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Asymptotic properties for median cross-validated nearest neighbor median estimate in nonparametric regression

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Science China Mathematics 1997年第6期40卷 585-597页

作者：杨瑛郑忠国 Department of probability and statistics Peking University Beijing 100871 China

Consider the nonparametric regression model Yni = g(xni) + eni, 1≤i≤n, where g is an unknown function to be estimated on [0,1], xni (1≤i≤n) are the fixed design points in the interval [0,1] and {eni,1≤i≤n} is a triangular array of row iid random variables having median zero. The nearest neighbor median estimator gn. h (xni )=m (Yi(1)(n)…Yi(h)(n) is taken as the estimator of the unknown function g(x). Median cross validation (mcv) criterion is employed to select the smoothing parameter h . Let hn * be the smoothing parameter chosen by mcv criterion. Under mild regularity conditions, the upper and lower bounds of hn* , the rate of convergence and the weak consistency of the median cross-validated estimate gn,hn* (xni) are obtained.

关键词： median cross-validation nearest neighbor median estimate weakly uniform consistency rate of convergence smoothing parameter selection.

Asymptotic theory of nonparametric regression estimates with censored data

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Science China Mathematics 2000年第6期43卷 574-580页

作者：施沛德王海燕张利华 Department of probability and statistics Peking University Beijing 100871china

For regression analysis, some useful information may have been lost when the responses are right censored. To estimate nonparametric functions, several estimates based on censored data have been proposed and their consistency and convergence rates have been studied in literature, but the optimal rates of global convergence have not been obtained yet. Because of the possible information loss, one may think that it is impossible for an estimate based on censored data to achieve the optimal rates of global convergence for nonparametric regression, which were established by Stone based on complete data. This paper constructs a regression spline estimate of a general nonparametric regression function based on right-censored response data, and proves, under some regularity conditions, that this estimate achieves the optimal rates of global convergence for nonparametric regression. Since the parameters for the nonparametric regression estimate have to be chosen based on a data driven criterion, we also obtain the asymptotic optimality of AIC, AICC, GCV, CP, and FPE criteria in the process of selecting the parameters.

关键词： nonparametric regression censored data regression spline optimal rates of convergence.

Stable Convergence of Square Integrable Martingale Array

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Acta Mathematica Sinica,English Series 1990年第4期6卷 334-348页

作者：程士宏 Department of probability and statistics Peking University

This paper is to give the necessary and sufficient conditions for a squareintegrable martingale array to converge stably, which improves the results given byJeganathan, Brown & Eagleson and others.

关键词： exp Stable Convergence of Square Integrable Martingale Array

PARAMETER ESTIMATION OF SPATIAL AR MODEL

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Chinese Annals of Mathematics,Series B 1991年第4期12卷 432-444页

作者：蒋继明 Department of probability and statistics Beijing University Beijing China

Consider a stable AR model of two parameter spatial series {Xt, t∈N2}, i. e. {Xtt∈N2} is homogeneous and satisfies the following difference equationXt-sum from n=s∈sXt-s=Wt （t∈N2）where {Wt, t∈N2} is a two parameter white noise and the notation1, k2）: 0≤k1≤p1, 0≤k2≤p2 but （k1, k2）≠（0, 0）}, and furthermore the two-variable polynomial1-sum from n=（s1,s2）∈1,s2） Z1k1Z2s2≠0（|Z1|≤1,|Z2|≤1）.In this paper, under frirly general conditions （it is required that {Wt} Satisfies the conditions of two-parameter martingale difference, which is much weaker than supposing {Wt} to be i. i. d.）, the author obtains strong consistency and asymptotic normality of the Y-W （LS） estimate of the AR parameters {as} whenever n1n2→∞, where n1 and denote the horizontal and vertical sampling width respectively.

关键词： consistency normality asymptotic notation furthermore whenever martingale polynomial proof weakly

Random Weighting Method for T Statistic

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Acta Mathematica Sinica,English Series 1989年第1期5卷 87-94页

作者：郑忠国 Department of probability and statistics Peking University

In this paper we deal with the measure model with independently distributed errors. Forestimating the error distribution of the sample mean, random weighting method is utthed. It is provedthat the accuracy of the appr... 详细信息

关键词： Random Weighting Method for T Statistic

Large Deviation Theorem for Hill's Estimator

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Acta Mathematica Sinica,English Series 1992年第3期8卷 243-254页

作者：程士宏 Department of probability and statistics Peking University

To estimate the exponent of a regularly varying d.f. F, the asymptotic behaviourof Hill’B estimator has been extensively discussed. Under the assumption that the d.f F iscontinuous, we obtain the large deviation theo... 详细信息

关键词： Large Deviation Theorem for Hill’s Estimator

Asymptotic theory for the MLE from randomly censored exponential samples

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Chinese Science Bulletin 1998年第13期43卷 1071-1076页

作者： WANG Qihua Department of probability and statistics, Peking University, Beijing 100871, China Department of probability and statistics Peking University Beijing China

The MLE of the parameter of the exponential population from the censored observations is considered. The Edgeworth expansions for the Studentized MLE are established by representing the relevant statistic as a U\|statistic plus a remainder. A semiparametric bootstrap method is introduced to the random censored model and the accuracy of bootstrap approximation of the MLE is investigated. Furthermore, it is shown that the MLE is asymptotically minimax efficient.

关键词： maximum likelihood estimate Edgeworth expansion bootstrap approximation asymptotic minimax efficiency law of iterated logarithm.

A COMPOUND POISSON MODEL FOR LEARNING DISCRETE BAYESIAN NETWORKS

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Acta Mathematica Scientia 2013年第6期33卷 1767-1784页

作者： Abdelaziz GHRIBI Afif MASMOUDI Laboratory of Physic-Mathematics University of Sfax Laboratory of probability and statistics University of Sfax

We introduce here the concept of Bayesian networks, in compound Poisson model, which provides a graphical modeling framework that encodes the joint probability distribution for a set of random variables within a directed acyclic graph. We suggest an approach proposal which offers a new mixed implicit estimator. We show that the implicit approach applied in compound Poisson model is very attractive for its ability to understand data and does not require any prior information. A comparative study between learned estimates given by implicit and by standard Bayesian approaches is established. Under some conditions and based on minimal squared error calculations, we show that the mixed implicit estimator is better than the standard Bayesian and the maximum likelihood estimators. We illustrate our approach by considering a simulation study in the context of mobile communication networks.

关键词： Bayesian network compound Poisson distribution multinomial distribution implicit approach mobile communication networks

GROUP CONTINGENCY TEST FOR TWO OR SEVERAL INDEPENDENT SAMPLES

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Journal of Systems Science & Complexity 2011年第6期24卷 1183-1192页

作者： Hexin ZHANG Xiangzhong FANG Xiaojing MA Department of probability and statistics School of Mathematical Sciences Peking University Beijing 100871 China

This paper proposes a new and distribution-free test called ＂Group Contingency＂ test （GC, for short） for testing two or several independent samples. Compared with traditional nonparametric tests, GC test tends to explore more information based on samples, and it＇s location-, scale-, and shapesensitive. The authors conduct some simulation studies comparing GC test with Wilcoxon rank sum test （W）, Kolmogorov-Smirnov test （KS） and Wald-Wolfowitz runs test （WW） for two sample case, and with Kruskal-Wallis （KW） for testing several samples. Simulation results reveal that GC test usually outperforms other methods.

关键词： Clustering group contingency test nonparametric test