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Inference on full or partial parameters based on the standardized signed log likelihood ratio. (English) Zbl 0605.62020
Suppose that a random sample of size n is drawn from a statistical model ${\cal M}$ of probability density p(x;$\omega)$ where the parameter $\omega$ is assumed to be uniquely expressed by $\chi \in R\sp d$ and $\Psi \in R\sp f$. Let ${\hat \omega}$ be the maximum likelihood estimator of $\omega$ under ${\cal M}$ and let 1($\omega)$ be the log likelihood. Further let ${\cal M}\sb{\Psi}$ be a submodel obtained by fixing $\Psi$ in ${\cal M}$. Suppose that (${\hat \omega}$,a${}\sb 0)$ is minimal sufficient under ${\cal M}$ such that $a\sb 0$ is approximately ancillary and the conditional density of ${\hat \omega}$ for given $a\sb 0$ satisfies $$ p({\hat \omega};\omega \vert a\sb 0)=[const.\vert j({\hat \omega})\vert\sp{1/2}\exp \{l(\omega)-l({\hat \omega})\}][1+O(n\sp{- 3/2})] $$ where $j(\omega)=\partial\sp 2l(\omega)/\partial \omega \partial \omega\sp t$. Then it is shown in this paper that there exists a statistic $r\sp*\sb{\Psi}$ of dimension f under ${\cal M}\sb{\Psi}$ which follows f-dimensional normal distribution N(0,I) to relative order $O(n\sp{-3/2})$. This is used to get confidence regions for the partial parameter $\Psi$, independently of the nuisance parameter $\chi$. When $f=1$, $r\sp*\sb{\Psi}$ is obtained by adjusting the mean and variance of the signed log likelihood ratio statistic given by $$ \{sgn({\hat \Psi}- \Psi)\}[2\{l({\hat \omega})-l({\hat \chi}\sb{\Psi},\Psi)\}]\sp{1/2}, $$ where ${\hat \chi}\sb{\Psi}$ stands for the maximum likelihood estimator of $\chi$ under the model ${\cal M}\sb{\Psi}$.
Reviewer: N.Sugiura

62F05Asymptotic properties of parametric tests
62F25Parametric tolerance and confidence regions
62F12Asymptotic properties of parametric estimators
62E20Asymptotic distribution theory in statistics