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.

中文翻译：

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区间值数据的自洽估计器

在区间值数据中，感兴趣的变量以具有下限和上限的区间的形式提供，而不是单个值。区间的单变量表示就其性质而言并不是唯一的，特别是当区间值数据是最小-最大 (MM) 类型时。研究人员专注于边缘直方图分布，它非常适合测量误差 (ME) 类型的间隔数据。文献中已经提出了两种估计量，即经验直方图估计量和非参数核估计量来估计边际直方图。在本文中，我们为区间值数据定义了一个新的单变量表示，称为自洽边际，并提出了一个自洽估计器（SCE）来估计它。我们在各种假设下从理论上和数值上研究了 SCE 的特性。