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Grouping corrections and maximum likelihood equations

Published online by Cambridge University Press:  24 October 2008

D. V. Lindley
D. V. Lindley
Affiliation:
Statistical LaboratoryCambridge
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Extract

1. Any mention of the word ‘grouping’ immediately brings to a statistician's mind the Sheppard corrections. These are usually used to make inferences about the underlying ungrouped population from observations made on the grouped population, but it is important to realize that, as stated and proved, they have nothing to do with sampling or inference and are merely expressions for the moments of one population in terms of the moments of another population derived from it. They can only be used for the inference problem when allied to the method of moments. This method, as formulated by K. Pearson, consists in taking for θ* the estimate of the population parameter θ, the same function of the sample moments mi that θ is of the population moments μi, each mi being an estimate of the corresponding μi. If the population is grouped the mi are estimates of the , the grouped population moments, so we require θ as a function of the to apply Pearson's method. This can be done since θ is known as a function of the µi and the µi are known as functions of the by the corrections. Use of the Sheppard corrections with any other inference method, even when this method, when applied to the continuous population, yields an estimate which is a sample moment, so far as I am aware, has not been examined except for the normal curve.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 46 , Issue 1 , January 1950 , pp. 106 - 110
Copyright
Copyright © Cambridge Philosophical Society 1950

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References

† It may not exist, for example, if f(n rh, θ) = 0, i.e. if the likelihood, ignoring the grouping, of a sample value is zero. This does not contradict the hypothesis since p(n rh, θ) may not vanish. We return to this point later.

‡ This we shall do throughout the rest of this paper.

† This result has been discussed by Fisher, R. A., ‘On the mathematical foundations of theoretical statistics,’ Philos. Trans. A, 222 (1922), 362.Google Scholar

‡ It is assumed that no value of n i is zero; we return to this later.

† We assume the range is infinite; the general case follows similarly.