International Journal of Research and Review DOI: https://doi.org/10.52403/ijrr.20210937 Vol.8; Issue: 9; September 2021 Website: www.ijrrjournal.com Short Communication E-ISSN: 2349-9788; P-ISSN: 2454-2237 International Journal of Research and Review (ijrrjournal.com) 275 Vol.8; Issue: 9; September 2021 Maximum Likelihood Estimation using the EM Algorithm Ahsene Lanani Department of Mathematics, Faculty of Exact Sciences, University Frères Mentouri Constantine 1, 25000 Algeria ABSTRACT This paper yields with the Maximum likelihood estimation using the EM algorithm. This algorithm is very used to solve nonlinear equations with missing data. We estimated the linear mixed model parameters and those of the variance-covariance matrix. The considered structure of this matrix is not necessarily linear. Key Words: Algorithm EM; Maximum likelihood; Mixed linear model 1. INTRODUCTION In the models using longitudinal data or repeated measurements, we are often confronted with missing data. This loss of data or information is due to several reasons, missing and often death of the corresponding experimental units among others. The pioneers in this field are Dempster et al. [1]. In addition, when we estimate the parameters of a mixed linear model using the maximum likelihood (ML) or the restricted maximum likelihood (REML) method, the normal equations are often nonlinear and consequently do not admit explicit solutions; from where the passage to iterative processes or algorithms. The EM algorithm can be used to estimate such parameters, like those generating the variance-covariance matrix of the model (Dempster et al. [2]; Jennrich and Schluchter [4]); Laird and Ware [5]. Thereafter, others works were developed on this subject [8-10]. An improvement of the convergence of the EM algorithm was carried out by Laird et al. [6]. Lindstrom and Bates [7] compared this algorithm with that of Newton - Raphson and some results were discussed. A recent study concerning the estimate of the parameters generating the variance-covariance matrix of a mixed model by using restricted maximum likelihood is given by Foulley et al. [3]. Section 2 of this paper describes the EM algorithm. An example of application is given in section 3. Section 4 relates to the results, especially those of the estimators of the parameters generating the variance- covariance matrix of the model. 2. EM ALGORITHM Let X be a random variable of density  where θ is an unknown parameter. Let us suppose that X is not completely observed; i.e. we observe a part Y of X. Let   , a random variable of density g(y/θ). Let t(x) be a vector of sufficient (exhaustive) statistics for θ. The purpose of the EM algorithm (E for expectation and M for maximization) is to find the value of θ which maximizes the likelihood g(y/θ) being given a value of y. This maximization (normal equations) gives the following equation:     (1) The EM algorithm uses two stages to solve this equation in θ. First stage. E-Step: We calculate the quantity:     (2) Secund stage. M-Step: We solve the equation in θ:    (3) In other words, in E-Step, given an initial value for ;   its value at the stage