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

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
Full Text:

### References:

 [1] ANSCOME, F. J. 1952. Large sample theory of sequential estimation. Proc. Cambridge Philos. Soc. 48 600 607. Z. · Zbl 0047.13401 [2] BHATTACHARy A, P. K. and GANGOPADHy AY, A. K. 1990. Kernel and nearest neighbor estimation of a conditional quantile. Ann. Statist. 18 1400 1415. Z. · Zbl 0706.62040 [3] BICKEL, P. J. and WICHURA, M. J. 1971. Convergence criteria for multiparameter stochastic processes and some applications. Ann. Math. Statist. 42 1656 1670. Z. · Zbl 0265.60011 [4] DOKSUM, K. 1969. Starshaped transformations and power of rank tests. Ann. Math. Statist. 40 1167 1176. Z. · Zbl 0188.50601 [5] GANGOPADHy AY, A. K. and SEN, P. K. 1992. Contiguity in nonparametric estimation of a Z conditional functional. In Nonparametric Statistics and Related Topics A. K. M. E.. Saleh, ed. 141 162. North-Holland, Amsterdam. Z. [6] GANGOPADHy AY, A. K. and SEN, P. K. 1993. Contiguity in Bahadur-ty pe representations of a conditional quantile and application in conditional quantile processes. In Statistics Z and Probability, a Raghu Raj Bahadur Festschrift J. K. Ghosh, S. K. Mitra, K. R.. Parthasarathy and B. L. S. Prakasa Rao, eds. 219 231. Wiley Eastern, New Delhi. Z. [7] GHOSH, M. and SEN, P. K. 1971. On a class of rank order tests for regression with partially informed stochastic predictors. Ann. Math. Statist. 42 650 661. Z. · Zbl 0215.54403 [8] GUTENBRUNNER, C. and JURECKOVA, J. 1992. Regression rank scores and regression quantiles. Ánn. Statist. 20 305 330. · Zbl 0759.62015 [9] HAJEK, J. and SIDAK, Z. 1967. Theory of Rank Tests. Academia, Prague. \' Ź. · Zbl 0161.38102 [10] JURECKOVA, J. 1992. Estimation in a linear model based on regression rank scores. J. Non ṕaramet. Statist. 1 197 203. Z. · Zbl 1263.62055 [11] JURECKOVA, J. and SEN, P. K. 1993. Asy mptotic equivalence of regression rank scores estima ťors and R-estimators in linear models. In Statistics and Probability, a Raghu Raj Z Bahadur Festschrift J. K. Ghosh, S. K. Mitra, K. R. Parthasarathy and B. L. S.. Prakasa Rao, eds. 279 291. Wiley Eastern, New Delhi. [12] JURECKOVA, J. and SEN, P. K. 1996. Robust Statistical Procedures: Asy mptotics and Interrela ťions. Wiley, New York. Z. [13] KOENKER, R. and BASSETT, G. 1978. Regression quantiles. Econometrica 46 33 50. Z. JSTOR: · Zbl 0373.62038 [14] PURI, M. L. and SEN, P. K. 1985. Nonparametric Methods in General Linear Models. Wiley, New York. Z. · Zbl 0569.62024 [15] SEN, P. K. 1981. Sequential Nonparametrics: Invariance Principles and Statistical Inference. Wiley, New York. Z. · Zbl 0583.62074 [16] SEN, P. K. 1993a. Regression rank scores estimation in ANOCOVA. Technical Report 2117, Institute of Statistics, Univ. North Carolina. Z. [17] SEN, P. K. 1993b. Perspectives in multivariate nonparametrics: conditional functionals and ANOCOVA models. Sankhy a Ser. A 55 516 532. Z. · Zbl 0806.62043 [18] SEN, P. K. 1994. Regression quantiles in nonparametric regression. J. Nonparamet. Statist. 3 237 253. Z. · Zbl 1380.62179 [19] SEN, P. K. and GHOSH, M. 1972. On strong convergence of regression rank statistics. Sankhy a Ser. A. 34 335 348. · Zbl 0269.62046 [20] CHAPEL HILL, NORTH CAROLINA 27599-7400 E-MAIL: pksen@sphrax.sph.unc.edu
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.