相依数据的若干统计模型及分析
|
摘要
在一些近代科学研究中，如生命科学和信息科学的研究，人们获得的
数据往往具有量大、高维和相依等特点。于是，关于相依随机变量的研究，
已引起人们的重视，取得一些研究成果（如陆传荣，林正炎（1997）及其它
所列文献），并提出了一些研究和加工相依变量的有效方法，如Bootstrap、
分块Bootstrap、Jackknife和分块Jackknife（Blocks of blocks jackknife）等方法
（Lahiri（1999）,Demitris and Joseph（1992））。然而，对各种具体的相依数据的统
计模型（如相依数据的非线性回归和相依数据的非参数回归模型）的研究
还不够充分和不够完备。为了进一步发展相依随机变量及相关统计模型
的理论，本文研究相依数据的若干统计模型及其分析，包括相依数据的
线性回归模型、非线性回归模型、半参数及非参数回归模型中参数估计
和函数拟合等。
我们在第一章研究了噪声为弱相依过程的固定设计或随机设计非参
数回归模型
Y_i=g(X_i)+ε_i，i=1，…，n,
其中作{ε_1，ε_2，…}是一均值为0方差为σ~2的宽平稳随机过程。在自协方差
R（k）=E（ε_iε_i+k）满足一些较弱的条件下，我们定义了固定设计模型函数g（·）
的两种类型的Bootstrap小波估计
同时定义了随机设计模型函数g（·）的Bootstrap小波估计
其中Y_i~*，t_i~*，X_i~*和A_i~*分别是观测值、设计点和设计区间的Bootstrap样本。
在自协方差R（k）=E（ε_iε_i+k）满足一些更弱的条件下，我们定义了固定设计
模型函数g（·）的两种类型的分块Bootstrap小波估计
tv
同时定义了随机设计模型函数g（·）的分块Bmtst＊p ／J’波估计
hi
j（t）二（hi）‘二 K”Em（t，X；）／f（t），
i二1
其中X＊，允X：和人分别是观测值、设计点和设计区间的分块Bootstra。样
本，l是数据块宽．我们得到固定设计模型中 BOOtstr叩小波和分块 BOOtSnap
／J’波估计量g＊一和g＊o的偏差和方差的渐近界及近似渐近正态性，建立
了随机设计模型中的Bootstrap ／J’波和分块Boot。。rap ／J’波估计量3＊的依概
率收敛性及偏差的渐近界，提出了选择数据块大小的原则．本文的理论结
果和数据模拟表明，在较弱的数据假设下，非参数回归模型中的B皿tstmp
小波和分块Bootstrap ／J’波方法是有效的．
第二章研究了相依数据的线性回归模型中分块拟回归方法．为了减
少计算复杂性，在线性回归的计算机试验设计中，最近人们提出了一种
新的统计方法～拟回归（O侧叫2000），AMM O一（2001））．在独立同分布的线
性回归模型中，拟回归不仅能提高计算速度，而且有较好的统计性质。
然而，对相关数据模型，这种方法的统计性质并不好．针对这一问题。
本文提出回归系数凡的分块拟回归估计：
Q
ITh
6。一 4 3T｝。i二 0、l…·．D—1．
其中Ti为适当的观测数据块，Q为数据块的个数．我们得到分块拟回归
的小样本和大样本性质，如无偏性、均方收敛性、强收敛性和渐近正态
性，并讨论了曲线拟合的性质．这些结果表明，分块拟回归比原拟回归
渐近有效．同时，我们还指出了分块拟回归（包括原拟回归）在高维问题
中的缺陷．为改善曲线拟合，我们提出一种修正分块拟回归．修正分块
拟回归只需将PO的估计调整为
IQM r＿lP一1
。＿．上 歹7风下7飞／。NJ 下7风。＿』。，＿。。2。＿
山二7了了｝二＿三＿＼1一7刁工工二人z“＼c门一1）L＋S厂I叭i一I〕L＋S）5
其中驯和闪分别为观测变量和设计变量，／为基函数．本文的理论结果
和数据模拟表明，在高维问题中，修正分块拟回归比原分块拟回归（包括
原拟回归）渐近有效．
S
v
我们在第三章研究了弱相依数据的半参模型中的分块Euclidean经验
似然方法．如果对问题的背景所知甚少，仅仅知道某无偏估计函数（如一
二矩），人们有时称此为半参模型．在独立同分布数据的半参模型种，人们
用经验似然进行参数估计（Owen（1988，1990），Qin。nd L娜less（1994））．在弱相
依数据的半参模型中，本文引入一种分块E皿lid阴n经验似然方法．这种
方法结构简单，其对数似然比为：
l。。（01＝HT（01’SZ‘（01T（01。
”HH＼”’一2“＼”｝“B’”）“＼”）’
其中穴0）和S以0）分别为
The
data gathered from the study of some modern sciences such as biology and information science is usually large, high dimensional and dependent. So, the theory of dependent random variables is an important subject in statistics. It is well known that one has obtained some theoretical and practical achievements in the field of dependent random variables recently (Lu and Lin (1997) and the literature listed in the reference). And some powerful tools for treating dependent random variables, for example, Bootstrap, blockwise Bootstrap, Jackknife and blocks of blocks jackknife, have been introduced in recent literature (Lahiri (1999), Demitris and Joseph (1992)). So far, however, the theory that studies specially the statistical models with dependent data such as nonlinear regression and nonparametric regression models is not very satisfactory. In order to develop and unprove the theory of dependent random variables and related statistical models, this paper mainly focuses on the statistical theory of parameter estimation and fitting of function in the statistical models with dependent data involving linear regression model, nonlinear regression model, semi-parametric model and nonparametric regression model.
In Chapter 1, we consider the following fixed design or random design nonparametric model with dependent noises:
where {c~, e2,.?.} is a stationary stochastic process with mean 0 and variaiice a2. Under some weak assumptions on auto-covariance R(k) = E(~ici), we define two Bootstrap wavelet estimators of gQ) in the fixed design model as
1 N N
g*(t) ~ ~ t~)n(t~) and g~(t) = ~ Y7 J Em(t, s)ds
i=1 j=i
and a Bootstrap wavelet estimator of g(.) in the random design model as
i= 1
where Y7, t~, X~, and A~ are respectively the Bootstrap samples of observations, design points and design intervals. Under weaker assumptions on auto-covariance than that of
lX
data i1l fixed desigIl model, we define two blockwise BootstraP wave1et estimators of g(.)
in the fixed design model as
1 + K*Em(t, t:)K(t:) and g*(t) = f K* l Em(t, s)d8
g*(t) = ai F K*Em(t, t:)K(t:) and g*(t) = Z K* l* Em(t, s)d8
l=1 i=l iA:
and a blockwise Bootstrap wavelet estimators of g(.) in the random design model as
bl
j(t) = (bl)--' Z K*Em(t, X;)/i(t),
i=l
where K*, t: 3 X; 5 and A: are re8pectively the blockwise Bootstfap samples of observations,
design points and design interals, l is the width of data block. In the fixed design model,
the asymptotic bounds of the biases and variances fOr the Bootstrap wavelet and blockwise
Bootstrap wave1et estimators g*(t) and g*(t) are obtained, their asymptotic normality for
a modified version are establi8hed. In the random design model, the consistency and the
asymptotic bound of the bias for the Bootstrap wavelet and blockwise Bootstrap wavelet
estimator g(t) are proved and a principle of selecting the width of data block is given.
These re8ults show that both the BootstraP wavelet and blockwise BootstraP wavelet
methods are valid in the models with weakly dependent processes.
In Chapter 2, we study the blockwise quasi--regression in the linear models with weakly
dependent data. Quasi-regression, a new method motivated by the problerns arising in
computer experiments, mainly focuses on speeding up evaluation (Owen (2000), An and
Owen (2001)). For i.i.d. observations, the quasi-regression method is valid not only i11
speeding l1p evaluation, but also in parametric statistical inference and fitting of function.
However, fOr dePendent observations, this method is invalid. To improve quasi--regression,
in this ChaPter, a blockwise quasi--regression estimator of the regression coefficient fij is
defined as
Q
1,Q
Pj = 6 gTf5j = 0,1,...,P-- l,
where Ti is the block of observations and Q is the flumber of the blocks. Some small sam-
ple and large sample superiority of this estimate 8uch as the unbiasedness, convergence,
asymptotic norlnality and the property of simulation, are obtained. We also find soIne de-
fects of blockwise quasi-r
数据往往具有量大、高维和相依等特点。于是，关于相依随机变量的研究，
已引起人们的重视，取得一些研究成果（如陆传荣，林正炎（1997）及其它
所列文献），并提出了一些研究和加工相依变量的有效方法，如Bootstrap、
分块Bootstrap、Jackknife和分块Jackknife（Blocks of blocks jackknife）等方法
（Lahiri（1999）,Demitris and Joseph（1992））。然而，对各种具体的相依数据的统
计模型（如相依数据的非线性回归和相依数据的非参数回归模型）的研究
还不够充分和不够完备。为了进一步发展相依随机变量及相关统计模型
的理论，本文研究相依数据的若干统计模型及其分析，包括相依数据的
线性回归模型、非线性回归模型、半参数及非参数回归模型中参数估计
和函数拟合等。
我们在第一章研究了噪声为弱相依过程的固定设计或随机设计非参
数回归模型
Y_i=g(X_i)+ε_i，i=1，…，n,
其中作{ε_1，ε_2，…}是一均值为0方差为σ~2的宽平稳随机过程。在自协方差
R（k）=E（ε_iε_i+k）满足一些较弱的条件下，我们定义了固定设计模型函数g（·）
的两种类型的Bootstrap小波估计
同时定义了随机设计模型函数g（·）的Bootstrap小波估计
其中Y_i~*，t_i~*，X_i~*和A_i~*分别是观测值、设计点和设计区间的Bootstrap样本。
在自协方差R（k）=E（ε_iε_i+k）满足一些更弱的条件下，我们定义了固定设计
模型函数g（·）的两种类型的分块Bootstrap小波估计
tv
同时定义了随机设计模型函数g（·）的分块Bmtst＊p ／J’波估计
hi
j（t）二（hi）‘二 K”Em（t，X；）／f（t），
i二1
其中X＊，允X：和人分别是观测值、设计点和设计区间的分块Bootstra。样
本，l是数据块宽．我们得到固定设计模型中 BOOtstr叩小波和分块 BOOtSnap
／J’波估计量g＊一和g＊o的偏差和方差的渐近界及近似渐近正态性，建立
了随机设计模型中的Bootstrap ／J’波和分块Boot。。rap ／J’波估计量3＊的依概
率收敛性及偏差的渐近界，提出了选择数据块大小的原则．本文的理论结
果和数据模拟表明，在较弱的数据假设下，非参数回归模型中的B皿tstmp
小波和分块Bootstrap ／J’波方法是有效的．
第二章研究了相依数据的线性回归模型中分块拟回归方法．为了减
少计算复杂性，在线性回归的计算机试验设计中，最近人们提出了一种
新的统计方法～拟回归（O侧叫2000），AMM O一（2001））．在独立同分布的线
性回归模型中，拟回归不仅能提高计算速度，而且有较好的统计性质。
然而，对相关数据模型，这种方法的统计性质并不好．针对这一问题。
本文提出回归系数凡的分块拟回归估计：
Q
ITh
6。一 4 3T｝。i二 0、l…·．D—1．
其中Ti为适当的观测数据块，Q为数据块的个数．我们得到分块拟回归
的小样本和大样本性质，如无偏性、均方收敛性、强收敛性和渐近正态
性，并讨论了曲线拟合的性质．这些结果表明，分块拟回归比原拟回归
渐近有效．同时，我们还指出了分块拟回归（包括原拟回归）在高维问题
中的缺陷．为改善曲线拟合，我们提出一种修正分块拟回归．修正分块
拟回归只需将PO的估计调整为
IQM r＿lP一1
。＿．上 歹7风下7飞／。NJ 下7风。＿』。，＿。。2。＿
山二7了了｝二＿三＿＼1一7刁工工二人z“＼c门一1）L＋S厂I叭i一I〕L＋S）5
其中驯和闪分别为观测变量和设计变量，／为基函数．本文的理论结果
和数据模拟表明，在高维问题中，修正分块拟回归比原分块拟回归（包括
原拟回归）渐近有效．
S
v
我们在第三章研究了弱相依数据的半参模型中的分块Euclidean经验
似然方法．如果对问题的背景所知甚少，仅仅知道某无偏估计函数（如一
二矩），人们有时称此为半参模型．在独立同分布数据的半参模型种，人们
用经验似然进行参数估计（Owen（1988，1990），Qin。nd L娜less（1994））．在弱相
依数据的半参模型中，本文引入一种分块E皿lid阴n经验似然方法．这种
方法结构简单，其对数似然比为：
l。。（01＝HT（01’SZ‘（01T（01。
”HH＼”’一2“＼”｝“B’”）“＼”）’
其中穴0）和S以0）分别为
The
data gathered from the study of some modern sciences such as biology and information science is usually large, high dimensional and dependent. So, the theory of dependent random variables is an important subject in statistics. It is well known that one has obtained some theoretical and practical achievements in the field of dependent random variables recently (Lu and Lin (1997) and the literature listed in the reference). And some powerful tools for treating dependent random variables, for example, Bootstrap, blockwise Bootstrap, Jackknife and blocks of blocks jackknife, have been introduced in recent literature (Lahiri (1999), Demitris and Joseph (1992)). So far, however, the theory that studies specially the statistical models with dependent data such as nonlinear regression and nonparametric regression models is not very satisfactory. In order to develop and unprove the theory of dependent random variables and related statistical models, this paper mainly focuses on the statistical theory of parameter estimation and fitting of function in the statistical models with dependent data involving linear regression model, nonlinear regression model, semi-parametric model and nonparametric regression model.
In Chapter 1, we consider the following fixed design or random design nonparametric model with dependent noises:
where {c~, e2,.?.} is a stationary stochastic process with mean 0 and variaiice a2. Under some weak assumptions on auto-covariance R(k) = E(~ici), we define two Bootstrap wavelet estimators of gQ) in the fixed design model as
1 N N
g*(t) ~ ~ t~)n(t~) and g~(t) = ~ Y7 J Em(t, s)ds
i=1 j=i
and a Bootstrap wavelet estimator of g(.) in the random design model as
i= 1
where Y7, t~, X~, and A~ are respectively the Bootstrap samples of observations, design points and design intervals. Under weaker assumptions on auto-covariance than that of
lX
data i1l fixed desigIl model, we define two blockwise BootstraP wave1et estimators of g(.)
in the fixed design model as
1 + K*Em(t, t:)K(t:) and g*(t) = f K* l Em(t, s)d8
g*(t) = ai F K*Em(t, t:)K(t:) and g*(t) = Z K* l* Em(t, s)d8
l=1 i=l iA:
and a blockwise Bootstrap wavelet estimators of g(.) in the random design model as
bl
j(t) = (bl)--' Z K*Em(t, X;)/i(t),
i=l
where K*, t: 3 X; 5 and A: are re8pectively the blockwise Bootstfap samples of observations,
design points and design interals, l is the width of data block. In the fixed design model,
the asymptotic bounds of the biases and variances fOr the Bootstrap wavelet and blockwise
Bootstrap wave1et estimators g*(t) and g*(t) are obtained, their asymptotic normality for
a modified version are establi8hed. In the random design model, the consistency and the
asymptotic bound of the bias for the Bootstrap wavelet and blockwise Bootstrap wavelet
estimator g(t) are proved and a principle of selecting the width of data block is given.
These re8ults show that both the BootstraP wavelet and blockwise BootstraP wavelet
methods are valid in the models with weakly dependent processes.
In Chapter 2, we study the blockwise quasi--regression in the linear models with weakly
dependent data. Quasi-regression, a new method motivated by the problerns arising in
computer experiments, mainly focuses on speeding up evaluation (Owen (2000), An and
Owen (2001)). For i.i.d. observations, the quasi-regression method is valid not only i11
speeding l1p evaluation, but also in parametric statistical inference and fitting of function.
However, fOr dePendent observations, this method is invalid. To improve quasi--regression,
in this ChaPter, a blockwise quasi--regression estimator of the regression coefficient fij is
defined as
Q
1,Q
Pj = 6 gTf5j = 0,1,...,P-- l,
where Ti is the block of observations and Q is the flumber of the blocks. Some small sam-
ple and large sample superiority of this estimate 8uch as the unbiasedness, convergence,
asymptotic norlnality and the property of simulation, are obtained. We also find soIne de-
fects of blockwise quasi-r
引文
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39 陈希孺，王松桂．(1987)．近代回归分析．合肥，安徽教育出版社．
40 Chen, X. R. and Wu, Y. H. (1989). Strong Law for a Mixing Sequence. Acta Math. Appl. Sinica. 5, 367-371.
41 洪圣岩．(1993)，半参数回归模型的参数估计的Bootstrap强逼近，中国科学(A辑)，23(3)，239-251．
42 洪圣岩，成平(1993)．半参数回归模型的参数估计的bootstrap逼近．中国科学(A辑)，23(3)，239-251．
43 Lin, L. (2000). Conservative Estimating Function in Nonlinear Model with Aggregated Data. Mathematica Acta Sinientia. 20(3), 335-340.
44 Lin, L. and Zhang, R. C. (2001). Blockwise Empirical Euclidean Likeilihood for Weakly Dependent Processes. Statistics and Probability Letters. (To appear).
45 Lin, L., Yang, G. J. and Zhang, R. C. (2001). Analysis and Improvement of Quasiregression. 泛华统计协会第五次国际学术会议，香港．
46 林路．非线性模型中拟似然估计的若干性质．(1999)．应用数学学报．22(2)，307-310，
47 林路．半相依非线性回归方程的近似得分函数和近似拟得分函数．(1998)．应用数学．11(4)，123-126．
48 林路．非线性模型中拟似然估计和约束拟似然估计．(1998)数学物理学报．18(增刊)，133-138．
49 林路．数据变换参数渐近置信域的曲率表示．(1998)．高校应用数学学报．13(2)，153-158．
50 林路，张润楚．假设检验的相对稳定性．应用数学学报，已接受，待发表．
51 刘维奇，潘晋孝．聚集数据的线性模型参数的改进Peter-Karsten估计．(1995)．工程数学学报．12(2)，115-119．
52刘维奇，潘晋孝．聚集数据的线性模型中一种新的参数估计．(1996)．工程数学学报，13(4)，340-356．
53陆传荣，林正炎．(1997)．混合相依变量的极限理论．北京，科学出版社．
54罗旭．(1994)．半参数模型的经验欧氏似然估计的大样本性质．应用概率统计，10(3)，344-352．
55罗旭．(1997)．两参数半参数模型的经验欧氏似然估计．应用概率统计．13(2)，133-141．
56钱伟民，柴根象．(1999)．半参数回归模型小波估计的强逼近．中国科学(A辑)，29(3)，233-240．
57秦永松，赵林成．(1998)．两样本分位数差异的半经验似燃比检验，应用数学学报．21(1)，103-112．
58王松桂．线性模型的理论及其应用，(1987)，合肥，安徽教育出版社，
59王松桂．线性回归系统回归系数的一种新估计，(1988)．中国科学(A辑)，18(10)，1033-1040．
60韦博成．(1994)，指数族非线性模型的几何渐近推断理论．中国科学(A辑)，24(12)，1270-1278．
61韦博成．(1989)．近代非线性回归分析．南京，东南大学出版社，
62Wei,B.C,Exponential Family Nonlinear Models,(1998).Singapore,Springer-verlag.
63张润楚，王兆军．(1994)．关于计算机试验设计的设计理论和数据分析．应用概率统计，10(4)，421-435．
64张润楚．(2000)近代试验设计的一些新近展，南开大学技术报告．
65朱仲义．非线性模型中拟似然估计的小样本性质，(1996)．工程数学学报．13(2)，59-64．
2 Antoniadis, A., Mckeague, I. W. (1994) . Wavelet Method for Estimation, JASA. 89, 1340-1353．
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18 Kerkgacyarian G.and Picard. (1992) . Density Estimation in Besov Space. Statistics and Probability Letters. 13, 15-24．
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20 Lahiri, S.H. (1999) . Theoretical Comparision of Block Bootstrap Methods, Annals of Statist. 27(1) , 386-404．
21 Li, B. and McCullagh, P. (1994) . Potential Functions and Conservative Estimating Functions. Annals of Statistics. 22, 341-356．
22 McCullagh, P. (1983) Quasi-Likelihood Functions. Annals of Statistics. 11, 59-67．
23 McCullagh, P. and Nelder, J. A.(1989) . Generalized Linear Models, Second Edition. London, Chmpman and Hall, 323-339．
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38 Yuichi, K. (1997). Empirical Likelihood Methods with Weakly Dependent Processes, Ann. Statist. 25, 2084-2102.
39 陈希孺，王松桂．(1987)．近代回归分析．合肥，安徽教育出版社．
40 Chen, X. R. and Wu, Y. H. (1989). Strong Law for a Mixing Sequence. Acta Math. Appl. Sinica. 5, 367-371.
41 洪圣岩．(1993)，半参数回归模型的参数估计的Bootstrap强逼近，中国科学(A辑)，23(3)，239-251．
42 洪圣岩，成平(1993)．半参数回归模型的参数估计的bootstrap逼近．中国科学(A辑)，23(3)，239-251．
43 Lin, L. (2000). Conservative Estimating Function in Nonlinear Model with Aggregated Data. Mathematica Acta Sinientia. 20(3), 335-340.
44 Lin, L. and Zhang, R. C. (2001). Blockwise Empirical Euclidean Likeilihood for Weakly Dependent Processes. Statistics and Probability Letters. (To appear).
45 Lin, L., Yang, G. J. and Zhang, R. C. (2001). Analysis and Improvement of Quasiregression. 泛华统计协会第五次国际学术会议，香港．
46 林路．非线性模型中拟似然估计的若干性质．(1999)．应用数学学报．22(2)，307-310，
47 林路．半相依非线性回归方程的近似得分函数和近似拟得分函数．(1998)．应用数学．11(4)，123-126．
48 林路．非线性模型中拟似然估计和约束拟似然估计．(1998)数学物理学报．18(增刊)，133-138．
49 林路．数据变换参数渐近置信域的曲率表示．(1998)．高校应用数学学报．13(2)，153-158．
50 林路，张润楚．假设检验的相对稳定性．应用数学学报，已接受，待发表．
51 刘维奇，潘晋孝．聚集数据的线性模型参数的改进Peter-Karsten估计．(1995)．工程数学学报．12(2)，115-119．
52刘维奇，潘晋孝．聚集数据的线性模型中一种新的参数估计．(1996)．工程数学学报，13(4)，340-356．
53陆传荣，林正炎．(1997)．混合相依变量的极限理论．北京，科学出版社．
54罗旭．(1994)．半参数模型的经验欧氏似然估计的大样本性质．应用概率统计，10(3)，344-352．
55罗旭．(1997)．两参数半参数模型的经验欧氏似然估计．应用概率统计．13(2)，133-141．
56钱伟民，柴根象．(1999)．半参数回归模型小波估计的强逼近．中国科学(A辑)，29(3)，233-240．
57秦永松，赵林成．(1998)．两样本分位数差异的半经验似燃比检验，应用数学学报．21(1)，103-112．
58王松桂．线性模型的理论及其应用，(1987)，合肥，安徽教育出版社，
59王松桂．线性回归系统回归系数的一种新估计，(1988)．中国科学(A辑)，18(10)，1033-1040．
60韦博成．(1994)，指数族非线性模型的几何渐近推断理论．中国科学(A辑)，24(12)，1270-1278．
61韦博成．(1989)．近代非线性回归分析．南京，东南大学出版社，
62Wei,B.C,Exponential Family Nonlinear Models,(1998).Singapore,Springer-verlag.
63张润楚，王兆军．(1994)．关于计算机试验设计的设计理论和数据分析．应用概率统计，10(4)，421-435．
64张润楚．(2000)近代试验设计的一些新近展，南开大学技术报告．
65朱仲义．非线性模型中拟似然估计的小样本性质，(1996)．工程数学学报．13(2)，59-64．