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Asymptotic confidence intervals for Poisson regression

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  • Kohler, Michael
  • Krzyzak, Adam

Abstract

Let (X,Y) be a -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 [alpha] 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 L1 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 [alpha], 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.

Suggested Citation

  • Kohler, Michael & Krzyzak, Adam, 2007. "Asymptotic confidence intervals for Poisson regression," Journal of Multivariate Analysis, Elsevier, vol. 98(5), pages 1072-1094, May.
    • Handle: RePEc:eee:jmvana:v:98:y:2007:i:5:p:1072-1094
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    Cited by:

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    2. José Santos & M. Neves, 2008. "A local maximum likelihood estimator for Poisson regression," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 68(3), pages 257-270, November.

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