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Nonparametric inference with generalized likelihood ratio tests

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Abstract

The advance of technology facilitates the collection of statistical data. Flexible and refined statistical models are widely sought in a large array of statistical problems. The question arises frequently whether or not a family of parametric or nonparametric models fit adequately the given data. In this paper we give a selective overview on nonparametric inferences using generalized likelihood ratio (GLR) statistics. We introduce generalized likelihood ratio statistics to test various null hypotheses against nonparametric alternatives. The trade-off between the flexibility of alternative models and the power of the statistical tests is emphasized. Well-established Wilks’ phenomena are discussed for a variety of semi- and non-parametric models, which sheds light on other research using GLR tests. A number of open topics worthy of further study are given in a discussion section.

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Authors and Affiliations

  1. Department of ORFE, Princeton University, Princeton, NJ, 08544, USA

    Jianqing Fan

  2. Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, USA

    Jiancheng Jiang

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  1. Jianqing Fan
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Correspondence to Jianqing Fan.

Additional information

This invited paper is discussed in the comments available at: http://dx.doi.org/10.1007/s11749-007-0081-7, http://dx.doi.org/10.1007/s11749-007-0082-6, http://dx.doi.org/10.1007/s11749-007-0083-5, http://dx.doi.org/10.1007/s11749-007-0084-4, http://dx.doi.org/10.1007/s11749-007-0085-3, http://dx.doi.org/10.1007/s11749-007-0086-2, http://dx.doi.org/10.1007/s11749-007-0087-1, http://dx.doi.org/10.1007/s11749-007-0088-0, http://dx.doi.org/10.1007/s11749-007-0089-z.

The work was supported by the NSF grants DMS-0354223, DMS-0532370 and DMS-0704337.

The paper was initiated when Jiancheng Jiang was a research fellow at Princeton University.

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Fan, J., Jiang, J. Nonparametric inference with generalized likelihood ratio tests. TEST 16, 409–444 (2007). https://doi.org/10.1007/s11749-007-0080-8

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