Transfer maps and nonexistence of joint determinant

Transfer Maps, sometimes called norm maps, for Milnor's $K$-theory were first defined by Bass and Tate (1972) for simple extensions of fields via tame symbol and Weil's reciprocity law, but their functoriality had not been settled until Kato (1980). On the other hand, functorial transfer maps for the Goodwillie group are easily defined. We show that these natural transfer maps actually agree with the classical but difficult transfer maps by Bass and Tate. With this result, we build an isomorphism from the Goodwillie groups to Milnor's $K$-groups of fields, which in turn provides a description of joint determinants for the commuting invertible matrices. In particular, we explicitly determine certain joint determinants for the commuting invertible matrices over a finite field, the field of rational numbers, real numbers and complex numbers into the respective group of units of given field.Comment: Some minor revisions have been made after the 1st versio

Maximum Score Estimation of Preference Parameters for a Binary Choice Model under Uncertainty

This paper develops maximum score estimation of preference parameters in the binary choice model under uncertainty in which the decision rule is affected by conditional expectations. The preference parameters are estimated in two stages: we estimate conditional expectations nonparametrically in the first stage and then the preference parameters in the second stage based on Manski (1975, 1985)'s maximum score estimator using the choice data and first stage estimates. The paper establishes consistency and derives rate of convergence of the two-stage maximum score estimator. Moreover, the paper also provides sufficient conditions under which the two-stage estimator is asymptotically equivalent in distribution to the corresponding single-stage estimator that assumes the first stage input is known. These results are of independent interest for maximum score estimation with nonparametrically generated regressors. The paper also presents some Monte Carlo simulation results for finite-sample behavior of the two-stage estimator