Asymptotic confidence intervals for Poisson regression

Authors:
Michael Kohler;Adam Krzyżak
Affiliations:
Fachrichtung 6.1-Mathematik, Universität des Saarlandes, Postfach 151150, D-66041 Saarbrücken, Germany;Department of Computer Science and Software Engineering, Concordia University, 1455 De Maisonneuve Blvd. West, Montreal, Que., Canada H3G 1M8
Venue:
Journal of Multivariate Analysis
Year:
2007

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Abstract

Let (X,Y) be a R^dxN"0-valued random vector where the conditional distribution of Y given X=x is a Poisson distribution with mean m(x). We estimate m by a local polynomial kernel estimate defined by maximizing a localized log-likelihood function. We use this estimate of m(x) to estimate the conditional distribution of Y given X=x by a corresponding Poisson distribution and to construct confidence intervals of level @a of Y given X=x. Under mild regularity conditions on m(x) and on the distribution of X we show strong convergence of the integrated L"1 distance between Poisson distribution and its estimate. We also demonstrate that the corresponding confidence interval has asymptotically (i.e., for sample size tending to infinity) level @a, and that the probability that the length of this confidence interval deviates from the optimal length by more than one converges to zero with the number of samples tending to infinity.