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