Negative $\lambda$-binomial regression in dose-effect relationship

This paper is concern to the problem of estimating the distribution function and its quantiles in the dose-effect relationships with nonparametric negative $\lambda$-binomial regression. Here, a kernel-based estimators of the distribution function are proposed, of which kernel is weighted by the negative $\lambda$-binomial random variable at each covariate. Our estimates are consistent, that is, they converge to their optimal values in probability as $n$, the number of observations, grow to infinity. It is shown that these estimates have a smaller asymptotic variance in comparison, in particular, with estimates of the Nadaray-Watson type and other estimates. Nonparametric quantiles estimators obtained by inverting a kernel estimator of the distribution function are offered. It is shown that the asymptotic normality of this bias-adjusted estimator holds under some regularity conditions. In the first part, the relations between the moments of the negative $\lambda$-binomial distribution are analyzed. A new characterization of the Poisson distribution is obtened.