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Asymptotic confidence intervals for Poisson regression

  • Kohler, Michael
  • Krzyzak, Adam
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    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.

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    Article provided by Elsevier in its journal Journal of Multivariate Analysis.

    Volume (Year): 98 (2007)
    Issue (Month): 5 (May)
    Pages: 1072-1094

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    Handle: RePEc:eee:jmvana:v:98:y:2007:i:5:p:1072-1094
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    1. Harro Walk, 2001. "Strong Universal Pointwise Consistency of Recursive Regression Estimates," Annals of the Institute of Statistical Mathematics, Springer, vol. 53(4), pages 691-707, December.
    2. Györfi, László & Walk, Harro, 1997. "On the strong universal consistency of a recursive regression estimate by Pál Révész," Statistics & Probability Letters, Elsevier, vol. 31(3), pages 177-183, January.
    3. Hudson, H. Malcolm & Lee, Thomas C. M., 1998. "Maximum likelihood restoration and choice of smoothing parameter in deconvolution of image data subject to Poisson noise," Computational Statistics & Data Analysis, Elsevier, vol. 26(4), pages 393-410, February.
    4. Algoet, Paul & Györfi, László, 1999. "Strong Universal Pointwise Consistency of Some Regression Function Estimates," Journal of Multivariate Analysis, Elsevier, vol. 71(1), pages 125-144, October.
    5. Michael Kohler, 2002. "Universal Consistency of Local Polynomial Kernel Regression Estimates," Annals of the Institute of Statistical Mathematics, Springer, vol. 54(4), pages 879-899, December.
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