On the resolution of a mixture of observations from two gamma distributions by the method of maximum likelihood
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The gamma distribution occurs very frequently in applied statistical work. This paper considers in detail the problem of assigning to their population of origin the members of a sample which is the result of mixing two random samples from two distinct gamma populations, and of providing estimates for the parameters (of scale and location) of these two populations.
Some attempts have previously been made to approach this problem using the usual maximum likelihood method, but have met with little success because of the extreme intractibility of the resulting maximum likelihood equations. Recently, a slightly different model has been considered from the maximum likelihood viewpoint in the cases of a mixture of Gaussian distributions, and of gamma distributions. In the later case, however, such severe constraints were placed on the population parameters that effectively only a trivial case was discussed.
This paper develops the range of results available for resolving a mixture of gamma distributions to an extent that covers most situations that are likely to arise in practice.
The final section of the paper considers the asymptotic properties of the estimators briefly in general, and in detail in a particular case.
KeywordsGaussian Distribution Random Sample Stochastic Process Probability Theory Economic Theory
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