Specifically, we define the likelihood ratio test statistic as -2 log (the null likelihood function
divided by the full likelihood function
Table 2 Stage 1 and 2 transitions probabilities and tests of hypotheses Log likelihood function
for Stage 1 is [P.
In practice, amending the likelihood function
(4) or the trivariate probit likelihood function
to incorporate fixed or random effects is difficult.
The renormalized likelihood function
is obtained by dividing the above maximum value into the likelihood function
This unobserved factor is integrated out of the likelihood function
by assuming a semiparametric form.
Equation [A1] below shows the log likelihood function
for the heterogeneous-preferences model, assuming that the error term is normally distributed.
The complete likelihood function
is the product of the likelihoods for the fully contested case in equation 7 as well as the partially contested cases derived in the Appendix and given in equations 12, 15, 16, 17, 18, and 19:
That is, the maximum likelihood approach to the mixture problem consists of maximizing the log likelihood function
, l([Theta]) of the sample [x.
To obtain the likelihood function
we condition on the number of coalescence events occurring since the time of divergence in the ancestry of the samples.
With the gamma heterogeneity, the unconditional likelihood function
of household i can be simplified to be
To maximize the likelihood function
, a set of equations obtained by taking the partial derivative of L with respect to [Xi] and p respectively are set to zero.
10 Notice that in the special case where these constants are all set to zero, the functional form suggested by Hamilton (1991) collapses to the traditional maximum likelihood function