Consistency, efficiency and robustness of conditional disparity methods

This report demonstrates the consistency and asymptotic normality of minimum-disparity estimators based on conditional density estimates. In particular, it provides L1 consistency results for conditional density estimators and conditional density estimators under homoscedastic model restrictions. It defines a generic formulation of disparity estimates in conditionally-specified models and demonstrates their asymptotic consistency and normality. In particular, for regression models with more than one continuous response or covariate, we demonstrate that disparity estimators based on unrestricted conditional density estimates have a bias that is larger than √ n, however for univariate homoscedastic models conditioned on a univariate continuous covariate, this bias can be removed. 1 Framework and Assumptions Throughout the following, we assume that we observe {Xn1(ω), Xn2(ω), Yn1(ω), Yn2)(ω), n ≥ 1} i.i.d. random variables with where Xn1(ω) ∈ Rx , Xn2(ω) ∈ Sx, Yn1(ω) ∈ Ry , Yn2(ω) ∈ Sy for countable sets Sx and Sy with joint distribution g(x1, x2, y1, y2) = P (X2 = x2, Y2 = y2)P (X1 ∈ dx1, Y1 ∈ dy1|x2, y2) and define the marginal and conditional densities h(x1, x2) = ∑ y2∈Sy ∫ g(x1, x2, y1, y2)dy1 f(y1, y2|x1, x2) = g(x1, x2, y1, y2) h(x1, x2) . on the support of (x1, x2). Further, for a scalar Yn1 ∈ R, removing Yn2 (or setting it to be always identically 0) we define the conditional expectation m(x1, x2) = ∫ y1f(y1|x1, x2)dy1 and note that in homoscedastic regression models we write for some density f∗(e), f(y1|x1, x2) = f(y1 −m(x1, x2)). (1.1)