Chapter 3 Multi-Parameter likelihoods

The models we have looked at so far in this course, and through most of MATH 235, have been extremely simple as they have each involved just a single parameter. Most statistical models are more complicated than this, often involving many unknown parameters. In this case, the flexibility and power of developing methods based on the likelihood function becomes much more apparent.

The model formulation takes the same form as in the single parameter case: it is assumed that there are observations $x_1,x_2,\mathrm{\dots },x_n$ which are independent realisations of random variables $X_1,X_2,\mathrm{\dots },X_n$.

We consider two cases, one when they are identically distributed each with probability (density or mass) function $f(x_i|\overrightarrow{\theta })$ and second when they are non-identically distributed with $X_i$ having probability function $f_i(x_i|\overrightarrow{\theta })$. The difference now is that the parameter $\overrightarrow{\theta }$ is a vector of parameters.