Robust Inference for Generalized Linear Models

By starting from a natural class of robust estimators for generalized linear models based on the notion of quasi-likelihood, we define robust deviances that can be used for stepwise model selection as in the classical framework. We derive the asymptotic distribution of tests based on robust deviances, and we investigate the stability of their asymptotic level under contamination. The binomial and Poisson models are treated in detail. Two applications to real data and a sensitivity analysis show that the inference obtained by means of the new techniques is more reliable than that obtained by classical estimation and testing procedures.

[1]  Cecil C. Craig,et al.  Note on the Distribution of Non-Central $t$ with an Application , 1941 .

[2]  A. Wald.  Tests of statistical hypotheses concerning several parameters when the number of observations is large , 1943 .

[3]  E. S. Pearson.  Note on an Approximation to the Distribution of Non-Central χ 2 , 1959 .

[4]  J. Imhof.  Computing the distribution of quadratic forms in normal variables , 1961 .

[5]  P. J. Huber.  The behavior of maximum likelihood estimates under nonstandard conditions , 1967 .

[6]  F. Hampel.  The Influence Curve and Its Role in Robust Estimation , 1974 .

[7]  R. W. Wedderburn.  Quasi-likelihood functions, generalized linear models, and the Gauss-Newton method , 1974 .

[8]  R. Davies.  The distribution of a linear combination of 2 random variables , 1980 .

[9]  K. Phelps,et al.  Use of the Complementary Log-Log Function to Describe Dose-Response Relationships in Insecticide Evaluation Field Trials , 1982 .

[10]  D. Pregibon.  Resistant fits for some commonly used logistic models with medical application. , 1982, Biometrics.

[11]  T. Caliński,et al.  Linear Statistical Inference , 1985 .

[12]  D. Ruppert,et al.  Optimally bounded score functions for generalized linear models with applications to logistic regression , 1986 .

[13]  B. R. Clarke.  Nonsmooth analysis and Fréchet differentiability of M-functionals , 1986 .

[14]  D. A. Williams,et al.  Generalized Linear Model Diagnostics Using the Deviance and Single Case Deletions , 1987 .

[15]  Peter J. Rousseeuw,et al.  Robust regression and outlier detection , 1987 .

[16]  C. Jennison,et al.  Robust Statistics: The Approach Based on Influence Functions , 1987 .

[17]  D. Ruppert.  Robust Statistics: The Approach Based on Influence Functions , 1987 .

[18]  Raymond J. Carroll,et al.  A Note on Asymmetry and Robustness in Linear Regression , 1988 .

[19]  R. Carroll,et al.  Conditionally Unbiased Bounded-Influence Estimation in General Regression Models, with Applications to Generalized Linear Models , 1989 .

[20]  P. McCullagh,et al.  Generalized Linear Models , 1992 .

[21]  Steve Steiner,et al.  Missing link , 1989 .

[22]  D. Lindenmayer,et al.  The conservation of arboreal marsupials in the Montane ash forests of the central highlands of Victoria, Southeast Australia: I. Factors influencing the occupancy of trees with hollows , 1990 .

[23]  S. Sheather,et al.  Robust Estimation and Testing , 1990 .

[24]  R. W. Farebrother,et al.  The Distribution of a Quadratic Form in Normal Variables , 1990 .

[25]  P. McCullagh,et al.  Generalized Linear Models, 2nd Edn. , 1990 .

[26]  D. G. Simpson,et al.  Breakdown robustness of tests , 1990 .

[27]  David B. Lindenmayer,et al.  The conservation of arboreal marsupials in the montane ash forests of the Central Highlands of Victoria, South-East Australia: III. The habitat requirements of leadbeater's possum Gymnobelideus leadbeateri and models of the diversity and abundance of arboreal marsupials , 1991 .

[28]  J. H. Schuenemeyer,et al.  Generalized Linear Models (2nd ed.) , 1992 .

[29]  S. Morgenthaler.  Least-Absolute-Deviations Fits for Generalized Linear Models , 1992 .

[30]  D. G. Simpson,et al.  Lower Bounds for Contamination Bias: Globally Minimax Versus Locally Linear Estimation , 1993 .

[31]  E. Ronchetti,et al.  Robust Bounded-Influence Tests in General Parametric Models , 1994 .

[32]  Elvezio Ronchetti,et al.  A Robust Version of Mallows's C P , 1994 .

[33]  Xuming He,et al.  Bounded Influence and High Breakdown Point Testing Procedures in Linear Models , 1994 .

[34]  N. L. Johnson,et al.  Continuous Univariate Distributions. , 1995 .

[35]  Kung-Yee Liang,et al.  Approximate likelihood ratios for general estimating functions , 1995 .

[36]  Richard M. Huggins,et al.  Variables Selection using the Wald Test and a Robust Cp , 1996 .

[37]  Elvezio Ronchetti,et al.  Robustness Aspects of Model Choice , 1997 .

[38]  C. Heyde,et al.  Quasi-likelihood and its application , 1997 .

[39]  Marianthi Markatou,et al.  Weighted Likelihood Equations with Bootstrap Root Search , 1998 .

[40]  J S Preisser,et al.  Robust Regression for Clustered Data with Application to Binary Responses , 1999, Biometrics.

[41]  E. Ronchetti,et al.  Robust inference with GMM estimators , 2001 .

[42]  Andreas Ruckstuhl,et al.  ROBUST FITTING OF THE BINOMIAL MODEL , 2001 .