Parameter estimation from interval-valued data using the expectation-maximization algorithm

Abstract

This paper investigates on the problem of parameter estimation in statistical model when observations are intervals assumed to be related to underlying crisp realizations of a random sample. The proposed approach relies on the extension of likelihood function in interval setting. A maximum likelihood estimate of the parameter of interest may then be defined as a crisp value maximizing the generalized likelihood function. Using the expectation-maximization (EM) to solve such maximizing problem therefore derives the so-called interval-valued EM algorithm (IEM), which makes it possible to solve a wide range of statistical problems involving interval-valued data. To show the performance of IEM, the following two classical problems are illustrated: univariate normal mean and variance estimation from interval-valued samples, and multiple linear/nonlinear regression with crisp inputs and interval output.

Keywords:

• maximum likelihood estimation
• EM algorithm
• generalized likelihood function
• interval-valued data
• regression analysis

Acknowledgements

The authors are grateful to the contributions of Prof. Thierry Denoeux for our work. The authors are also grateful for the contributions of the editors and the anonymous referees. This work is supported by the National Natural Science Foundation of China (Nos. 51106025 and 51076027). The discussion section is suggested by the referees.