宇世航, 趙世舜
(1. 齊齊哈爾大學 理學院, 黑龍江 齊齊哈爾 161006; 2. 吉林大學 數(shù)學學院, 長春 130012)
基于替代數(shù)據(jù)和核實樣本推斷的研究目前已有許多結(jié)果[1-10]. Sepanski等[1]研究了基于核實數(shù)據(jù)的非線性EV模型; Wolvreton等[11]提出了f(x)的遞歸型核密度估計:
由于遞歸型核密度估計在添加樣本點時, 不必重新計算所有項, 只需計算添加項, 因此使計算更方便. 基于此, 本文考慮借助于核實數(shù)據(jù), 構(gòu)造一遞歸型概率密度估計量, 并研究其漸近正態(tài)性.
于是在一些正則條件下,f(x)可被如下遞歸核估計量一致估計:
定義
AppendixA條件:
(A·f):f(x)是k階有界可導(dǎo)的;
(A·K):K(·)在有界支撐集上是k階非負有界的核函數(shù);
定理1在AppendixA條件下, 有
證明:
(3)
這里C為任意常數(shù), 且在不同處可取不同的值. 于是, 由式(3)~(6)有
則
從而
其中
而
及條件(A·bn,ηn), 可得
(8)
同理, 有
(10)
于是由式(2)~(10), 有
(11)
同理, 由(A·f),(A·K),(A·hj), 得
(13)
綜上所述, 有
令
由條件(A·K)和(A·h), 有
由式(15)~(17), 顯然有
I1→N(0,θ1σ2(x)),
(18)
I2→N(0,θ2σ2(x)),
(19)
而
故結(jié)合式(15),(18)~(20)可得
(n,N)=(20,100),(50,100),(50,300),(100,300),
圖1 n=20, N=100時的模擬結(jié)果Fig.1 Simulation result for n=20, N=100
圖2 n=50, N=100時的模擬結(jié)果Fig.2 Simulation result for n=50, N=100
圖3 n=50, N=300時的模擬結(jié)果Fig.3 Simulation result for n=50, N=300
圖4 n=100, N=300時的模擬結(jié)果Fig.4 Simulation result for n=100, N=300
由圖1~圖4可見, 給定樣本總數(shù)N的情況下, 模擬效果隨核實數(shù)據(jù)樣本容量n的增加而漸好; 當固定核實數(shù)據(jù)樣本容量n時, 頂部隨樣本總量N的增加模擬效果漸好, 尾部變差; 如果同時增大N和n, 模擬結(jié)果更趨近于f(x), 并且也更平滑.
[1] Sepanski J H, Lee L F. Semiparametric Estimation of Nonlinear Error-in-Variables Models with Validation Study [J]. J Nonparametric Statist, 1995, 4: 365-394.
[2] WANG Qi-hua. Estimation of Partial Linear Error-in-Variables Model with Validation Data [J]. J Multivaiate Anal, 1999, 69: 30-64.
[3] WANG Qi-hua. Estimation of Linear Error-in-Covariables Models with Validation Data under Random Censorship [J]. J Multivariate Anal, 2000, 74(2): 245-266.
[4] WANG Qi-hua, Rao J N K. Empirical Likelihood-Based Inference in Linear Error-in-Corariables Models with Validation Data [J]. Bioraetrika, 2002, 89(2): 345-358.
[6] WANG Qi-hua, YU Ke-ming. Likelihood-Based Kernel Estimation in Semiparametric Errors-in-Covariables Models with Validation Data [J]. Journal of Multivariate Analysis, 2007, 98(3): 455-480.
[7] XUE Liu-gen. Empirical Likelihood Inference in Nonlinear Semiparanetric EV Models with Validation Data [J]. Acta Mathematica Sinica: Chinese Series, 2006, 49(1): 145-154. (薛留根. 核實數(shù)據(jù)下非線性半?yún)?shù)EV模型的經(jīng)驗似然推斷 [J]. 數(shù)學學報: 中文版, 2006, 49(1): 145-154.)
[8] DAI Peng-jie, SUN Zhi-hua, WANG Peng. Model Checking for General Linear Error-in-Covariables Model with Validation Data [J]. Acta Mathematica Sinica, 2010, 23(6): 1153-1166.
[9] WANG Qi-hua. Dimension Reduction in Partly Linear Error-in-Response Models with Validation Data [J]. Journal of Multivariate Analysis, 2003, 85(2): 234-252.
[10] DU Li-lun, ZOU Chang-liang, WANG Zhao-jun. Nonparmetric Regression Function Estimation for Errors-in-Variables Models with Validation Data [J]. Statistica Sinica, 2011, 21: 1093-1113.
[11] Wolvreton C T, Wagner T J. Asymptotically Optimal Discriminant Functions for Pattern Classification [J]. IEEE Trans, 1969, 15(2): 258-265.
[12] Rao B L S P. Nonparametric Functional Estimation [M]. London: Academic Press, 1983.