核密度估计的对称检验在L_1(R~d)下的中偏差（英文）
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摘要
设f_n是基于一个核函数K和取值于R~d的独立同分布随机变量列的一个非参数核密度估计.本文证明了{f_n(x)-f_n(-x),n≥1}在L_1(R~d)空间下的两个中偏差定理.
Let f_n be a non-parametric kernel density estimator based on a kernel function K and a sequence of independent and identically distributed random variables taking values in R~d. In this paper we prove two moderate deviation theorems in L_1(R~d) for {f_n(x)-f_n(x), n≥1}.
Let f_n be a non-parametric kernel density estimator based on a kernel function K and a sequence of independent and identically distributed random variables taking values in R~d. In this paper we prove two moderate deviation theorems in L_1(R~d) for {f_n(x)-f_n(x), n≥1}.
引文
[1]HE X X, GAO F Q. Moderate deviations and large deviations for a test of symmetry based on kernel density estimator[J]. Acta Math Sci Ser B, 2008, 28(3):665-674.
[2]GAO F Q. Moderate deviations and large deviations for kernel density estimators[J]. J Theoret Probab, 2003, 16(2):401-418.
[3]GINE E, GUILLOU A. On consistency of kernel density estimators for randomly censored data:rates holding uniformly over adaptive intervals[J]. Ann Inst H Poincare Probab Statist, 2001, 37(4):503-522.
[4]DIALLO A O K, LOUANI D. Moderate and large deviation principles for the hazard rate function kernel estimator under censoring[J]. Statist Probab Lett, 2013, 83(3):735-743.
[5]LOUANI D. Large deviations limit theorems for the kernel density estimator[J]. Scand J Statist,1998, 25(1):243-253.
[6]DEMBO A, ZEITOUNI O. Lar.ge Deviations Techniques and Applications[M]. 2nd ed. New York:Springer, 1998.
[7]XU M Z, ZHOU Y Z. Moderate deviations and large deviations for a test of symmetry based on kernel density estimator in Rd[J]. Math Pract Theory, 2015, 45(23):209-215.(in Chinese)
[8]XU M Z, ZHOU Y Z. Large deviations for a test of symmetry based on kernel density estimator in R~d[J]. Chinese J Appl Probab Statist, 2015, 31(3):238-246.
[9]GAO F Q. Moderate deviations and law of the iterated logarithm in L_1(R~d)for kernel density estimators[J]. Stochastic Process Appl, 2008, 118(3):452-473.
[10]DEVROYE L. The equivalence of weak strong and complete convergence in L_1 for kernel density estimates[J]. Ann Statist, 1983, 11(3):896-904.
[11]NOLAN D, POLLARD D. U-processes:rates of convergence[J]. Ann Statist, 1987, 15(2):780-799.
[2]GAO F Q. Moderate deviations and large deviations for kernel density estimators[J]. J Theoret Probab, 2003, 16(2):401-418.
[3]GINE E, GUILLOU A. On consistency of kernel density estimators for randomly censored data:rates holding uniformly over adaptive intervals[J]. Ann Inst H Poincare Probab Statist, 2001, 37(4):503-522.
[4]DIALLO A O K, LOUANI D. Moderate and large deviation principles for the hazard rate function kernel estimator under censoring[J]. Statist Probab Lett, 2013, 83(3):735-743.
[5]LOUANI D. Large deviations limit theorems for the kernel density estimator[J]. Scand J Statist,1998, 25(1):243-253.
[6]DEMBO A, ZEITOUNI O. Lar.ge Deviations Techniques and Applications[M]. 2nd ed. New York:Springer, 1998.
[7]XU M Z, ZHOU Y Z. Moderate deviations and large deviations for a test of symmetry based on kernel density estimator in Rd[J]. Math Pract Theory, 2015, 45(23):209-215.(in Chinese)
[8]XU M Z, ZHOU Y Z. Large deviations for a test of symmetry based on kernel density estimator in R~d[J]. Chinese J Appl Probab Statist, 2015, 31(3):238-246.
[9]GAO F Q. Moderate deviations and law of the iterated logarithm in L_1(R~d)for kernel density estimators[J]. Stochastic Process Appl, 2008, 118(3):452-473.
[10]DEVROYE L. The equivalence of weak strong and complete convergence in L_1 for kernel density estimates[J]. Ann Statist, 1983, 11(3):896-904.
[11]NOLAN D, POLLARD D. U-processes:rates of convergence[J]. Ann Statist, 1987, 15(2):780-799.