- Email: [email protected]
Asymptotic linearity and limit distributions, approximations
ARTICLE IN PRESS Journal of Statistical Planning and Inference 140 (2010) 353–357
Contents lists available at ScienceDirect
Journal of Statistical Planning and Inference journal homepage: www.elsevier.com/locate/jspi
Asymptotic linearity and limit distributions, approximations ~ T. Mexia a, Manuela M. Oliveira b, Joao a
FCT Nova University of Lisbon, Department of Mathematics, Quinta da Torre, 2825 Monte da Caparica, Portugal ~ ´nio Verney, Rua Romao University of Evora, Department of Mathematics and CIMA – Center for Research on Mathematics and its Applications, Cole´gio Luı´s Anto ´ vora, Portugal Ramalho 59, 7000-671 E
b
a r t i c l e in fo
abstract
Article history: Received 20 January 2009 Received in revised form 13 September 2009 Accepted 14 September 2009 Available online 19 September 2009
Linear and quadratic forms as well as other low degree polynomials play an important role in statistical inference. Asymptotic results and limit distributions are obtained for a class of statistics depending on m þ X, with X any random vector and m non-random vector with JmJ- þ 1. This class contain the polynomials in m þ X. An application to the case of normal X is presented. This application includes a new central limit theorem which is connected with the increase of non-centrality for samples of fixed size. Moreover upper bounds for the suprema of the differences between exact and approximate distributions and their quantiles are obtained. & 2009 Elsevier B.V.. All rights reserved.
Keywords: Asymptotic linearity Linear and quadratic forms Polynomials Normal distributions Central limit theorems
Contents 1. 2. 3. 4. 5. 6.
Introduction . . . . . . . . . . . . . . Notations and concepts . . . . . Asymptotic results . . . . . . . . . Application: the normal case Final remarks . . . . . . . . . . . . . Uncited reference. . . . . . . . . .
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353 354 354 356 357 357
1. Introduction Linear and quadratic forms as well as other low degree polynomials play an important role in statistics (e.g. Ito, 1980; Lazraq and Cleroux, 2001; Oliveira and Mexia, 2007a, 2007b; Sabatier and Traissac, 1994; Scheffe´, 1959). In Areia et al. (2008) it is pointed out that low degree polynomials in normal independent variables, with sufficiently small variation coefficients, are approximately normal. Moreover, also in Areia et al. (2008), are presented simulations suggesting that, when JlJ- þ 1, a polynomial Pðl þ XÞ has a leading component linear in X. This linear component will be normal. We will obtain asymptotic approximations and limit distributions for a class of statistics containing Pðl þ XÞ. In the next section we will present the concept of asymptotic linear functions which will play a central role in our study. We will also introduce the relevant notations and point out that polynomials are asymptotically linear functions. Next we
Corresponding author. Tel.: þ351 266 745300; fax: þ351 266 744546.
E-mail address: [email protected] (M.M. Oliveira). 0378-3758/$ - see front matter & 2009 Elsevier B.V.. All rights reserved. doi:10.1016/j.jspi.2009.09.014
ARTICLE IN PRESS 354
J.T. Mexia, M.M. Oliveira / Journal of Statistical Planning and Inference 140 (2010) 353–357
use a variant of the d method (e.g. Lehmann and Casella, 2001) to obtain asymptotic and limit results. In the last section we will consider the case when m þ X will be normal with mean vector m. This variant will display a good control of the approximation errors both for distribution functions and their quantiles. If Vþ is the Moore–Penrose matrix inverse of the variance–covariance matrix V of m þ X, the quadratic form ðm þ t þ XÞ V ðm þ XÞ will have a chi-square distribution with r ¼ rankðVÞ degrees of freedom and non-centrality parameter d ¼ mt Vþ m, (e.g. Mexia, 1990). Thus we may use d to measure the non-centrality of the sample. In our study of the normal case we will obtain a central limit theorem, for fixed size samples whose non-centrality parameter diverges to þ1. Moreover d may be used to measure sample non-centrality whenever V, the variance–covariance matrix of X is defined, and d! þ 1 will imply JlJ- þ 1. Since our asymptotic and limit results are derived assuming that JlJ- þ 1, they are connected with the increase of non-centrality in samples, normal or not, with fixed size. When the components X1 ; . . . ; Xm of X are i.i.d. with mean value m and variance s2 we have l ¼ m1m and V ¼ s2 Im so Vþ ¼ ð1=s2 ÞIm and d ¼ mðm2 =s2 Þ ¼ m=VC 2 , with VC ¼ s=m the variation coefficient. Thus when the X1 ; . . . ; Xm are observations with low variation coefficients we have a sample with large d. The variation coefficient with high precision observations are used. 2. Notations and concepts Given a sufficiently regular function g : Rk -R, let g and g be the gradient and the Hessian matrix of g. Then, with rd ðxÞ the supremum of the spectral radius rðyÞ of g ðyÞ, when Jy xJrd, the function g is asymptotically linear, if whatever d40, we have kd ðuÞ - 0; u-1
with (
) rd ðxÞ ; JxJZu : Jg ðxÞJ
kd ðuÞ ¼ sup
Since the first and second order partial derivatives of polynomials are themselves polynomials with degree decreasing with successive derivations, polynomials will be asymptotically linear. Given a random vector X and an asymptotically linear function gðÞ, we will show that, when JlJ-1, the distribution FY of Y ¼ gðl þ XÞ approaches the distribution of Z ¼ gðlÞ þ g ðlÞt X: With lp the p th quantile of the random variable L, given Z3 ¼
ðZ gðlÞÞ Jg ðlÞJ
Y3 ¼
ðY gðlÞÞ Jg ðlÞJ
and
we will obtain upper bounds for jy3p z3p j. Moreover, when 1 g ðlÞ - b JmJ-1 Jg ðlÞJ t
we will establish that FZ 3 converges to FZ33 , with Z33 ¼ b X. As we shall see these results are easy to apply when X is normal. Then Z33 will also be normal and a central limit theorem will apply. 3. Asymptotic results If g : Rk -R is sufficiently regular we have, see Khuri (2003) Z 1 Y ¼ gðl þ XÞ ¼ gðlÞ þ g ðlÞt X þ ð1 hÞðXt g ðl þ hXÞXÞ dh: 0
Thus, when JXJrd, Z 1 jY Zj ¼ ð1 hÞðXt g ðl þ hXÞXÞ dhrrd ðlÞd2 0
ARTICLE IN PRESS J.T. Mexia, M.M. Oliveira / Journal of Statistical Planning and Inference 140 (2010) 353–357
355
as well as jY 3 Z 3 jrkd ðJlJÞd2 : We now establish a:s
Proposition 3.1. If gðÞ is asymptotically linear, then jY 3 Z 3 j - 0, where a.s. indicates almost sure convergence. JmJ-1
Proof. If gðÞ is asymptotically linear, whatever e40 and d40, there is uðeÞ such that, for u4uðeÞ, kd ðuÞoe. Thus, when JlJZuðeÞ, JXJod implies jY 3 Z 3 jre. Whatever d40, we may choose d40 to ensure that PrðJXJodÞZ1 d, thus when JlJZuðeÞ, we will also have PrðjY 3 Z 3 jreÞZ1 d which establishes the thesis. & Since kd ðuÞ decreases with u we may define uðd; eÞ ¼ Minfu; kd ðuÞd2 reg thus, with lq the p th quantile of the distribution of L ¼ JXJ2 , whenever Lrlq, JlJZuq ðeÞ ¼ uðlq ; eÞ implies jY 3 Z 3 jre: So, if qðeÞ is the probability of the event AðeÞ ¼ fjY 3 Z 3 jreg we see that qðeÞZq, when JlJ4uq ðeÞ. Let Ac ðeÞ be the complement of AðeÞ and qc ðeÞ ¼ 1 qðeÞ. We now have Proposition 3.2. Whatever z and e40, FZ3 ðz eÞ qc ðeÞrFY 3 ðzÞrFZ3 ðz þ eÞ þ qc ðeÞ. Proof. The thesis follows from FY 3 ðzÞ ¼ PrðY 3 rzÞ ¼ qðeÞPrðY 3 rzjAðeÞÞ þ qc ðeÞPrðY 3 rzjAc ðeÞÞrqðeÞPrðZ 3 rz þ ejAðeÞÞ þ qc ðeÞ and from FZ 3 ðz eÞ qc ðeÞrqðeÞPrðZ 3 rz ejAðeÞÞrqðeÞPrðY 3 rzjAðeÞÞrPrðY 3 rzÞ ¼ FY 3 ðzÞ:
&
Corollary 3.1. When FZ3 ðzÞ has density fZ 3 ðzÞ bounded by C, we have SupfjFY FZ jg ¼ SupfjFY 3 FZ 3 jgrdðeÞ with dðeÞ ¼ 2ðC e þ qðeÞÞ. Proof. Since FY ðxÞ ¼ FY 3
x gðlÞ Jg ðlÞJ
FZ ðxÞ ¼ FZ 3
x gðlÞ Jg ðlÞJ
!
and !
we have SupfjFY FZ jg ¼ SupfjFY 3 FZ3 g. To complete the proof we have only to apply Proposition 3.1 remembering that, since fZ3 is bounded by C; FZ 3 ðz þ eÞ FZ 3 ðz eÞr2C e. & Corollary 3.2. When fZ3 is bounded by C and gðÞ is asymptotically linear, SupfjFY FZ jg
-
0.
JmJ-þ1
This last corollary gives conditions for FZ to approach FY uniformly when JlJ!1. Next we have Proposition 3.3. If fZ 3 is bounded by C and if fZ3 ðzÞZmðdÞ40, whenever z3ddðeÞ rzrz31dþdðeÞ, we have dðeÞ : Supfjy3p z3p j; drpr1 dgr2 e þ mðdÞ Proof. According to Corollary 3.1 of Proposition 3.2, we have FZ 3 ðy3p eÞ dðeÞrFY 3 ðy3p Þ ¼ prFZ3 ðy3p þ eÞ þ dðeÞ so FZ3 ðy3p eÞrp þ dðeÞ and p dðeÞrFZ3 ðy3p þ eÞ. Thus y3p erzpþdðeÞ and z3pdðeÞ ry3p þ e, so z3pdðeÞ ery3p rz3pþdðeÞ þ e. To complete the proof we have only to point out that z3pþdðeÞ z3pdðeÞ r
2dðeÞ : mðdÞ
&
Corollary 3.3. If JlJZuq ðeÞ, fZ3 is bounded by C and fZ 3 ðzÞZmðdÞ40, then whenever z3ddðeÞ rzrz31dþdðeÞ, we have Supfjy3p z3p j; drpr1 dgr2ðe þ dq ðeÞÞmðdÞÞ; with dq ðeÞ ¼ 2ðC e þ qc Þ where qc ¼ 1 q.
ARTICLE IN PRESS 356
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Proof. The thesis follows from Proposition 3.3 since uq ðeÞ4q, when JlJ4uq ðeÞ.
&
We also have Proposition 3.4. If 1 g ðlÞ - b; JmJ-1 Jg ðlÞJ
FZ 3 - FZ33 JmJ-1
this convergence being uniform when fZ3 is bounded by C and gð:Þ is asymptotically linear. Proof. The first part of the thesis follows from !t 1 a:s: g ðlÞ b X - 0 JmJ-1 Jg ðlÞJ whenever 1 g ðlÞ - b: JmJ-1 Jg ðlÞJ The second part of the thesis follows from Proposition 3.3 since uq ðeÞ4q, when JlJ4uq ðeÞ.
&
4. Application: the normal case Let X be normal with null mean vector and regular variance–covariance matrix M. Then 2
fZ 3 ðzÞ ¼
2
t
eJg ðlÞJ ðzgðlÞÞ =g ðlÞ Mg ðlÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Jg ðlÞJ 2pg ðlÞt Mg ðlÞ
is bounded by Jg ðlÞJ CðlÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2pg ðlÞt Mg ðlÞ If yk is the smallest eigenvalue of M we have 1 CðlÞrC ¼ pffiffiffiffiffiffiffiffiffiffiffi : 2pyk With xp the quantile for probability p of the standardized normal density, the quantiles for fZ3 will be qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g ðlÞt Mg ðlÞ z3p ðlÞ ¼ g ðlÞ þ xp ; Jg ðlÞJ while the minimum of fZ 3 in ½z3ddðeÞ ðlÞ; z31dþdðeÞ ðlÞ will be x2
=2
e ddðeÞ mðdjlÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Jg ðlÞJ 2pg ðlÞt Mg ðlÞ so that CðlÞ x2 =2 ¼ e ddðeÞ mðdjlÞ and 0 1 ! qc ðeÞJg ðlÞJ C dðejlÞ 2ðCðlÞe þ qc ðeÞÞ qc ðeÞ x2ddðeÞ =2 B x2 =2 ¼ ¼ 2e e þ pffiffiffiffiffiffiffiffiffiffiffi : @e þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAr2e ddðeÞ mðdjlÞ mðdjlÞ 2pyk 2pg ðlÞt Mg ðlÞ We thus obtain bounds that do not depend on l, both for fZ3 ðzÞ and for dðejlÞ=mðdjlÞ. It is now easy to apply the previous results to this case. Moreover if X is normal with null mean vector and variance–covariance matrix M and if y1 Z; . . . ; Zyh are the eigenvalues of M with multiplicities g1 ; . . . ; gh , L ¼ JXJ2
P may be written (see Imhof, 1961) as a linear combination hj¼1 yj w2gj , of independent chi-square variates. When M is known it is easy to compute the quantiles lp (Fonseca et al., 2007).
ARTICLE IN PRESS J.T. Mexia, M.M. Oliveira / Journal of Statistical Planning and Inference 140 (2010) 353–357
357
5. Final remarks In this work we derived asymptotic and limit results for samples, with fixed size, whose non-centrality diverges to þ1. In the case of normal samples a central limit theorem of a new type is established. Our approach is a variant of the d method. This variant will apply to asymptotically linear statistics gðl þ XÞ such as the polynomials Pðl þ XÞ and leads to a good control of the approximation involved (see Corollary 3.1, Proposition 3.2 and Corollary 3.3). References Areia, A., Oliveira, M.M., Mexia, J.T., 2008. Models for series of studies based on geometrical representation. Statistical Methodology 1, 277–288. Fonseca, M., Mexia, J.T., Zmys´lony, R., 2007. Jordan Algebras. Generating pivot variables and orthogonal normal models. Journal of Interdisciplinary Mathematics 1, 305–326. Imhof, J.P., 1961. Computing the distribution of quadratic forms in normal variables. Biometrics 48 (3–4), 419–426. Ito, K., 1980. Robustness of anova and manocova test procedures. In: Krishnaiah, P.R. (Ed.), Handbook of Statistics, vol. I. North Holland, Amsterdam. Khuri, A.I., 2003. Advanced Calculus with Applications in Statistics, second ed. Wiley, New York. Lazraq, A., Cleroux, R., 2001. The PLS multivariate regression model: testing the significance of successive PLS components. Journal of Chemometrics 15, 523–536. Lehmann, E.L., Casella, G., 2001. Theory of Point Estimation. Springer, Berlin. Mexia, J.T., 1990. Best linear unbiased estimates, duality of F tests and the Scheffe´ multiple comparison method in presence of controlled heteroscedasticity. Computational Statistics & Data Analysis 10, 3. Oliveira, M.M., Mexia, T.J., 2007a. ANOVA like analysis of matched series of studies with a common structure. Journal of Statistical Planning and Inference 137, 1862–1870. Oliveira, M.M., Mexia, J.T., 2007b. Modelling series of studies with a common structure. Computational Statistics and Data Analysis 51, 5876–5885. Sabatier, R., Traissac, P., 1994. The ACT (STATIS method). Computational Statistics & Data Analysis 18, 97–119. Scheffe´, H., 1959. The Analysis of Variance. Wiley, New York.
Contents lists available at ScienceDirect
Journal of Statistical Planning and Inference journal homepage: www.elsevier.com/locate/jspi
Asymptotic linearity and limit distributions, approximations ~ T. Mexia a, Manuela M. Oliveira b, Joao a
FCT Nova University of Lisbon, Department of Mathematics, Quinta da Torre, 2825 Monte da Caparica, Portugal ~ ´nio Verney, Rua Romao University of Evora, Department of Mathematics and CIMA – Center for Research on Mathematics and its Applications, Cole´gio Luı´s Anto ´ vora, Portugal Ramalho 59, 7000-671 E
b
a r t i c l e in fo
abstract
Article history: Received 20 January 2009 Received in revised form 13 September 2009 Accepted 14 September 2009 Available online 19 September 2009
Linear and quadratic forms as well as other low degree polynomials play an important role in statistical inference. Asymptotic results and limit distributions are obtained for a class of statistics depending on m þ X, with X any random vector and m non-random vector with JmJ- þ 1. This class contain the polynomials in m þ X. An application to the case of normal X is presented. This application includes a new central limit theorem which is connected with the increase of non-centrality for samples of fixed size. Moreover upper bounds for the suprema of the differences between exact and approximate distributions and their quantiles are obtained. & 2009 Elsevier B.V.. All rights reserved.
Keywords: Asymptotic linearity Linear and quadratic forms Polynomials Normal distributions Central limit theorems
Contents 1. 2. 3. 4. 5. 6.
Introduction . . . . . . . . . . . . . . Notations and concepts . . . . . Asymptotic results . . . . . . . . . Application: the normal case Final remarks . . . . . . . . . . . . . Uncited reference. . . . . . . . . .
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353 354 354 356 357 357
1. Introduction Linear and quadratic forms as well as other low degree polynomials play an important role in statistics (e.g. Ito, 1980; Lazraq and Cleroux, 2001; Oliveira and Mexia, 2007a, 2007b; Sabatier and Traissac, 1994; Scheffe´, 1959). In Areia et al. (2008) it is pointed out that low degree polynomials in normal independent variables, with sufficiently small variation coefficients, are approximately normal. Moreover, also in Areia et al. (2008), are presented simulations suggesting that, when JlJ- þ 1, a polynomial Pðl þ XÞ has a leading component linear in X. This linear component will be normal. We will obtain asymptotic approximations and limit distributions for a class of statistics containing Pðl þ XÞ. In the next section we will present the concept of asymptotic linear functions which will play a central role in our study. We will also introduce the relevant notations and point out that polynomials are asymptotically linear functions. Next we
Corresponding author. Tel.: þ351 266 745300; fax: þ351 266 744546.
E-mail address: [email protected] (M.M. Oliveira). 0378-3758/$ - see front matter & 2009 Elsevier B.V.. All rights reserved. doi:10.1016/j.jspi.2009.09.014
ARTICLE IN PRESS 354
J.T. Mexia, M.M. Oliveira / Journal of Statistical Planning and Inference 140 (2010) 353–357
use a variant of the d method (e.g. Lehmann and Casella, 2001) to obtain asymptotic and limit results. In the last section we will consider the case when m þ X will be normal with mean vector m. This variant will display a good control of the approximation errors both for distribution functions and their quantiles. If Vþ is the Moore–Penrose matrix inverse of the variance–covariance matrix V of m þ X, the quadratic form ðm þ t þ XÞ V ðm þ XÞ will have a chi-square distribution with r ¼ rankðVÞ degrees of freedom and non-centrality parameter d ¼ mt Vþ m, (e.g. Mexia, 1990). Thus we may use d to measure the non-centrality of the sample. In our study of the normal case we will obtain a central limit theorem, for fixed size samples whose non-centrality parameter diverges to þ1. Moreover d may be used to measure sample non-centrality whenever V, the variance–covariance matrix of X is defined, and d! þ 1 will imply JlJ- þ 1. Since our asymptotic and limit results are derived assuming that JlJ- þ 1, they are connected with the increase of non-centrality in samples, normal or not, with fixed size. When the components X1 ; . . . ; Xm of X are i.i.d. with mean value m and variance s2 we have l ¼ m1m and V ¼ s2 Im so Vþ ¼ ð1=s2 ÞIm and d ¼ mðm2 =s2 Þ ¼ m=VC 2 , with VC ¼ s=m the variation coefficient. Thus when the X1 ; . . . ; Xm are observations with low variation coefficients we have a sample with large d. The variation coefficient with high precision observations are used. 2. Notations and concepts Given a sufficiently regular function g : Rk -R, let g and g be the gradient and the Hessian matrix of g. Then, with rd ðxÞ the supremum of the spectral radius rðyÞ of g ðyÞ, when Jy xJrd, the function g is asymptotically linear, if whatever d40, we have kd ðuÞ - 0; u-1
with (
) rd ðxÞ ; JxJZu : Jg ðxÞJ
kd ðuÞ ¼ sup
Since the first and second order partial derivatives of polynomials are themselves polynomials with degree decreasing with successive derivations, polynomials will be asymptotically linear. Given a random vector X and an asymptotically linear function gðÞ, we will show that, when JlJ-1, the distribution FY of Y ¼ gðl þ XÞ approaches the distribution of Z ¼ gðlÞ þ g ðlÞt X: With lp the p th quantile of the random variable L, given Z3 ¼
ðZ gðlÞÞ Jg ðlÞJ
Y3 ¼
ðY gðlÞÞ Jg ðlÞJ
and
we will obtain upper bounds for jy3p z3p j. Moreover, when 1 g ðlÞ - b JmJ-1 Jg ðlÞJ t
we will establish that FZ 3 converges to FZ33 , with Z33 ¼ b X. As we shall see these results are easy to apply when X is normal. Then Z33 will also be normal and a central limit theorem will apply. 3. Asymptotic results If g : Rk -R is sufficiently regular we have, see Khuri (2003) Z 1 Y ¼ gðl þ XÞ ¼ gðlÞ þ g ðlÞt X þ ð1 hÞðXt g ðl þ hXÞXÞ dh: 0
Thus, when JXJrd, Z 1 jY Zj ¼ ð1 hÞðXt g ðl þ hXÞXÞ dhrrd ðlÞd2 0
ARTICLE IN PRESS J.T. Mexia, M.M. Oliveira / Journal of Statistical Planning and Inference 140 (2010) 353–357
355
as well as jY 3 Z 3 jrkd ðJlJÞd2 : We now establish a:s
Proposition 3.1. If gðÞ is asymptotically linear, then jY 3 Z 3 j - 0, where a.s. indicates almost sure convergence. JmJ-1
Proof. If gðÞ is asymptotically linear, whatever e40 and d40, there is uðeÞ such that, for u4uðeÞ, kd ðuÞoe. Thus, when JlJZuðeÞ, JXJod implies jY 3 Z 3 jre. Whatever d40, we may choose d40 to ensure that PrðJXJodÞZ1 d, thus when JlJZuðeÞ, we will also have PrðjY 3 Z 3 jreÞZ1 d which establishes the thesis. & Since kd ðuÞ decreases with u we may define uðd; eÞ ¼ Minfu; kd ðuÞd2 reg thus, with lq the p th quantile of the distribution of L ¼ JXJ2 , whenever Lrlq, JlJZuq ðeÞ ¼ uðlq ; eÞ implies jY 3 Z 3 jre: So, if qðeÞ is the probability of the event AðeÞ ¼ fjY 3 Z 3 jreg we see that qðeÞZq, when JlJ4uq ðeÞ. Let Ac ðeÞ be the complement of AðeÞ and qc ðeÞ ¼ 1 qðeÞ. We now have Proposition 3.2. Whatever z and e40, FZ3 ðz eÞ qc ðeÞrFY 3 ðzÞrFZ3 ðz þ eÞ þ qc ðeÞ. Proof. The thesis follows from FY 3 ðzÞ ¼ PrðY 3 rzÞ ¼ qðeÞPrðY 3 rzjAðeÞÞ þ qc ðeÞPrðY 3 rzjAc ðeÞÞrqðeÞPrðZ 3 rz þ ejAðeÞÞ þ qc ðeÞ and from FZ 3 ðz eÞ qc ðeÞrqðeÞPrðZ 3 rz ejAðeÞÞrqðeÞPrðY 3 rzjAðeÞÞrPrðY 3 rzÞ ¼ FY 3 ðzÞ:
&
Corollary 3.1. When FZ3 ðzÞ has density fZ 3 ðzÞ bounded by C, we have SupfjFY FZ jg ¼ SupfjFY 3 FZ 3 jgrdðeÞ with dðeÞ ¼ 2ðC e þ qðeÞÞ. Proof. Since FY ðxÞ ¼ FY 3
x gðlÞ Jg ðlÞJ
FZ ðxÞ ¼ FZ 3
x gðlÞ Jg ðlÞJ
!
and !
we have SupfjFY FZ jg ¼ SupfjFY 3 FZ3 g. To complete the proof we have only to apply Proposition 3.1 remembering that, since fZ3 is bounded by C; FZ 3 ðz þ eÞ FZ 3 ðz eÞr2C e. & Corollary 3.2. When fZ3 is bounded by C and gðÞ is asymptotically linear, SupfjFY FZ jg
-
0.
JmJ-þ1
This last corollary gives conditions for FZ to approach FY uniformly when JlJ!1. Next we have Proposition 3.3. If fZ 3 is bounded by C and if fZ3 ðzÞZmðdÞ40, whenever z3ddðeÞ rzrz31dþdðeÞ, we have dðeÞ : Supfjy3p z3p j; drpr1 dgr2 e þ mðdÞ Proof. According to Corollary 3.1 of Proposition 3.2, we have FZ 3 ðy3p eÞ dðeÞrFY 3 ðy3p Þ ¼ prFZ3 ðy3p þ eÞ þ dðeÞ so FZ3 ðy3p eÞrp þ dðeÞ and p dðeÞrFZ3 ðy3p þ eÞ. Thus y3p erzpþdðeÞ and z3pdðeÞ ry3p þ e, so z3pdðeÞ ery3p rz3pþdðeÞ þ e. To complete the proof we have only to point out that z3pþdðeÞ z3pdðeÞ r
2dðeÞ : mðdÞ
&
Corollary 3.3. If JlJZuq ðeÞ, fZ3 is bounded by C and fZ 3 ðzÞZmðdÞ40, then whenever z3ddðeÞ rzrz31dþdðeÞ, we have Supfjy3p z3p j; drpr1 dgr2ðe þ dq ðeÞÞmðdÞÞ; with dq ðeÞ ¼ 2ðC e þ qc Þ where qc ¼ 1 q.
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Proof. The thesis follows from Proposition 3.3 since uq ðeÞ4q, when JlJ4uq ðeÞ.
&
We also have Proposition 3.4. If 1 g ðlÞ - b; JmJ-1 Jg ðlÞJ
FZ 3 - FZ33 JmJ-1
this convergence being uniform when fZ3 is bounded by C and gð:Þ is asymptotically linear. Proof. The first part of the thesis follows from !t 1 a:s: g ðlÞ b X - 0 JmJ-1 Jg ðlÞJ whenever 1 g ðlÞ - b: JmJ-1 Jg ðlÞJ The second part of the thesis follows from Proposition 3.3 since uq ðeÞ4q, when JlJ4uq ðeÞ.
&
4. Application: the normal case Let X be normal with null mean vector and regular variance–covariance matrix M. Then 2
fZ 3 ðzÞ ¼
2
t
eJg ðlÞJ ðzgðlÞÞ =g ðlÞ Mg ðlÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Jg ðlÞJ 2pg ðlÞt Mg ðlÞ
is bounded by Jg ðlÞJ CðlÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2pg ðlÞt Mg ðlÞ If yk is the smallest eigenvalue of M we have 1 CðlÞrC ¼ pffiffiffiffiffiffiffiffiffiffiffi : 2pyk With xp the quantile for probability p of the standardized normal density, the quantiles for fZ3 will be qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g ðlÞt Mg ðlÞ z3p ðlÞ ¼ g ðlÞ þ xp ; Jg ðlÞJ while the minimum of fZ 3 in ½z3ddðeÞ ðlÞ; z31dþdðeÞ ðlÞ will be x2
=2
e ddðeÞ mðdjlÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Jg ðlÞJ 2pg ðlÞt Mg ðlÞ so that CðlÞ x2 =2 ¼ e ddðeÞ mðdjlÞ and 0 1 ! qc ðeÞJg ðlÞJ C dðejlÞ 2ðCðlÞe þ qc ðeÞÞ qc ðeÞ x2ddðeÞ =2 B x2 =2 ¼ ¼ 2e e þ pffiffiffiffiffiffiffiffiffiffiffi : @e þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAr2e ddðeÞ mðdjlÞ mðdjlÞ 2pyk 2pg ðlÞt Mg ðlÞ We thus obtain bounds that do not depend on l, both for fZ3 ðzÞ and for dðejlÞ=mðdjlÞ. It is now easy to apply the previous results to this case. Moreover if X is normal with null mean vector and variance–covariance matrix M and if y1 Z; . . . ; Zyh are the eigenvalues of M with multiplicities g1 ; . . . ; gh , L ¼ JXJ2
P may be written (see Imhof, 1961) as a linear combination hj¼1 yj w2gj , of independent chi-square variates. When M is known it is easy to compute the quantiles lp (Fonseca et al., 2007).
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5. Final remarks In this work we derived asymptotic and limit results for samples, with fixed size, whose non-centrality diverges to þ1. In the case of normal samples a central limit theorem of a new type is established. Our approach is a variant of the d method. This variant will apply to asymptotically linear statistics gðl þ XÞ such as the polynomials Pðl þ XÞ and leads to a good control of the approximation involved (see Corollary 3.1, Proposition 3.2 and Corollary 3.3). References Areia, A., Oliveira, M.M., Mexia, J.T., 2008. Models for series of studies based on geometrical representation. Statistical Methodology 1, 277–288. Fonseca, M., Mexia, J.T., Zmys´lony, R., 2007. Jordan Algebras. Generating pivot variables and orthogonal normal models. Journal of Interdisciplinary Mathematics 1, 305–326. Imhof, J.P., 1961. Computing the distribution of quadratic forms in normal variables. Biometrics 48 (3–4), 419–426. Ito, K., 1980. Robustness of anova and manocova test procedures. In: Krishnaiah, P.R. (Ed.), Handbook of Statistics, vol. I. North Holland, Amsterdam. Khuri, A.I., 2003. Advanced Calculus with Applications in Statistics, second ed. Wiley, New York. Lazraq, A., Cleroux, R., 2001. The PLS multivariate regression model: testing the significance of successive PLS components. Journal of Chemometrics 15, 523–536. Lehmann, E.L., Casella, G., 2001. Theory of Point Estimation. Springer, Berlin. Mexia, J.T., 1990. Best linear unbiased estimates, duality of F tests and the Scheffe´ multiple comparison method in presence of controlled heteroscedasticity. Computational Statistics & Data Analysis 10, 3. Oliveira, M.M., Mexia, T.J., 2007a. ANOVA like analysis of matched series of studies with a common structure. Journal of Statistical Planning and Inference 137, 1862–1870. Oliveira, M.M., Mexia, J.T., 2007b. Modelling series of studies with a common structure. Computational Statistics and Data Analysis 51, 5876–5885. Sabatier, R., Traissac, P., 1994. The ACT (STATIS method). Computational Statistics & Data Analysis 18, 97–119. Scheffe´, H., 1959. The Analysis of Variance. Wiley, New York.