A self-consistent estimator for interval-valued data

Abstract

In interval-valued data, the variable of interest is provided in the form of an interval with lower and upper bounds, not a single value. An univariate representation for the interval is not unique by its nature, in particular when interval-valued data are of the min-max (MM) type. Researchers focus on the marginal histogram distribution which is well suited to the measurement error (ME) type interval data. Two estimators, the empirical histogram estimator and nonparametric kernel estimator, have been proposed for the estimation of the marginal histogram in the literature. In this paper, we define a new univariate representation, named as a self-consistent marginal, for interval-valued data, and propose a self-consistent estimator (SCE) to estimate it. We theoretically and numerically investigate the properties of the SCE under various assumptions. We further illustrate the advantages of the SCE over the two existing estimators with empirical examples.