Maximum Likelihood Estimation of Isotonic Modal Regression

Abstract

For each $t$ in an index set $T$, let $P_t$ be a probability measure with mode $M(t)$. In this paper we consider a maximum likelihood nonparametric estimator $\hat{M}_n(t)$ of $M(t)$ subject to the constraint that $\hat{M}_n(\cdot)$ be isotonic with respect to an order on $T$. The estimator is a solution to a minimization problem with zero-one loss. The estimator is not a max-min or min-max representation of "naive" estimators. Naive modal estimators are used but they are not linear in the sense of Robertson and Wright (1975) nor do they have the Cauchy mean value property. Consistency results are given for the cases of $T$ finite, $P_t$ discrete; $T$ infinite, $P_t$ continuous; $T$ finite, $P_t$ continuous. An efficient and economical quadratic-time dynamic programming algorithm is presented for computations. In the case of $T$ finite, $P_t$ discrete, the algorithm operates on a matrix of frequency counts, "backing up" one column at a time as the optimal completion cells for the modal estimates are searched for. Illustrative simulations suggest that the estimator performs well in small samples and is robust to certain kinds of contamination perturbations.