Maximum Likelihood and Minimum | chi 2 Estimates of the Logistic Function

Abstract

It is well known that both types of estimates are "efficient" and "best asymptotically normal" for the theory of large samples. This paper presents a sampling investigation of the bias and variance of these estimates in small samples, the only real statistical samples. The results indicate that the minimum chi-square (MC) estimate has a uniformly lower variance than the maximum likelihood estimate, although the bias of the former is usually somewhat larger. The biases, however, are small and for all practical purposes can be ignored. The conclusions hold for a wide variety of sample sizes, numbers of dose points and spacings of doses. The author also discussed Rao-Blackwellized MC estimates, the problem of a definition of the MC estimate, methods of sampling and calculation, the case of zero survivors, and the sufficiency of the estimates.