Implementation of EM algorithm based on non-precise observations

Document Type : Research Paper

Author

Department of Statistics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran

Abstract

The EM algorithm is a powerful tool and generic useful device in a variety of problems for maximum likelihood estimation with incomplete data which usually appears in practice. Here, the term ``incomplete" means a general state and in different situations it can mean different meanings, such as lost data, open source data, censored observations, etc. This paper introduces an application of the EM algorithm in which the meaning of ``incomplete" data is non-precise or fuzzy observations. The proposed approach in this paper for estimating an unknown parameter in the parametric statistical model by maximizing the likelihood function based on fuzzy observations. Meanwhile, this article presents a case study in the electronics industry, which is an extension of a well-known example used in introductions to the EM algorithm and focuses on the applicability of the
algorithm in a fuzzy environment. This paper can be useful for graduate students to understand the subject in fuzzy environment and moreover to use the EM algorithm in more complex examples.

Keywords


[1] G.M. Cordeiro, E.M.M. Ortega, and A.J. Lemonte, The exponential{Weibull lifetime distribution, Journal of Statistical Computation and Simulation 84 (2013) 2592{2606.
[2] A.P. Dempster, N.M. Laird, and D.B. Rubin, Maximum likelihood from incomplete data via the EM algorithm, Journal of the Royal Statistical Society 39 (1977) 1{38.
[3] T. Denoeux, Maximum likelihood estimation from fuzzy data using the EM algorithm, Fuzzy Sets and Systems 183 (2011) 72{91.
[4] B. Flury, and A. Zoppe, Exercises in EM, The American Statistician 54 (2000) 207{209.
[5] K. Knight, Mathematical Statistics, Chapman & Hall, New York, 2000.
[6] A. Parchami, EM.Fuzzy: EM algorithm for maximum likelihood estimation by non-precise information, R package version 1.0 (2018). URL: https://CRAN.R-project.org/package=EM.Fuzzy.
[7] R. Pourmousa, On truncated measures of income inequality from a fuzzy perspective, Iranian Journal of Fuzzy Systems 15 (2018) 123{137.
[8] L.A. Zadeh, Probability measures of fuzzy events, Journal of Mathematical Analysis and Applications 23 (1968) 421{427.