A modified Reed-Muench method of estimation in dose-effect relationship

This paper revises the estimators for a distribution function in the dose-effect relationship as Reed and Muench (1938) by proposing a new statistics to estimate the distribution function when the main variable is not necessarily normally distributed. We prove the consistency and asymptotic normality of this estimates without assuming the form of a parametric family. The initial statistics of Reed and Mench designed to estimate the median dose of ED$_{50}$ are modified in such a way that it is possible to construct stable estimates of the distribution function and effective doses of ED$_{100\lambda}$ in a wide range of $\lambda$: from 0.05 to 0.95. A stochastic approximation algorithm for estimating of the distribution function is designed. The convergence theorem of this algorithm is proved. To illustrate the practical utility of our approach, the techniques developed in the paper are used for computing the mean age of eruption of the premolars in boys. The statistical data are from the Hayes&Mantel (1958) work. Finally, a Monte Carlo exercise is performed based on the on simulated data. The results show that the nonparametric estimates of the distribution function considered in this paper work well in practice, in some cases even for relatively small sample sizes.