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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

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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