Discretized Likelihood Methods

Summary

Suppose that X₁, X₂, . . ., X_n, . . .; is a sequence of i.i.d. random variables with a density f(x, θ). Let c_n be a maximum order of consistency. We consider a solution ${{\rm{\hat \theta }}_{\rm{n}}}$ of the discretized likelihood equation

$$\sum\limits_{{\rm{i}} = 1}^{\rm{n}} {\log {\rm{f}}\left( {{{\rm{x}}_1},{{{\rm{\hat \theta }}}_{\rm{n}}} + \mathop {{\rm{r}}{{\rm{c}}_{\rm{n}}}}\nolimits^{ - 1} } \right) - } \sum\limits_{{\rm{i}} = 1}^{\rm{n}} {\log {\rm{f(}}{{\rm{x}}_1},{{{\rm{\hat \theta }}}_{\rm{n}}}{\rm{)}}} = {{\rm{a}}_{\rm{n}}}({{\rm{\hat \theta }}_{\rm{n}}},{\rm{r}})$$

where a_n(θ, r) is chosen so that ${{\rm{\hat \theta }}_{\rm{n}}}$ is asymptotically median unbiased (AMU). Then the solution ${{\rm{\hat \theta }}_{\rm{n}}}$ is called a discretized likelihood estimator (DLE). In this chapter it is shown in comparison with DLE that a maximum likelihood estimator (MLE) is second order asymptotically efficient but not always third order asymptotically efficient in the regular case. Further, it shall be seen that the asymptotic efficiency (including higher order cases) may be systematically discussed by discretized likelihood methods.