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Estimates of characteristic functions of certain random variables with application to \(\omega ^ 2\)-statistics. I. (English. Russian original) Zbl 0601.60032
Theory Probab. Appl. 29, 488-503 (1985); translation from Teor. Veroyatn. Primen. 29, No. 3, 474-487 (1984).
Let there be defined on the probability space (\(\Omega\),\({\mathcal F},P)\) a sequence of independent random variables \(X_ 1,X_ 2,...\), with values in some measurable space (\({\mathcal X},{\mathcal A})\), having on it the same distribution P. Assume there is a symmetric function \(\Phi\) : \({\mathcal X}\times {\mathcal X}\to R^ 1\) and a symmetric matrix \(\{c_{ij}\}\), \(i,j=1,...,n\). Introduce into consideration a random variable of the form \[ \theta_ n=\sum^{n}_{i=1}\sum^{n}_{j=1}c_{ij}\Phi (X_ i,X_ j). \] The necessity of investigating properties of the distribution of the random variable \(\theta_ n\) is dictated by a number of probabilistic and statistical problems. For example, if \(c_{ij}=n^{-2}\), \(i,j=1,...,n\), then \(\theta_ n\) becomes the von Mises functional; if \(\Phi (x,y)=xy\), where \(x,y\in R^ 1\), then \(\theta_ n\) becomes a quadratic form; if \(c_{ii}=0\), \(c_{ij}=(C^ 2_ n)^{-1}\) when \(i\neq j\), then \(\theta_ n\) is a U-statistic. The question of the limit distribution of the \(\theta_ n\) has been analyzed in many works, but the most general results were derived by H. Rubin and R. A. Vitale [Ann. Stat. 8, 165-170 (1980; Zbl 0422.62016)].
The present paper is devoted to estimates for the characteristic function \(\phi_ n(t)=E \exp \{it\theta_ n\}\), which under certain conditions are nontrivial and are useful in estimating the convergence rate and in obtaining an asymptotic decomposition. Using the derived inequalities, it is shown that the convergence rate for the \(\omega^ 2\)-statistics of Ayne, Anderson-Darling, Beran, Watson, von Mises-Smirnov and Rao have order \(O(n^{-1})\), \(n\to \infty\). If the sample size is random and distributed by the Poisson law with parameter \(\lambda\), then the convergence rate for \(\omega^ 2_{\lambda}\)-statistics has order \(O(\lambda^{-1})\), \(\lambda\to \infty.\)
These results can be extended also to other tests whose statistics can be expressed in the form of von Mises functionals or U-statistics. In this connection, note that the general results of F. Götze [Z. Wahrscheinlichkeitstheor. Verw. Geb. 50, 333-355 (1979; Zbl 0405.60009)] imply for a certain class of \(\omega^ 2\)-statistics, the validity of a convergence rate estimate of order \(O(n^{-1+\epsilon})\), where \(\epsilon >0\).

MSC:
60F17 Functional limit theorems; invariance principles
62G05 Nonparametric estimation
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