广义估计方程根的强相合性_尹长明
2012年12月 广西师范学院学报:自然科学版
第29卷第4期 Journal of Guangxi Teachers Education University:Natural Science Edition
Dec. 2012Vol. 29No. 4
文章编号:1002-8743(2012) 04-0034-03
广义估计方程根的强相合性
尹长明, 陈莉莉, 颜然
*
(广西大学数学与信息科学学院, 广西南宁530004)
摘 要:在广义估计方程有正确的假设下, 当个体数目趋于无穷且重复观测次数也趋于无穷时, 利用Wu (1981) 引理2证明了广义估计方程根的强相合性1这一结果是对文献中的相应结果进行的改进1
关键词:广义估计方程; 根; 强相合性中图分类号:O21211 文献标识码:A
1 引言
广义估计方程(GEE) 理论是对广义线性模型理论的推广1自从Liang 和Zenger 在生物学上引入此方法以来, GEE 已被应用到许多领域1
假设(y ij , x ij ) 是第i 个个体的第j 次观测值, j =1, , , m , i =1, , , n, y ij 是一维响应变量, x ij 是p @1维协变量, m i 是第i 个个体的观测次数1假设不同个体的观测值是独立的, 而同一个体内部的观测值是相关的1y i =(y i 1, , , y im i ) T , x i =(x i 1, , , x i mi ) T 1Liang 和Zenger 用广义线性模型去建立y ij 的边缘密度为
f y ij x ij , B ,
T
y ij H ij -a(H ij ) +b(y ij )
P
/, (1)
其中:H ij =u (x ij B ) , u 为已知单射函数1B I R 是我们所要估计的未知参数, B 0是它的真值,
L ) =E(y i j x ij , B ,
记g (t) =(a c b u ) 同时记
2
A i (B ) =diag (R 2i 1, , , R im i ) ,
-1
(2) (3)
(t ) , 则g(L ij (B ) ) =x ij B , 函数g (t ) 被称为联系函数1定义:
T
T L ) =E (y i X i , B ,
$i (B ) =diag (u c (x T ) , , , u c (x T ) ) , i 1B im B i D i (B ) =A i (B ) $i (B ) X i ,
/2/2V i (B , A ) =A 1) R i (A ) A 1) , i (B i (B
其中对任意的向量v , diag (v ) 表示由v 的对角元素构成的对角矩阵1当R i (A ) =R i 时, V i (B 0, A ) =r i (B 0) , 其中R i 为真实协方差阵(通常是未知的) 1
文献[1]建议求解以下方程(广义估计方程, GEE ) :g nm (B ) =
i=1
E g m , i (B ) =E D i (B ) V i
i
n n
-1
i =1
(B , A ) (y i -L i (B ) ) =0(4)
*
收稿日期:2012-11-02
*基金项目:国家自然科学基金资助项目(11061102)
( , , ,
第4期 尹长明, 等:广义估计方程根的强相合性
C
# 35#
的解B nm 1注意到解方程(4) 未用到个体观测值的联合分布1由于方程(4) 只依赖于个体观测值的边际期望和方差, 因此更一般地可以忽略(1) 的边缘密度假设, 只要有期望和方差的正确假设即(2) 和(3) 就行了1
为了叙述简洁, 我们定义
M nm (B ) =cov (g nm (B ) ) =
i =1
E D i (B ) V i
T
n
-1
(B , A ) r i (B ) V -i 1(B , A ) D i (B ) ,
9g nm (B )
, 9B
(5)
D nm (B ) =-其中M nm (B ) 为正定阵, 但是D nm (B ) 不是对称阵1
在本文中, c 在不同的位置代表不同的独立于n 的正的常数1#表示Fresenius 范数, K min (A ) 和K max (A ) 表示矩阵A 的最小和最大特征根, g nm (B 0) 和D nm (B 0) 等简写为g nm 和D nm 等1A 矩阵A 的平方根, 并满足(A 1/2) 2=A 及A -C
1/2
表示正定
1/2
=(A 1/2) -11
关于广义估计方程根B nm 的强相合性, 在[1]及以后的文献中多在当个体之间相互独立及个体数目n y ], m i 为第i 个个体的重复观测次数有界或者以有限的速度, 随着n y ]的情形下讨论1例如可参见文献[1]~[7]1其中Xie 和Yang [6]在n y ], m y ], g nm 为强无穷小双阵序列或者n y ], m 有界等条件下, 比较了g nm 和log n 之间的收敛速度, 从而证明了广义估计方程根的强相合性1本文将条件减弱为:当n y ], m y ]时, 无需g nm 为强无穷小双阵序列, 也无需估计g nm 与log n 比较收敛速度, 在其他假设条件成立的情况下, 利用Wu (1981) 引理2[9], 证明了广义估计方程根的强相合性1
我们对广义估计方程建立如下主要结果:定理1 假定B 0是真参数, 下列条件成立:(A1) 对D >0, n y ], m y ]有
K m i n (M nm ) y ];
(A2) 存在一个正的常数c, 使得
T
K D nm (B ) K \c K max (M nm )
(6) (7) (8)
且对P E >0, 以及当B I N (E ) ={B B B -B 0[E }时,
D nm (B ) 几乎处处非奇异,
则存在一随机变量序列B nm 和一随机数n 0, 使得
P g nm B nm =0, n \n 0=1
并且当n y ]时,
B nm y B 0 a. s. 1
C
C
C
(9) (10)
2 定理1的证明
为了证明定理1, 我们需要下面的一些引理1
引理1[8] 设H (B ) 是R P y R P 的连续单射, H (B 0) =a, 又
H (B ) -a B B -B 0=c >r ,
则对任何的b, b -a [r , 存在B , B -B 0=c 使得H (B ) =b 1
引理2
[9]
{X n }为随机序列, EX i =0,
i=1
E var (X i )
n
y ], 有
E X i
n
# 36#
广西师范学院学报:自然科学版 第29卷
引理3 若(A1) 成立, 则
e g nm
y 0, 1/2+D
K m ax (M nm )
其中e 为任意p @1维向量, 且e T e =11
证明 由(4) 和(5) 知
v ar (e T g nm ) =e T M nm e [K max (M nm ) ,
E(e T g nm ) =0,
由(6) 及(12) 知
i=1
T
(11)
(12) (13)
E var (e
n
T
g nm ) y ], (14)
由(13) 、(14) 及引理2知(11) 成立1
引理4[6] 设c 为p @p 矩阵, 任意的p @1维向量K , K =1, 我们有
T T T K c c K \(K c K ) 21
(15)
定理的证明 由向量值中值定理可得
g nm (B ) -g nm =D nm ( B ) (B -B 0) , 其中 B 在B 与B 0之间的连线上1从而
inf ) g nm (B ) -g nm =B I 9N (E
=
inf ) E B I 9N (E inf
(B -B 0) T
E
T
nm (
B ) (B -B 0) =inf ) D nm ( B I 9N (E
(B -B 0)
E
1/2
T T
(B -B B ) D nm ( B ) (B -B 0) inf 0) D nm ( B I 9N (E )
1/2
B ) D nm ( B )
T T K D B ) D nm ( B ) \B I inf nm ( N (E )
1/2
,
由(7) 和(15) 知
B I 9N (E )
T 1/2+D
g nm (B ) -g nm \E (K D nm ( B ) K ) \c K max (M nm ) ,
(16)
其中c =c 0E 1
由(11) 知
1/2+D
e T g nm [c K max (M nm ) ,
(17) (18)
由(16) 和(17) 得
inf g nm (B ) -g nm \e g nm #B I 9N (E )
C
C
C
T
由(8) 可知, T B B y g nm (B ) 是从N (E ) 到T (N (E ) ) 的一个连续单射, 所以由(18) 及引理1可知存在B nm I N (E ) 使得g nm B nm =0几乎处处成立, 因此(9) 成立, 而由B nm I N (E ) ={B B B -B 0[E }知, (10) 显然成立1定理全部得证1B nm -B 0[E
C
参考文献:
[1]L IAN G K Y, ZEN GER S L. L ong itudinal data analysis using gener alized linear models[J].Biometrika, 1986, 73:13-22. [2]CROW DER M. On consistency and inconsistency of estimat ing equations[J]. Econometric T heory , 1986, 2:305-330. [3]L I B A. M inimax approach to consistency and efficiency equations [J]. Ann Statist, 1996, 24:1283-1297. [4]G RA M ER H. M athematical M ethods of Statistics[M ]. Pr inceton:Princeton U niv Press, 1947:113-321.
[5]Y UA N K H, JENN RICH R I. Asy mptotic of estimating equations under natural conditions[J].M ultiv ar iate Anal, 1998,
65:245-260.
[6]XI E M inge, Y AN G Y aning. Asy mptotic for generalized estimating equations w ith large cluster size[J]. T he Annals of
Statistics, 2003, 31(1) :310-347.
[7]BA LA N R M , KAT INA I S. Asy mptotic results with generalized estimating equations for longitudinal data[J].T he An -nals of Statistics, 2005, 33(2) :525-541.
[8]DU GU N DJI J. T opolog y[M ]. Boston:Ellyn and Bacon, 1996.
[9]W U C F. Asy mptotic theory of nonlinear least squar es est imation[J]. Ann Statist, 9:501-513.
[页]
第3期 车良革, 等:1991-2009年南流江流域植被覆盖时空变化及其与地质相关分析
# 59#
Spatial and Temporal Variations of Vegetation Cover Degrees
in the Nanliujiang River Watershed from 1991to 2009
CHE Liang -ge, HU Bao -qing, LI Yue -lian
(School of Resources and Environmental Science,
Guang xi Teachers Education U niversity, Nanning 530001, China)
Abstract:With 1991, 2000and 2009, the three years of TM im ages as a data source, this paper calcu -lates and analyses the characteristics of the change and spatial distribution about the vegetation in the Nanl-i ujiang River w atershed in 1991-2009by using the dimidiate pix el model based on NDV I. Overlaying the
vegetation coverage m ap and the divided lithology geological map, the impact on vegetation coverage exerted by geological structure is analyzed. The results can be described in the follow ing three aspects:(1) Vegeta -tion coverage of the Nanliujiang River w atershed changes significantly during the recent 19years. The river w atershed vegetation quality declined in the first 10years, but the average vegetation coverag e in the river w atershed is on the rise since 2000. (2) Vegetation coverag e and geological strata w ithin the river watershed are closely linked. Vegetation coverage of g ranite (C ) formation is hig her than other strata, w hile in the area of karst, vegetation coverage is lower and vegetation is prone to degraded, but recovers slow ly; the vegetation coverage in the Quaternary strata of this basin is the lowest. (3) the Nanliujiang River watershed vegetation mainly keeps stable and being restored, w hich has been g reatly im proved. The change of vegeta -tion coverage . s spatial distribution is significant in the Nanliujiang River w atershed.
Key Words:vegetation fractional cover; NDVI; dim idiate pixel model; the Nanliujiang w atershed
[责任编辑:黄天放]
[上接第36页]
Strong Consistency of Generalized Estimation Equation Root
YIN Chang -ming, CHEN L-i li, YAN Ran
(School of Mathematics and Information Science, Guangxi U niversity, Nanning 530004, P. R. China) Abstract:The paper studies strong consistency of generalized estimation equation root. If the genera-l ized estimating equation has correct assum ption, w hen the number of subjects and cluster sizes both g o to infinity, by using Lemma 2of Wu(1981) this paper proves strong consistency of generalized estimation e -quation root. The result is the improvement of the same location of the literature.
Key Words:generalized estimation equation; root; strong consistency
[责任编辑:班秀和]