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Kendall's Advanced Theory of Statistics, Volume 1: Distribution Theory

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As a grad student, the 472 page first volume of Kendell and Stuart was the book I relied on in order to learn how to calculate the unbiased estimator of a population of statistical data. In elementary texts on statistics and data reduction you are given an inkling of this problem with regard to calculating a quantity such as the mean of a finite distribution. A real distribution differs from an ideal distribution in that its number of elements is finite rather than infinite. In order to compensate for the fact that the real distribution contains as few as N elements, the sum of a given value (e.g. position) for each element is divided by N-1 (instead of N for an ideal or infinite distribution) in order to better estimate the mean. In order to properly compensate for the finite number of elements of a real distribution, however, one needs to calculate the unbiased estimator of that distribution. The books teaches the reader the complex techniques, concepts, and statistical and populations parameters that are used in compensating for the finite nature of real data.

Populations consist of an infinite number of events, and statistics to the finite number of events that correspond to actual data. In order to bridge this gap between the idealized world of populations and the everyday world of statistical data, Kendall and Student introduce the reader to a variety of mathematical entities, some used to characterize abstract populations; other belonging to statistics. In addition to the familiar moments characterizing populations such as the mean, the authors develop the concept of cumulants, which are the logarithmic analogues of moments. Being an logarithmic entity, the cumulant is independent of the choice of origin. As a result, by expressing a moment in terms of cumulants, the researcher is able to set the origin to zero and thereby allow odd moments to also assume the value of zero--thus greatly simplifying the expression. The expectation value of a product of cumulants can then, in turn, be expressed as a k-statistic, which can be formulated in terms of augmented symmetric functions and power sum statistics.

I should note that it is my experience that the need for the complicated mathematical machinery discussed in this book is not always obvious when first calculating a statistical quantity, which often consists of a sum of moments to the second or forth power. The problem, however, has a tendency to become more difficult when the researcher needs to calculate the statistical variance of the quantity in question. If a given statistic includes a forth moment, for example, its variance will include an eighth moment. Calculation the unbiased estimator of this eighth moment will certainly require use and understanding of all of the population and statistical parameters development in this book.