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Asymptotic distributions of M-estimators in a spatial regression model under some fixed and stochastic spatial sampling designs

  • Regression
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Abstract

In this paper, we consider M-estimators of the regression parameter in a spatial multiple linear regression model. We establish consistency and asymptotic normality of the M-estimators when the data-sites are generated by a class of deterministic as well as a class of stochastic spatial sampling schemes. Under the deterministic sampling schemes, the data-sites are located on a regular grid but may have aninfill component. On the other hand, under the stochastic sampling schemes, locations of the data-sites are given by the realizations of a collection of independent random vectors and thus, are irregularly spaced. It is shown that scaling constants of different orders are needed for asymptotic normality under different spatial sampling schemes considered here. Further, in the stochastic case, the asymptotic covariance matrix is shown to depend on the spatial sampling density associated with the stochastic design. Results are established for M-estimators corresponding to certain non-smooth score functions including Huber’s ψ-function and the sign functions (corresponding to the sample quantiles).

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Authors and Affiliations

  1. Department of Statistics, Iowa State University, 102 Snedecor Hall, 50011-1210, Ames, IA, USA

    S. N. Lahiri

  2. Department of Statistics and Applied Probability, National University of Singapore, 10 Kent Ridge Crescent, 119260, Singapore, Singapore

    Kanchan Mukherjee

Authors
  1. S. N. Lahiri
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  2. Kanchan Mukherjee
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Additional information

Research of Lahiri is partially supported by NSF grant no. DMS-0072571. Research of Mukherjee is partially supported by the Academic Research Grant R-155-000-003-112 from the National University of Singapore.

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Lahiri, S.N., Mukherjee, K. Asymptotic distributions of M-estimators in a spatial regression model under some fixed and stochastic spatial sampling designs. Ann Inst Stat Math 56, 225–250 (2004). https://doi.org/10.1007/BF02530543

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  • DOI: https://doi.org/10.1007/BF02530543

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