# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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

##### MSC:
 62F05 Asymptotic properties of parametric tests 62F25 Parametric tolerance and confidence regions 62F12 Asymptotic properties of parametric estimators 62E20 Asymptotic distribution theory in statistics