Estimation of the binary response model using a mixture of distributions estimator (MOD)

In this paper, we develop a semiparametric sieve estimator, which is termed a mixture of distributions estimator (MOD), to estimate a binary response model when the distribution of the errors is unknown. The estimator of the distribution function is composed of a mixture of smooth distributions, where the number of mixtures increases with the sample size. The model is semiparametric because it is assumed that a parametric index type restriction holds. Optimal rates of convergence are established for the distribution function under the L2 norm, and conditions are derived under which estimates of the parametric component are asymptotically normal. An appealing feature about MOD is that it is possible to restrict the estimator of the distribution function, a priori, to be smooth, nonnegative, nondecreasing, and to integrate to one. This has important practical and theoretical implications.