Nonparametric estimation of multivariate convex-transformed densities

We study estimation of multivariate densities $p$ of the form $p(x)=h(g(x))$ for $x\in \mathbb {R}^d$ and for a fixed monotone function $h$ and an unknown convex function $g$. The canonical example is $h(y)=e^{-y}$ for $y\in \mathbb {R}$; in this case, the resulting class of densities [\mathcal {P}(e^{-y})={p=\exp(-g):g is convex}] is well known as the class of log-concave densities. Other functions $h$ allow for classes of densities with heavier tails than the log-concave class. We first investigate when the maximum likelihood estimator $\hat{p}$ exists for the class $\mathcal {P}(h)$ for various choices of monotone transformations $h$, including decreasing and increasing functions $h$. The resulting models for increasing transformations $h$ extend the classes of log-convex densities studied previously in the econometrics literature, corresponding to $h(y)=\exp(y)$. We then establish consistency of the maximum likelihood estimator for fairly general functions $h$, including the log-concave class $\mathcal {P}(e^{-y})$ and many others. In a final section, we provide asymptotic minimax lower bounds for the estimation of $p$ and its vector of derivatives at a fixed point $x_0$ under natural smoothness hypotheses on $h$ and $g$. The proofs rely heavily on results from convex analysis.Comment: Published in at http://dx.doi.org/10.1214/10-AOS840 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Imputation Estimators Partially Correct for Model Misspecification

Inference problems with incomplete observations often aim at estimating population properties of unobserved quantities. One simple way to accomplish this estimation is to impute the unobserved quantities of interest at the individual level and then take an empirical average of the imputed values. We show that this simple imputation estimator can provide partial protection against model misspecification. We illustrate imputation estimators' robustness to model specification on three examples: mixture model-based clustering, estimation of genotype frequencies in population genetics, and estimation of Markovian evolutionary distances. In the final example, using a representative model misspecification, we demonstrate that in non-degenerate cases, the imputation estimator dominates the plug-in estimate asymptotically. We conclude by outlining a Bayesian implementation of the imputation-based estimation.Comment: major rewrite, beta-binomial example removed, model based clustering is added to the mixture model example, Bayesian approach is now illustrated with the genetics exampl

Imputation Estimators Partially Correct for Model Misspecification

Inference problems with incomplete observations often aim at estimating population properties of unobserved quantities. One simple way to accomplish this estimation is to impute the unobserved quantities of interest at the individual level and then take an empirical average of the imputed values. We show that this simple imputation estimator can provide partial protection against model misspecification. We illustrate imputation estimators’ robustness to model specification on three examples: mixture model-based clustering, estimation of genotype frequencies in population genetics, and estimation of Markovian evolutionary distances. In the final example, using a representative model misspecification, we demonstrate that in non-degenerate cases, the imputation estimator dominates the plug-in estimate asymptotically. We conclude by outlining a Bayesian implementation of the imputation-based estimation.