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Empirical likelihood ratio confidence regions. (English) Zbl 0712.62040
X$${}_ 1,X_ 2,..$$. are independent p-dimensional random variables with common cumulative distribution function $$F_ 0$$. $$F_ n$$ denotes the empirical cumulative distribution function. L(F) denotes the likelihood function that $$F_ n$$ maximizes, and R(F) denotes the empirical likelihood ratio function $$L(F)/L(F_ n)$$. If T is a statistical functional, R(F) is used to construct nonparametric confidence regions and tests for $$T(F_ 0)$$. Most of the discussion is for the mean. The notation $$F\ll F_ n$$ means that F is supported in the sample. The following theorem is proved.
Let $$X,X_ 1,X_ 2,..$$. be i.i.d. p-dimensional random variables, with $$u_ 0$$ denoting the mean vector of X. Suppose the rank of the covariance matrix of X is a positive value q. Let Y be a scalar random variable with a chi-square distribution with q degrees of freedom. For r in (0,1) let C(r,n) denote $$\{\int X dF |$$ $$F\ll F_ n$$, R(F)$$\geq r\}$$. Then C(r,n) is a convex set and $$\lim_{n\to \infty}P(C(r,n)$$ contains $$u_ 0)=P(Y\leq -2 \log r)$$. If $$E(\| X\|^ 4)$$ is finite, $| P(C(r,n)\quad contains\quad u_ 0)-P(Y\leq -2 \log r)| =O(n^{-1/2}).$ Analogous results are given for other functionals, and algorithms are described.
Reviewer: L.Weiss

##### MSC:
 62G20 Asymptotic properties of nonparametric inference 62H99 Multivariate analysis 62E20 Asymptotic distribution theory in statistics 62G30 Order statistics; empirical distribution functions 62G15 Nonparametric tolerance and confidence regions 62H12 Estimation in multivariate analysis 62H15 Hypothesis testing in multivariate analysis
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