In the literature, there is, no specific robust method for estimating Poisson regression parame- ters. However, the general robust methods for generalized linear ...
Weighted Maximum Likelihood Estimates in Poisson Regression S. Hosseinian and S. Morgenthaler 1
Ecole polytechnique f´ed´erale de Lausanne (EPFL), FSB IMA, Statition 8, 1015 Lausanne, Switzerland
Keywords: Robustness, Generalized linear model, Poisson model, Influence function, Maximum likelihood
Abstract In the literature, there is, no specific robust method for estimating Poisson regression parameters. However, the general robust methods for generalized linear models can be applied to Poisson models. The conditionally unbiased bounded influence estimator of K¨ unsch et al. (1989) is one of the first robust models that can be used to estimate the parameters for these models. The Lq quasi-likelihood estimator of Morgenthaler (1992) can be also applied to Poisson regression models. More recently, a Mallows quasi-likelihood estimator of of Cantoni and Ronchetti (2001) has been applied in particular to Poisson regression. In ICORS 2008 we introduced a robust estimator for the binary regression. The resulting estimating equation consists in a simple modification of the familiar maximum likelihood equation with the weights wq (µ). We generalize this estimator to Poisson regression. The resulting estimating equation is a weighted maximum likelihood with weights that depend on µ and two constants c1 and c2 . This presents an improvement compared to other robust estimates discussed in the literature that typically have weights, which depend on the couple (xi , yi ) rather than on µi = h(xTi β) alone. Let the response variable have a Poisson distribution yi ∼ P(µi ) and let the explanatory variables be xi ∈ Rp . The estimating equation for the weighted maximum likelihood estimator WMLEM H is given by n X h0 (ηi ) M H W (µi )(yi − µi )xi = 0 , (1) µi i=1 where µi = h(ηi ), h0 (ηi ) = ∂µi /∂ηi and weight function v 1, c1 < µi < c1 v; c1 µi , µ < cv1 ; W M H (µi ) = c2vv−µi i , c1 v < µi < c2 v; v 0, otherwise. where v is the median of µ. In this research we chose some arbitrary the value of c1 = 2 and c2 = 3. This estimator is easy to compute and results in resistant estimates. In this talk we discuss its robustness properties and show how to compute the asymptotic variance covariance matrix ˆ As an illustration we discuss computational aspects of two examples and compare to other of β. estimators that have been proposed. We investigate the influence function of these estimates and we compare their sensitivity to the maximum likelihood estimate.
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References E. Cantoni and E. Ronchetti (2001). Robust inference for generalized linear models. J. Amer. Statist. Assoc. , Vol. 96(455), pp 10221030. H. R. K¨ unsch, L. A. Stefanski and R. J. Carroll (1989) Conditionally unbiased bounded-influence estimation in general regression models, with applications to generalized linear models. Journal of the American Statistical Association, Vol. 84, pp 460-466. S. Morgenthaler (1992). Least-absolute-deviation fits for generalized linear models. Biometrika, Vol. 79, pp. 747-754. D. Pregibon (1982). Resistant fits for some commonly used logistic models with medical applications. Biometrics, Vol. 38, pp. 485-498.
