Maximum Likelihood Estimation of Dose-Response Functions Subject to Absolutely Monotonic Constraints

Abstract

Statistical properties are derived for maximum likelihood estimates of dose-response functions in which the response probability is related to the dose by means of a polynomial of unknown degree with nonnegative coefficients. Dose-response functions of this form are predicted by the multistage model of carcinogenesis. We first establish necessary and sufficient conditions for strong consistency of the estimates. For these results no assumptions are made about the polynomial degree, so the number of coefficients to be estimated is effectively infinite. Under some additional assumptions, which do involve restrictions on the polynomial degree, we obtain the asymptotic distribution of the vector of maximum likelihood estimates about the true vector of polynomial coefficients. Because the coefficients are constrained to be nonnegative, the limiting distribution will generally not be normal.