The definitions of likelihood given in Chapter 2 are unchanged in the multi-parameter case; we simply need to replace by . In general, for observed data , having probability (density or mass) function , the likelihood is defined simply as
where is the observed value of . In particular, for independent data such that is the realisation of having probability (density or mass) function , we define the likelihood
Specializing further to the case where each of the has the same distribution , the likelihood becomes
As in the one-parameter case, it is often more convenient to work with the log-likelihood, which, in the latter case, becomes
The maximum likelihood estimator (MLE), of , is the value that maximises (or ). Notice, though, that this now requires maximization in -dimensional space, where is the length of the parameter vector.
Calculating the MLE
For multiparameter models, we still maximise the log-likelihood with respect to the parameters, , to find the MLE, . Assuming the log-likelihood function is differentiable at the MLE then the maximum will be a turning point, and we find it by solving the set of simultaneous equations
Note: For some models (particularly those where the support of the density function depends on one or more of the parameters) the log-likelihood function is not differentiable at the MLE (recall the Uniform example from MATH235). In these cases plotting the log-likelihood surface can help. Naturally, R is useful here.