Regression rank scores estimation in ANOCOVA. (English) Zbl 0898.62087

Sen, Pranab Kumar

Ann. Stat. 24, No. 4, 1586-1601 (1996).

To motivate semiparametric analysis of covariance (ANOCOVA) models, consider first the conventional model where for the \(i\) th observation, \(Y_i\), \({\mathbf Z}_t\), and \({\mathbf t}_i\) stand for the primary, (stochastic) concomitant and (nonstochastic) design variates, respectively, and, conditional on \({\mathbf Z}_i= {\mathbf z}_i\), \[ Y_i= {\beta}'{\mathbf t}_i+ {\gamma}'{\mathbf z}_i+ e_i, \qquad i=1,\dots, n,\tag{1} \] where \({\beta}\) and \({\gamma}\) are the regression parameter vectors for the fixed and random effects components, and the \(e_i\) are independent and identically distributed random variables (i.i.d. r.v.’s) having a normal distribution with mean 0 and a finite (conditional) variance \(\sigma^2\). The \({\mathbf t}_i\) are given \(p\)-vectors not all equal, \({\mathbf T}= ({\mathbf t}_1,\dots, {\mathbf t}_n)'\) and the \({\mathbf Z}_i\) are stochastic \(q\)-vectors, so that there are \(p+q\) regression parameters and an additional scale parameter \(\sigma\). The assumed joint normality of \(({\mathbf Z}_i, e_i)\) yields homoscedasticity, linearity of regression as well as normality of the conditional distribution in (1). Without this joint normality, a breakdown may occur in each of these three basic postulations. On the other hand, the design vectors may still pertain to a linear regression function. Thus, there is a need to examine thoroughly the robustness aspects of mixed-effects models with due emphasis on all these factors. The role of regression rank scores in robust estimation of fixed-effects parameters as well as covariate regression functionals is critically appraised, and the relevant asymptotic theory is presented.

MSC:

62J05	Linear regression; mixed models
62G05	Nonparametric estimation
62G20	Asymptotic properties of nonparametric inference

Keywords:

heteroscedasticity; regression quantiles; R-estimators; stochastic predictors; weak invariance principles; semiparametric analysis; mixed-effects models; regression rank scores; robust estimation; covariate regression functionals

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