Results of Cosslett: Part II
Cosslett (1981b) summarized results obtained elsewhere, especially from his earlier papers (Cosslett, 1978, 1981a). He also included a numerical evaluation of the asymptotic bias and variance of various estimators. We shall first discuss CBMLE of the generalized choice-based sample model with unknown /and known Q. Cosslett (1981b) merely stated the consistency, asymptotic normality, and asymptotic efficiency of the estimator, which are proved in Cosslett (1978). The discussion here will be brief because the results are analogous to those given in the previous subsection.
The log likelihood function we shall consider in this subsection is similar to
(9.5.47) except that the last term is simplified because Q is now known. Thus
log L2 = 2 log Р(ЛІХ» Л + 2 log/(*<) (9.5.63)
– 2 ^ ,oe &>(*).
which is to be maximized with respect to fi and /subject to the constraints
It is shown that this constrained maximization is equivalent to maximizing with respect to kj, j = 0, 1, 2,. . . , m,
J-o subject to the contraint ’S. jL^jQqU) = 1. Consistency, asymptotic normality, and asymptotic efficiency can be proved in much the same way as in the preceding subsection.
Next, we shall report Cosslett’s numerical analysis, which is in the same spirit as that reported in Section 9.5.2 concerning the Manski-Lerman estimator. Cosslett compared RSMLE, CBMLE, WMLE, and MME in the simple choice-based sample model with/unknown and Q known.20 Three binary QR models (logit, probit, and arctangent) were considered. In each model there is only one independent variable, which is assumed to be normally distributed. The asymptotic bias and the asymptotic variance of the estimators are evaluated for different values of j? (two coefficients), Q(l), Я(1), and the mean of the independent variable. The optimal design is also derived for each estimator. Cosslett concluded that (1) RSMLE can have a large asymptotic bias; (2) CBMLE is superior to WMLE and MME; (3) a comparison between WMLE and MME depends on parameters, especially 6(1); (4) the choice of Я(1) = 0.5 was generally quite good, leading to only a small loss of efficiency compared to the optimal design. (This last finding is consistent with the results of our numerical analysis reported in Section 9.5.2.)