse Gaussian
- for y = 0, 1, 2, ...
for y=r/m, r=0, 1, 2,..., m
|Some terms in the density function have been dropped in the log-likelihood
function since they do not affect the
estimation of the mean and scale parameters.|
SAS/INSIGHT software uses a ridge stabilized
Newton-Raphson algorithm to maximize the
log-likelihood function l( , ; y)
with respect to the regression parameters.
On the rth iteration, the algorithm
updates the parameter vector b by
where H is the Hessian matrix and u is
the gradient vector, both evaluated at
b(r) = b(r-1) - H-1(r-1) u(r-1)
The Hessian matrix H can be expressed as
where X is the design matrix, Wo is
a diagonal matrix with ith diagonal element
H = - X' Wo X
where gi is the link function,
Vi is the variance function,
and the primes denote derivatives of
g and V with respect to .All values are evaluated at the current
mean estimate .,where wi is the prior weight
for the ith observation.
SAS/INSIGHT software uses either the full Hessian matrix
H = - X' Wo X
or the Fisher's scoring method in the
In the Fisher's scoring method, Wo
is replaced by its expected value We
with ith element wei.
H = X' We X
The estimated variance-covariance matrix
of the parameter estimates is
where H is the Hessian matrix
evaluated at the model parameter estimates.
The estimated correlation matrix of the parameter
estimates is derived by scaling the estimated
variance-covariance matrix to 1 on the diagonal.
|A warning message appears when the specified model fails to converge.
The output tables, graphs, and variables are based on the results from the last iteration.|
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.