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A Robust Procedure in Nonlinear Models for Repeated Measurements

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Nonlinear regression models arise when definite information is available about the form of the relationship between the response and predictor variables. Such information might involve direct knowledge of the actual form of the true model or might be
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  Communications in Statistics—Theory and Methods , 38: 138–155, 2009Copyright © Taylor & Francis Group, LLCISSN: 0361-0926 print/1532-415X onlineDOI: 10.1080/03610920802074836 A Robust Procedure in Nonlinear Modelsfor Repeated Measurements ANTONIO SANHUEZA 1 , PRANAB K. SEN 2 ,AND VÍCTOR LEIVA 3 1 Department of Mathematics and Statistics,Universidad de La Frontera, Temuco, Chile 2 Department of Biostatistics, University of North Carolina,North Carolina, USA 3 Department of Statistics, CIMFAV,Universidad de Valparaíso, Chile  Nonlinear regression models arise when definite information is available about the form of the relationship between the response and predictor variables. Suchinformation might involve direct knowledge of the actual form of the true modelor might be represented by a set of differential equations that the model must satisfy. We develop M-procedures for estimating parameters and testing hypothesesof interest about these parameters in nonlinear regression models for repeated measurement data. Under regularity conditions, the asymptotic properties of the M-procedures are presented, including the uniform linearity, normality and consistency. The computation of the M-estimators of the model parameters is performed with iterative procedures, similar to Newton–Raphson and Fisher’sscoring methods. The methodology is illustrated by using a multivariate logisticregression model with real data, along with a simulation study. Keywords  Consistency; M-estimators; M-tests; Normality; Uniform asymptoticlinearity; Wald-type tests. Mathematics Subject Classification  Primary 62J02; Secondary 62G35. 1. Introduction Repeated measurements occur frequently in many scientific fields where statisticalmodels are employed. For instance, in agriculture, agriculture-crops yield indifferent fields over different years; in biology, growth curves; in education, student Received May 23, 2007; Accepted March 24, 2008Address correspondence to Antonio Sanhueza, Departamento de Matemática yEstadística, Universidad de La Frontera, Casilla 54-D, Temuco, Chile; E-mail: asanhueza@ufro.cl 138   Robustness in Nonlinear Models 139 progress under various learning conditions; in medicine, successive periods of illness and recovery under different treatment regimens; and so on. The primarycharacteristic of repeated measurements that distinguishes them is the fact thatmore than one observation on the same response variable is available on eachobservational unit.The analysis of nonlinear models for repeated measurements generallyconsists of estimating the unknown parameters and testing hypothesis aboutthese parameters. In many cases, estimators of the parameters in univariate andmultivariate nonlinear models are based on classical methods of estimation, such asmaximum likelihood (ML) and least squares (LS) procedures. However, estimatorsand test statistics based on these methods are usually non robust to outliers ordepartures from the specified distribution of the error term in the model. Thus,from considerations of robustness (again outlier or for the error contaminationin a nonlinear model), the classical procedures are not very desirable and robustestimation procedures, such as M-procedures (ML type), are better in this respect.In the univariate linear model, a variety of robust methods based on M-, L-, andR- estimators have been proposed; see Jurecková and Sen (1996). The M-estimatorsinclude ML and LS methods as special cases and also allow for the constructionof estimators and tests that are robust against departures from normality.For the multivariate linear model, robust M-estimators have been proposedfor estimating its unknown parameters and for testing hypotheses about theseparameters. Maronna (1976) proposed a simultaneous M-estimation procedure forthe multivariate location and scatter parameters, under the assumption of ellipticalsymmetry of the underlying distribution. Singer and Sen (1985) developed two typesof M-estimators for multivariate linear models, coordinatewise M-estimators and anextension of the Maronna-type M-estimator.The objective of this study is to consider M-procedures for estimating andtesting the parameters of the nonlinear regression model for repeated measurements.The proposed M-procedures are formulated along the lines of generalized leastsquare procedures. It may be seen as an extension of the M-procedures that wasstudied in univariate nonlinear models (see Sanhueza and Sen, 2001, 2004), andalso as a generalization of the robust methods for the multivariate linear model(see Maronna, 1976). Under some regularity conditions, the asymptotic propertiesof the M-procedures are presented, such as uniform asymptotic linearity, normalityand consistency. The computation of the M-estimators of the model parameters isperformed by using iterative procedures, similar to Newton–Raphson and Fisher’sscoring methods. The methodology is illustrated by means of an analysis of repeatedmeasurement data from a study of the combined effects of hepatotoxins in whichbetween and within subject measurements are collected. Also, a simulation study iscarried out assuming a multivariate nonlinear model.In Sec. 2, we define the M-estimators of the parameters of the generalmultivariate nonlinear model. In Sec. 3, we develop the asymptotic properties of theM-estimators. In Sec. 4, we consider hypothesis testing of interest on the parametersof the considered model by using robust M-tests. Also, we present an iterativecomputational method for computing the M-estimators. Finally, Sec. 5 illustratesa numerical application of the developed M-procedures by using a multivariatelogistic regression model with real data. Furthermore, a small simulation study isconducted in order to examine the performance of the developed M-estimators.  140 Sanhueza et al. 2. Definition of the Robust Estimators We consider the nonlinear regression model for repeated measurements: Y i  = f  i    + e i  i = 1 n  (1)where  Y i  = Y  i 1 Y  ip  ⊤  is a  p × 1 vector of repeated measurements;  f  i    = f x i 1   f x ip    ⊤  is a  p × 1 vector of nonlinear functions of     of specifiedform;  x ij   = x ij  1 x ijm  ⊤  is a  m × 1 vector of known regression constants (maybe time-dependent covariates);   =  1  s  ⊤  is a  s × 1 vector of unknownparameters; and  e i  = e i 1 e ip  ⊤  is the error vector, which is assumed to beindependent and identically distributed with cumulative distribution function (cdf) G , defined on   p , with mean vector zero and covariance matrix   . The modelincorporates both between and within unit measurements, where the within designedunit is the same for all  n  units.In the following, we use the notation   y  2 A  = y ⊤ Ay , where  A  is a positivedefinite (p.d.) matrix. Definition 2.1.  The M-estimator of     given in Eq. (1) is defined as: ˆ  n  = Argmin   n  i = 1  h  u i     2  − 1     ∈  ⊆  s    (2)where  h  u i    = hu i 1   hu ip    ⊤  with  h ·   being a real valued function, u ij     = Y  ij   − f x ij     , for  j   = 1 p  and   = Var  h  u i    .In particular, if we let  hz = z  in Eq. (2), we have the LS method for estimating  . In the conventional setup of robust methods (see Hampel et al., 1986; Huber, 1981,and Jurecková and Sen, 1996), we primarily use bounded and monotone functions h ·  . In this respect, the so-called Huber-score function corresponds to: hz =  1 √  2  z  if    z ≤ k  k  z −  k 2 2  12   if    z  > k (3)for suitable chosen  k , with 0  < k <  . Remark 2.1.  The minimization in Eq. (2) is equivalent to the robust estimatingequations given by: n  i = 1   Y i  ˆ  n  = 0   (4)where   Y i    = X ⊤ i     − 1   u i    X i    =    ⊤ f  i    =  ˙ f    x i 1    ˙ f    x ip     ⊤    Robustness in Nonlinear Models 141 ˙ f    x ij     =    f x ij     =    1 f x ij      s f x ij      ⊤    u i    =  u i 1   u ip     ⊤ u ij     = 2 hu ij    h ′ u ij      and  h ′ u =  dd uhu Here   ·   is called the score function.We may rewrite Eq. (4) by using matrix notation as:  ∗  ˆ  n  = X ∗⊤  ˆ  n   ∗    ∗  ˆ  n  = 0   (5)where X ∗    =  X ⊤ 1     X ⊤ n      ⊤ np × s   ∗  = diag  np × np and  ∗    =   ⊤  u 1     ⊤  u n     ⊤ np × 1  In order to study the asymptotic properties of the M-estimator defined inEq. (2), we make the following sets of regularity assumptions concerning: [A] thecdf   G , [B] the score function   ·  , and [C] the function  f ·   defined in Eq. (1).[A] The cdf   G  is absolutely continuous with an absolutely continuous densityfunction  g ·   having a finite Fisher information, that is:    +−  g ′ vgv  2 dGv <    where  g ′ v =  dd vgv [B1] The score function   ·   is non constant, absolutely continuous, anddifferentiable with respect to   j  , for  j   = 1 s .[B2]   u = 0 and   u 2  <  ,    ′ u <   and    ′ u 2  <  , with  ′ u =  dd u u .[B3] lim  → 0   sup   ≤   u  +   − u     = 0, where    is defined in Eq. (2)and lim  → 0   sup   ≤    ′ u  +   −  ′ u     = 0.[C1] The function  f x     is continuous and twice differentiable with respect to  ∈  , where    is a compact subset of    s .[C2] lim  → 0  sup   ≤     j  f x    +     k f x    +   −   j  f x       k f x      = 0 andlim  → 0  sup   ≤     2  j    k f x   +   −   2  j    k f x     = 0, for  jk = 1 s , where   is given in Eq. (2). 3. Asymptotic Theory for M-Procedures Here we present the uniform asymptotic linearity of the M-estimator defined inEq. (4). Also, the existence of a solution of Eq. (4) that is a  √  n -consistent estimator  142 Sanhueza et al. of     is proven, which admits an asymptotic representation. Finally, we present theasymptotic normality of the developed M-estimators. The proofs of the followingtheorems are presented in Appendix. Theorem 3.1.  Under the regularity conditions  [A], [B1]–[B3] , and   [C1]–[C2]  given inSec. 2, we have: sup  t ≤ C   1 √  n n  i = 1    Y i   +  t √  n  −   Y i     +  1 n  n  t  = o p  1   as  n →  (6) where   n  =  ni = 1  X ⊤ i     − 1 WX i    , with  W = diag   and    =     ′ vfvdv . Theorem 3.2.  Under the regularity conditions  [A], [B1]–[B3] , and   [C1]–[C2]  given inSec. 2, we have: sup  t ≤ C  vec  K  ≤ D  1 √  n n  i = 1    Y i   +  t √  n +  K √  n  −   Y i     +  1 n  n  t  = o p  1  (7) as  n → , where  K  is a positive definite matrix and    n  is as defined in Eq. (6). Theorem 3.3.  Under the regularity conditions  [A], [B1]–[B3]  and   [C1]–[C2]  given inSec. 2, there exists a sequence  ˆ  n  of solutions of Eq. (4) such that: √  n  ˆ  n −   = O p  1   as  n →   (8) where  ˆ  n  =  +  1 n   n n  − 1   ni = 1   Y i    + o p n − 12  . Theorem 3.4.  Under the regularity conditions  [A], [B1]–[B3] , and   [C1]–[C2]  given inSec. 2 and   ch 1   i    ni = 1  i  − 1  −→ 0  as  n −→ , we have: 1 √  n n  i = 1   Y i    −→ N s   0     (9) where   = lim n → 1 n  ni = 1  i  and    = Var    u i    =  kl    are p.d. matrices, with  i  = X ⊤ i     − 1   − 1 X i    , for   kl = 1 p , and    kl  =  u ik   u il    . Corollary 3.1.  Under the regularity conditions  [A], [B1]–[B3] , and   [C1]–[C2]  given inSec. 2, we have: √  n  ˆ  n −   −→ N s   0    − 1  − 1   (10) where   = lim n → 1 n  n  is a p.d. matrix, with   n  =  ni = 1  X ⊤ i     − 1 WX i    , and  W = diag  , with   =     ′ vfvdv .
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