Negative binomial regression in dose-effect relationships

This paper is devoted to problem on estimating the distribution function and its quantiles in the dose-effect relationships with nonparametric negative binomial regression. Most of the mathematical researches on dose-response relationships concerned models with binomial regression, in particular, models with binary data. Here we propose a kernel-based estimates for the distribution function, the kernels of which are weighted by a negative binomial random variable at each covariate. These covariates are quasirandom van der Corput and Halton low-discrepancy sequences. Our estimates are consistent, that is, they converge to their optimal values in probability as the number of observations $n$ grows to infinity. The proposed estimats are compared by their mean-square errors. We show that our estimates have a smaller asymptotic variance in comparison, in particular, with estimates of the Nadaraya-Watson type and other estimates. We present nonparametric estimates for the quantiles obtained by inverting a kernel estimate of the distribution function. We show that the asymptotic normality of these bias-adjusted estimates is preserved under some regularity conditions. We also provide a multidimensional generalization of the obtained results.