Computational Statistics

, Volume 33, Issue 2, pp 731–756 | Cite as

A lack-of-fit test for generalized linear models via single-index techniques

Original Paper


A generalized partially linear single-index model (GPLSIM) is proposed in which the unknown smooth function of single index is approximated by a spline function that can be expressed as a linear combination of B-spline basis functions. The regression coefficients and the unknown smooth function are estimated simultaneously via a modified Fisher-scoring method. It can be shown that the estimators of regression parameters are asymptotically normally distributed. The asymptotic covariance matrix of the estimators can be estimated directly and consistently by using the least-squares method. As an application, the proposed GPLSIM can be employed to assess the lack of fit of a postulated generalized linear model (GLM) based on the comparison of the goodness of fit of the GPLSIM and postulated GLM to construct a likelihood ratio test. An extensive simulation study is conducted to examine the finite-sample performance of the likelihood ratio test. The practicality of the proposed methodology is illustrated with a real-life data set from a study of nesting horseshoe crabs.


B-spline Bootstrap Generalized linear model Generalized partially linear single-index model Likelihood estimator Likelihood ratio test Monte Carlo 



The authors express their thanks to an associate editor and two referees whose constructive comments improved the presentation.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Division of Biostatistics, Department of Public Health SciencesUniversity of CaliforniaDavisUSA
  2. 2.School of Community Health SciencesUniversity of NevadaRenoUSA

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