The Annals of Statistics

On the Convergence Properties of the EM Algorithm

C. F. Jeff Wu

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Two convergence aspects of the EM algorithm are studied: (i) does the EM algorithm find a local maximum or a stationary value of the (incomplete-data) likelihood function? (ii) does the sequence of parameter estimates generated by EM converge? Several convergence results are obtained under conditions that are applicable to many practical situations. Two useful special cases are: (a) if the unobserved complete-data specification can be described by a curved exponential family with compact parameter space, all the limit points of any EM sequence are stationary points of the likelihood function; (b) if the likelihood function is unimodal and a certain differentiability condition is satisfied, then any EM sequence converges to the unique maximum likelihood estimate. A list of key properties of the algorithm is included.

Article information

Ann. Statist., Volume 11, Number 1 (1983), 95-103.

First available in Project Euclid: 12 April 2007

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Zentralblatt MATH identifier


Primary: 62F10: Point estimation
Secondary: 90C30: Nonlinear programming

EM algorithm GEM algorithm incomplete data curved exponential family maximum likelihood estimate


Wu, C. F. Jeff. On the Convergence Properties of the EM Algorithm. Ann. Statist. 11 (1983), no. 1, 95--103. doi:10.1214/aos/1176346060.

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