Maximum Likelihood Estimation from Incomplete Data

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Abstract
This paper is concerned with the fitting of frequency distributions to counts which are "incomplete." The most important cases of incompleteness covered are: (1) "Missing Frequencies" (Truncation). These arise for instance when the "zero-class count" cannot be observed. An example of this kind is dealt with in detail and is concerned with a chromosome breakage study (Sampford 1955, Biometrica 42, 58) in which susceptible cells showing no breakage are not distinguishable from cells not susceptible to breaks. (2) Grouped or Pooled Frequencies (Censoring). These arise for example when the number of counts exceeding a tolerance value have all been pooled in one group. The fitting of frequency distributions to such data by the method of maximum likelihood is considerably simplified by a new iterative procedure akin to the missing plot technique in analysis of variance: Trial values are estimated for the missing frequencies and these estimates are iteratively improved by their maximum likelihood estimates until, at convergence, the solution agrees with the (computationally more complex) solution of the maximum likelihood equations. New methods for estimating the variances of maximum likelihood estimates are also developed and illustrated with numerical examples.

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