# 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)
Empirical likelihood ratio confidence intervals for a single functional. (English) Zbl 0641.62032
Let $(X\sb 1,...,X\sb n)$ be a random sample, its components $X\sb i$ are observations from a distribution-function $F\sb 0$. The empirical distribution function $F\sb n$ is a nonparametric maximum likelihood estimate of $F\sb 0$. $F\sb n$ maximizes $$L(F)=\prod\sp{n}\sb{i=1}\{F(X\sb i)-F(X\sb i-)\}$$ over all distribution functions F. Let $R(F)=L(F)/L(F\sb n)$ be the empirical likelihood ratio function and T(.) any functional. It is shown that sets of the form $$\{T(F)\vert R(F)\ge c\}$$ may be used as confidence regions for some $T(F\sb 0)$ like the sample mean or a class of M-estimators (especially the quantiles of $F\sb 0)$. These confidence intervals are compared in a simulation study to some bootstrap confidence intervals and to confidence intervals based on a t-statistic for a confidence coefficient $1-\alpha =0.9$. It seems that two of the bootstrap intervals may be recommended but the simulation is based on 1000 runs only.
Reviewer: D. Rasch

##### MSC:
 62G15 Nonparametric tolerance and confidence regions 62G30 Order statistics; empirical distribution functions 62G05 Nonparametric estimation
Full Text: