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This paper considers extensions of minimum-disparity estimators to theproblem of estimating parameters in a regression model that is conditionallyspecified; that is where a parametric model describes the distribution of aresponse $y$ conditional on covariates $x$ but does not specify thedistribution of $x$. We define these estimators by estimating a non-parametricconditional density estimates and minimizing a disparity between this estimateand the parametric model averaged over values of $x$. The consistency andasymptotic normality of such estimators is demonstrated for a broad class ofmodels in which response and covariate vectors can take both discrete andcontinuous values and incorportates a wide set of choices for kernel-basedconditional density estimation. It also establishes the robustness of theseestimators for a broad class of disparities. As has been observed in Tamura andBoos (J. Amer. Statist. Assoc. 81 (1986) 223--229), minimum disparityestimators incorporating kernel density estimates of more than one dimensioncan result in an asymptotic bias that is larger that $n^{-1/2}$ and wecharacterize a similar bias in our results and show that in specialized casesit can be eliminated by appropriately centering the kernel density estimate. Wealso demonstrate empirically that bootstrap methods can be employed to reducethis bias and to provide robust confidence intervals. In order to demonstratethese results, we establish a set of $L_1$-consistency results for kernel-basedestimates of centered conditional densities.

Consistency, efficiency and robustness of conditional disparity methods