NA随机变量递归密度核估计的渐近性质
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摘要
设{x_n，n≥1}为同分布样本序列，f(x)为X_1的概率密度函数，基于样本X_1，…，X_n，1969年Wolverton and Wagner([1])提出f(x)的递归型核估计
f_n(x)=1/n sum from j=1 to n (1/(h_j) K((x-X_j)/h_j))，由于经过简单的化简我们有
f_n(x)=(n-1)/nf_(n-1)(x)+1/(nh_n))K((x-X_n)/h_n)。从这个特点我们知道用递归核估计去估计未知密度时，在添加样本点的情况下，不必重新计算所有项，只需计算添加项，再经简单计算即可。这比用普通型的密度核估计
(?)_n(x)=1/(nh_n) sum from j=1 to n (K((x-X_j)/h_n))去估计未知密度，从计算方面而言具有很大方便，因此很多学者对之进行了研究。
在独立样本情形，Wolverton and Wagner([1]，1969),Wegman and Davis([2]，1979)等对某些形式的递归密度核估计作了深入研究，其结果比较完美；在相依样本情形，Masry([3]，[4]1986，1987)分别对ρ-混合和渐近不相关情形讨论f_n(x)的均方收敛和渐近正态性；蔡宗武教授([5]，[6]1990，1992)分别讨论在(α，β)混合下的强收敛速度和ρ-混合下的渐近正态性，杨善朝教授([7]1997)在ρ-混合下，对递归密度核估计的强收敛速度作了进一步的讨论，改进了Masry([3])的结论。
然而，NA(负相协)和PA(正相协)变量在可靠性理论、概率过程、随机过程、多元统计、空间统计等领域中有广泛的应用，而且在大气、地质、海洋、生物、经济等领域也有应用具有广泛应用，故在NA和PA下对密度核估计的渐近性质的研究也甚为重要。因此引起了学者们的重视，如韦来生教授([15]，2001)，杨善朝教授([16]，2000)，Roussas([17]，2000)，等人在NA下对普通型的密度核估计做了相合性和渐
近正态性的研究．而在NA和PA下尚未见有人对递归型密度核估计进行研究．因
此，本文主要在＊A相依样本下，对核函数们叫，未知密度函数人X）和窗宽札满
足一定的条件下，具体地讨论了递归型密度核估计人k）所具有的大样本性质，即
研究其渐近方差，r－阶平均相合性，逐点强相合性和渐近正态性．作为可靠性问
题中的应用，构造了生存函数和失效率函数户k）＝‘－刊d）－八K＞d）和叶d）＝Fd
，lerw T、《XI
P以川“二7．JIx；＞川；7”。“于下下－7，
n Hd““P、lx）
讨论了r。k）的逐点强相合性．
我们的主要假设和结论如下：
1：主要假设
（AI）设厂。，。2 2｝为同分布 NA序列，满足
（i）局与儿付的联合密度八。材存在且满足
suP山x，y，幻一八川州 S啊＜＋co；
Z＆
门Z、LDn 上下叫／r、＿p／L＿上y／巫L＼p—1＿p．／＋＿n。、＿＋＿
卜“J“’二十小h 上J八人／’——一’n 上J＼人J‘”p一且—一，一’“11——·
（iii）K（l）有限可微亚 且 K。LI，jR K（。）d。二 l，suP（1＋IJ）IK（x）＜十co。
（tV） 存在正增常数列｛Cfd C。＜。，使得 h。C。＞0，hJ‘。（C。）＋ 0·
（AZ）（i）K E LI，jBK（。）du＝1，SliP（1＋问）IK（1）l＜＋OO；
（n in＊If＊）＊＊二＊，in＊＊If＊）＊＊＜ ＋co·
2：主要结论
定理2＊ 在（M条件下，对八X）任意连续点。；。有
Inlrn从。＊。叶人（x），人（川）＝＊（x，川．
16八X）厂”（叫叫 当X二队
具甲川兄功＝（
IO，gx ／ y．
定理3＊ 设厂＊n三 2｝为同分布平稳 NA序列，满足…）＊条件，且有
1，一、＾．．、、＿1，。、，－一、n，1、
h、上 0、2＞”hZ＝O（hZ），＞六ICOv（XI，Xj）＝O（六），
‘“”＋”’L－’”’一一’””””L－I12’—一“－“’－“””hZ”’
0＜，＜2，那么
＊） 当＊I乙义＋十co时，有thlr EIjn＊卜／（叫刀＝ 0，
（n） 当hn二n一会日，有＊ （J）一f（x）l厂二O（n一i）．
2
定理 4．1 设（X＊。三 2｝为同分布 NA随机变量序列，满足条件AI（n）和 AZ，进
一步设川叫非降或非升，且存在口＞0，使得太－以丫1，则对八x）的任一连续点
。有
人一）一j（X）．
定理 5．1 设体。，。2 2｝为平稳的 NA随机变量序列，在K）条件下，又设存在
P。。，q。。
Let Xn,n 1 be a sample sequence with identical distribution ,having unknown probability density function fn(x). based on the sample Xt, ,Xn,the recursive density estimator for f(x) that
was introduced by Wolverton and Wagner in 1969,it can be calculated recursively,i.e.
From the practical point of view,the recursive density estimator has a clear advantage over non-recursive estimators in that it can be updated with each additional observation,in contrast to the nonrecursive case of
where the estimator must be entirely recomputed. Since it is more convenient to use the recursive density estimator fn(x),there has been a significant number of research papers on this subject matter. The most relevant papers are these by Wolverton and Wagner(1969),and Weg-man and Davis (1979) for independent samples those are very ideal;Masry(1986,1987) discussed both quadratic-mean consistency and asymptotic normality for dependent sample and asymptotically uncorrelated;Professor Cai Zongwu(1990,1992) discussed both rate of strong convergence and asymptotic normality for (a, 3)-dependent and p-dependent;Professor Yang Shanchao (1997) further discussed rate of strong convergence for -dependent,mended the result of Masry's. In a snapshot,one may say that negative association and positive association occur often in cetain reliability theory problems,filter theory problems,and multianalysis theory problems.Many more applications are to be anticipated in a host of other areas and ,in particular,those areas whe
re spatial statistics play an important role,such area are ,for example,analysis of agricultural field experiments,geostatistical annalysis,image annalysis,oceanographic applications,signal processing in radar and sonars,and stereology. so it is more important to study asymptotic propertis of density estimation for positive and negative association,arousing our considerable regard. For example,Professor Wei Laiseng (2001),Professor Yang Shanchao (2000) studied the density estimation n(z) on consistency and asymptotic normality for negative association. However,up to
the present,there is no paper discussed asymptotic propertis of the recursive kernel estimate of probability density function under negative association and positive association. In this paper,the author investigates some asymptotic propertis of fn(x) under negative association that includes the consistency in r-order mean,the strong consistency and the asymptotic normality under certain conditions for the kernel function K(x) ,the unknown density function f(x) and the bandwidth hn. As applications in reliability theory problems,the hazard rate and survial function,
will be estimated by
and present the strongly consistency of rn (x) . The following is our main assumption and results 1 The main assumption
(Al) The r.v.s X ,X2, ,are nagetive association sequence with identical distribution,satisfing (i) If f(x,y,k) is the joint p.d.f. of the r.v.s X ,Xi+k,then
(iii) K(x) have bounded derivative,and
(iv) There exists positive,increase constant
2 The main result
Theorem 2.1 Suppose the assumption of (Al) are fulfilled ,if x and y is arbitrary continuity of (x),then
Thereinto H(x,y) =
Theorems. 1 Lct{Xn,n > 2} be stationary NA sequence with identical distribution,satisfiug assumption of (A2)(i) ,further if
Theorem4.1 Let {Xn,n > 2} be NA sequence with identical distribution,satisfing the assumption of Al(ii) and A2,further,if K(u) isn't increase or decrease,and exist a > 0,satisfing
at each point x of continuity f(x).
Theorem 5.1 Let{Xn,n > 1} be stationary NA sequence with identical distribution,in the condition of(Al),and let pn,qn,pn + qn < n and kn = ] exist,fulfilling
Corollary5.1 Suppose all the assumption of the theorems. 1 and (A2) are fulfilled ,and,in addition,the second-order derivative (x) exists and is bounded ,and let
Then
Theorem 6.1 Let {Xn,n > 2} be stationary NA sequence with identical distribution,satisfing assumption of theorem 4.1,then for all x c( (z)),we have
f_n(x)=1/n sum from j=1 to n (1/(h_j) K((x-X_j)/h_j))，由于经过简单的化简我们有
f_n(x)=(n-1)/nf_(n-1)(x)+1/(nh_n))K((x-X_n)/h_n)。从这个特点我们知道用递归核估计去估计未知密度时，在添加样本点的情况下，不必重新计算所有项，只需计算添加项，再经简单计算即可。这比用普通型的密度核估计
(?)_n(x)=1/(nh_n) sum from j=1 to n (K((x-X_j)/h_n))去估计未知密度，从计算方面而言具有很大方便，因此很多学者对之进行了研究。
在独立样本情形，Wolverton and Wagner([1]，1969),Wegman and Davis([2]，1979)等对某些形式的递归密度核估计作了深入研究，其结果比较完美；在相依样本情形，Masry([3]，[4]1986，1987)分别对ρ-混合和渐近不相关情形讨论f_n(x)的均方收敛和渐近正态性；蔡宗武教授([5]，[6]1990，1992)分别讨论在(α，β)混合下的强收敛速度和ρ-混合下的渐近正态性，杨善朝教授([7]1997)在ρ-混合下，对递归密度核估计的强收敛速度作了进一步的讨论，改进了Masry([3])的结论。
然而，NA(负相协)和PA(正相协)变量在可靠性理论、概率过程、随机过程、多元统计、空间统计等领域中有广泛的应用，而且在大气、地质、海洋、生物、经济等领域也有应用具有广泛应用，故在NA和PA下对密度核估计的渐近性质的研究也甚为重要。因此引起了学者们的重视，如韦来生教授([15]，2001)，杨善朝教授([16]，2000)，Roussas([17]，2000)，等人在NA下对普通型的密度核估计做了相合性和渐
近正态性的研究．而在NA和PA下尚未见有人对递归型密度核估计进行研究．因
此，本文主要在＊A相依样本下，对核函数们叫，未知密度函数人X）和窗宽札满
足一定的条件下，具体地讨论了递归型密度核估计人k）所具有的大样本性质，即
研究其渐近方差，r－阶平均相合性，逐点强相合性和渐近正态性．作为可靠性问
题中的应用，构造了生存函数和失效率函数户k）＝‘－刊d）－八K＞d）和叶d）＝Fd
，lerw T、《XI
P以川“二7．JIx；＞川；7”。“于下下－7，
n Hd““P、lx）
讨论了r。k）的逐点强相合性．
我们的主要假设和结论如下：
1：主要假设
（AI）设厂。，。2 2｝为同分布 NA序列，满足
（i）局与儿付的联合密度八。材存在且满足
suP山x，y，幻一八川州 S啊＜＋co；
Z＆
门Z、LDn 上下叫／r、＿p／L＿上y／巫L＼p—1＿p．／＋＿n。、＿＋＿
卜“J“’二十小h 上J八人／’——一’n 上J＼人J‘”p一且—一，一’“11——·
（iii）K（l）有限可微亚 且 K。LI，jR K（。）d。二 l，suP（1＋IJ）IK（x）＜十co。
（tV） 存在正增常数列｛Cfd C。＜。，使得 h。C。＞0，hJ‘。（C。）＋ 0·
（AZ）（i）K E LI，jBK（。）du＝1，SliP（1＋问）IK（1）l＜＋OO；
（n in＊If＊）＊＊二＊，in＊＊If＊）＊＊＜ ＋co·
2：主要结论
定理2＊ 在（M条件下，对八X）任意连续点。；。有
Inlrn从。＊。叶人（x），人（川）＝＊（x，川．
16八X）厂”（叫叫 当X二队
具甲川兄功＝（
IO，gx ／ y．
定理3＊ 设厂＊n三 2｝为同分布平稳 NA序列，满足…）＊条件，且有
1，一、＾．．、、＿1，。、，－一、n，1、
h、上 0、2＞”hZ＝O（hZ），＞六ICOv（XI，Xj）＝O（六），
‘“”＋”’L－’”’一一’””””L－I12’—一“－“’－“””hZ”’
0＜，＜2，那么
＊） 当＊I乙义＋十co时，有thlr EIjn＊卜／（叫刀＝ 0，
（n） 当hn二n一会日，有＊ （J）一f（x）l厂二O（n一i）．
2
定理 4．1 设（X＊。三 2｝为同分布 NA随机变量序列，满足条件AI（n）和 AZ，进
一步设川叫非降或非升，且存在口＞0，使得太－以丫1，则对八x）的任一连续点
。有
人一）一j（X）．
定理 5．1 设体。，。2 2｝为平稳的 NA随机变量序列，在K）条件下，又设存在
P。。，q。。
Let Xn,n 1 be a sample sequence with identical distribution ,having unknown probability density function fn(x). based on the sample Xt, ,Xn,the recursive density estimator for f(x) that
was introduced by Wolverton and Wagner in 1969,it can be calculated recursively,i.e.
From the practical point of view,the recursive density estimator has a clear advantage over non-recursive estimators in that it can be updated with each additional observation,in contrast to the nonrecursive case of
where the estimator must be entirely recomputed. Since it is more convenient to use the recursive density estimator fn(x),there has been a significant number of research papers on this subject matter. The most relevant papers are these by Wolverton and Wagner(1969),and Weg-man and Davis (1979) for independent samples those are very ideal;Masry(1986,1987) discussed both quadratic-mean consistency and asymptotic normality for dependent sample and asymptotically uncorrelated;Professor Cai Zongwu(1990,1992) discussed both rate of strong convergence and asymptotic normality for (a, 3)-dependent and p-dependent;Professor Yang Shanchao (1997) further discussed rate of strong convergence for -dependent,mended the result of Masry's. In a snapshot,one may say that negative association and positive association occur often in cetain reliability theory problems,filter theory problems,and multianalysis theory problems.Many more applications are to be anticipated in a host of other areas and ,in particular,those areas whe
re spatial statistics play an important role,such area are ,for example,analysis of agricultural field experiments,geostatistical annalysis,image annalysis,oceanographic applications,signal processing in radar and sonars,and stereology. so it is more important to study asymptotic propertis of density estimation for positive and negative association,arousing our considerable regard. For example,Professor Wei Laiseng (2001),Professor Yang Shanchao (2000) studied the density estimation n(z) on consistency and asymptotic normality for negative association. However,up to
the present,there is no paper discussed asymptotic propertis of the recursive kernel estimate of probability density function under negative association and positive association. In this paper,the author investigates some asymptotic propertis of fn(x) under negative association that includes the consistency in r-order mean,the strong consistency and the asymptotic normality under certain conditions for the kernel function K(x) ,the unknown density function f(x) and the bandwidth hn. As applications in reliability theory problems,the hazard rate and survial function,
will be estimated by
and present the strongly consistency of rn (x) . The following is our main assumption and results 1 The main assumption
(Al) The r.v.s X ,X2, ,are nagetive association sequence with identical distribution,satisfing (i) If f(x,y,k) is the joint p.d.f. of the r.v.s X ,Xi+k,then
(iii) K(x) have bounded derivative,and
(iv) There exists positive,increase constant
2 The main result
Theorem 2.1 Suppose the assumption of (Al) are fulfilled ,if x and y is arbitrary continuity of (x),then
Thereinto H(x,y) =
Theorems. 1 Lct{Xn,n > 2} be stationary NA sequence with identical distribution,satisfiug assumption of (A2)(i) ,further if
Theorem4.1 Let {Xn,n > 2} be NA sequence with identical distribution,satisfing the assumption of Al(ii) and A2,further,if K(u) isn't increase or decrease,and exist a > 0,satisfing
at each point x of continuity f(x).
Theorem 5.1 Let{Xn,n > 1} be stationary NA sequence with identical distribution,in the condition of(Al),and let pn,qn,pn + qn < n and kn = ] exist,fulfilling
Corollary5.1 Suppose all the assumption of the theorems. 1 and (A2) are fulfilled ,and,in addition,the second-order derivative (x) exists and is bounded ,and let
Then
Theorem 6.1 Let {Xn,n > 2} be stationary NA sequence with identical distribution,satisfing assumption of theorem 4.1,then for all x c( (z)),we have
引文
[1] Wolverton, C. T. and Wagner, T, J., Asymptotically optimal discriminant functions for pattern classification, IEEE Trans. Th., 1969, IT15: 258-265.
[2] Wegman. E. T. and Dvis, H. I., Remarks on some recusive estimator of aprobability density, Ann. Statist., 1979, 7: 316-327.
[3] Masry, E., Recursive probability density estimation for weakly dependent statinary processes, IEEE Trans. Inform. Th., 1986, IT32: 254-267.
[4] Masry, Strong consisteny an rates for rescursive probability density estimators of stationary processes, J. Multivar. Anal. 1987, 22: 79-93.
[5] 蔡宗武，相依随机变量的密度估计的强收敛速度，系统科学与数学．1990, 10(4)：360-370．
[6] 蔡宗武，相依随机变量的密度函数的递归核估计的渐近正态性．应用概率统计1992，9(2)：123-129．
[7] 杨善朝，混合序列距不等式和非参数估计，数学学报，1997, 40(2)：271-279．
[8] Joag-Dev. K and Proschan F.. Negative association of random variables with application. Ann. statist. 1983, 11: 286-295.
[9] Block H W, Savits T H, Shaked M. Some concepts of negative dependence. Ann. Probab. 1982, 10: 765-772.
[10] Roussas G. G. Positive and negative dependence with some statistical application [J]. in: Ghosh, S. (Ed), Aymptotics, Nonparametrics and time Series. Marcel Dekker, New York, 1999, 757-788.
[11] Cai Z. and Roussas G. G. Kaplan-Meier estimator under association. J. Multivariate Anal. 1998, 67: 318-348.
[12] Cai Z. and Roussas G. G. Weak convergence for a smooth estimator of a distribution function under association. Stochsstic Anal. Appl. 1999a, 17: 145-168.
[13] Cai Z. and Roussas G. G. Berry-Esseen bounds for smooth estimator of a function under association. J. Nonparametric Statist. 1999b, 10: 79-106.
[14] 杨善朝，王岳宝，NA样本回归函数估计的强相合性．应用数学学报，1999，22(4)：522-530．
[15] 韦来生，NA样本概率密度函数核估计的相合性．系统科学与数学，2000, 21(1)：79-87．
[16] 杨善朝，非参数估计一致强相合性．2001，待发．
[17] George G. Roussas, Asymptotic normality of the kernel estimate of a probability density function under association [J], Statistics. and prob. lett. 2000, 50: 1-12.
[18] 王小明，NA序列部分和的完全收敛性．应用数学学报，1999，22(3),407-412．
[19] 胡舒合，回归函数递归估计相合的充要条件，高校应用数学，1991, 12：511-519．
[20] 苏淳、赵林城、王岳宝，NA序列的矩不等式与弱收敛，1996：1091-1099．
[21] Roussas G. G. Fix design regression for time series: Asymptotic normality, J. M. A. 1992, 40: 262-291.
[22] 杨善朝，随机变量部分和的矩不等式，中国科学，2000, 30(3)：218-223．
[23] 杨善朝，NA样本最近邻估计的相合性，应用数学学报，2002，已录用．
[24] Han-ying liang, Complete convergence for weighted sum of negatively association random variable. Stati. and Prb. Letters, 2000(48), 317-325.
[25] 方前胜，一类概率密度函数估计的渐近正态性，数理统计与应用概率1998, 13(3)：183-187．
[26] 林正炎，相依样本情形时密度的核估计，科学通报1983, 12：709-713．
[27] Pan JM., On the covergence rates in the central limit theorem for negative associated sequences. Chinese Jounal of Appl. prob. and stat..1997,13: 183-192.
[28] Matula P. A note on the almost sure convergence of sum of negative dependent random variables[J]. Statist. Prob. Lett., 1992, 15:209-213.
[29] Yuan Ming, Su Chun, Weak convergence for empirical processes of Negative Associated sequences, Chinese Journal of Applied probability and statics, 2000, 16(1): 45-56.
[2] Wegman. E. T. and Dvis, H. I., Remarks on some recusive estimator of aprobability density, Ann. Statist., 1979, 7: 316-327.
[3] Masry, E., Recursive probability density estimation for weakly dependent statinary processes, IEEE Trans. Inform. Th., 1986, IT32: 254-267.
[4] Masry, Strong consisteny an rates for rescursive probability density estimators of stationary processes, J. Multivar. Anal. 1987, 22: 79-93.
[5] 蔡宗武，相依随机变量的密度估计的强收敛速度，系统科学与数学．1990, 10(4)：360-370．
[6] 蔡宗武，相依随机变量的密度函数的递归核估计的渐近正态性．应用概率统计1992，9(2)：123-129．
[7] 杨善朝，混合序列距不等式和非参数估计，数学学报，1997, 40(2)：271-279．
[8] Joag-Dev. K and Proschan F.. Negative association of random variables with application. Ann. statist. 1983, 11: 286-295.
[9] Block H W, Savits T H, Shaked M. Some concepts of negative dependence. Ann. Probab. 1982, 10: 765-772.
[10] Roussas G. G. Positive and negative dependence with some statistical application [J]. in: Ghosh, S. (Ed), Aymptotics, Nonparametrics and time Series. Marcel Dekker, New York, 1999, 757-788.
[11] Cai Z. and Roussas G. G. Kaplan-Meier estimator under association. J. Multivariate Anal. 1998, 67: 318-348.
[12] Cai Z. and Roussas G. G. Weak convergence for a smooth estimator of a distribution function under association. Stochsstic Anal. Appl. 1999a, 17: 145-168.
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