Inference on a new sigmoid regression model with unknown support and unbounded likelihood function

In this paper we introduce a non-regular sigmoid shape regression model with boundaries of support for the response variable being two unknown parameters; the likelihood function is unbounded, for which the global maximizers are not consistent estimators. Although the two sample extremes, the smallest and the largest observations, are consistent estimators of the two unknown boundaries, they have slow convergence rate and are asymptotically biased. Improved estimators are developed by correcting for the asymptotic biases of the two sample extremes for the one sample case; but these proposed estimators do not obtain the optimal convergence rate either. To obtain efficient estimation, we resort to the local maximizers of the likelihood function, i.e., the solution to the likelihood equations that is obtained by setting to zero the gradient of the log-likelihood function with respect to the parameters. We prove that, with probability approaching one as the sample size goes to infinity, there exists a solution to the likelihood equation that is consistent at the rate of the square root of the sample size and it is asymptotically normally distributed.