Maximum likelihood in small samples: Estimation in the presence of nuisance parameters

The paper discusses various pivotal quantities associated with the application of maximum likelihood to small samples to estimate a parameter θ in the presence of other unknown parameters θ 2 ,…,θ k . This extends the previous work of Sprott (1973, 1975). The criterion of normality of the relative likelihood, applicable to the single parameter case, is here replaced by the normality of the relative likelihood maximized over θ 2 ,…θ k . This gives a more objective criterion for the application of standard maximum likelihood methods to estimate θ 1 than merely the numerical size of the sample. As for the single parameter case, it is necessary to emphasize that the normality of the maximum relative likelihood need not entail, nor be entailed by, a large sample size; it requires expressing the problem, if possible, in terms of a parameter φ=φ(θ 1 ), the maximum relative likelihood of which is approximately normal. Comparisons with exact results are given. But the practicality of such methods arises in more complicated cases when exact solutions are not easily available.