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Nonparametric tests for multi-parameter M-estimators

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  • Kolassa, John E.
  • Robinson, John

Abstract

We consider likelihood ratio like test statistics based on M-estimators for multi-parameter hypotheses for some commonly used parametric models where the assumptions on which the standard test statistics are based are not justified. The nonparametric test statistics are based on empirical exponential families and permit us to give bootstrap methods for the tests. We further consider saddlepoint approximations to the tail probabilities used in these tests. This generalizes earlier work of Robinson et al. (2003) in two ways. First, we generalize from bootstraps based on resampling vectors of both response and explanatory variables to include bootstrapping residuals for fixed explanatory variables, resulting in a surprising result for the weighted resampling. Second, we obtain a theorem for tail probabilities under weak conditions providing essential justification for the approximation to bootstrap results for both cases. We use as examples linear regression, non-linear regression and generalized linear models under models with independent and identically distributed residuals or vectors of observations, giving numerical illustrations of the results.

Suggested Citation

  • Kolassa, John E. & Robinson, John, 2017. "Nonparametric tests for multi-parameter M-estimators," Journal of Multivariate Analysis, Elsevier, vol. 158(C), pages 103-116.
    • Handle: RePEc:eee:jmvana:v:158:y:2017:i:c:p:103-116
    DOI: 10.1016/j.jmva.2017.04.004
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    References listed on IDEAS

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    1. Lô, Serigne N. & Ronchetti, Elvezio, 2009. "Robust and accurate inference for generalized linear models," Journal of Multivariate Analysis, Elsevier, vol. 100(9), pages 2126-2136, October.
    2. Chris Field & John Robinson & Elvezio Ronchetti, 2008. "Saddlepoint approximations for multivariate M-estimates with applications to bootstrap accuracy," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 60(1), pages 205-224, March.
    3. Cantoni E. & Ronchetti E., 2001. "Robust Inference for Generalized Linear Models," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1022-1030, September.
    4. Chris Field & John Robinson & Elvezio Ronchetti, 2008. "Saddlepoint approximations for multivariate M-estimates with applications to bootstrap accuracy," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 60(1), pages 225-227, March.
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    Cited by:

    1. Ronchetti, Elvezio, 2020. "Accurate and robust inference," Econometrics and Statistics, Elsevier, vol. 14(C), pages 74-88.

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