- Series: Kendall's Advanced Theory of Statistics (Book 1)
- Hardcover: 704 pages
- Publisher: Hodder Education Publishers; 6 edition (June 2, 1994)
- Language: English
- ISBN-10: 0340614307
- ISBN-13: 978-0340614303
- Product Dimensions: 9.7 x 1.6 x 6.8 inches
- Shipping Weight: 3.5 pounds
- Average Customer Review: 4 customer reviews
- Amazon Best Sellers Rank: #3,084,551 in Books (See Top 100 in Books)
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Kendall's Advanced Theory of Statistics: Volume 1: Distribution Theory 6th Edition
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'As a comprehensive resource, ... it is unsurpassed.' -- International Statistics Review 'The general authority of this work, together with its lasting value as a reference source, make it the Bible which no statistician should be without.' -- The Statistician 'Will occupy a central place in the statistical literature.' -- Nature --This text refers to an out of print or unavailable edition of this title.
From the Back Cover
This major revision contains a largely new chapter 7 providing an extensive discussion of the bivariate and multivariate versions of the standard distributions and families. Chapter 16 has been enlarged to cover mulitvariate sampling theory, an updated version of material previously found in the old Volume 3. The previous chapters 7 and 8 have been condensed into a single chapter providing an introduction to statistical inference. Elsewhere, major updates include new material on skewness and kurtosis, hazard rate distributions, the bootstrap, the evaluation of the multivariate normal integral and ratios of quadratic forms. This new edition includes over 200 new references, 40 new exercises and 20 further examples in the main text. In addition, all the text examples have been given titles and these are listed at the front of the book for easier reference.
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The more general focus of this book is that of distribution theory, a discipline dedicated to describing the statistical distribution of the values associated with the members of any group of individuals or events, be they atoms, workers in a given industry, deaths due to smallpox, or pencils in a can on your desk. The concept of population (sometimes called a parent population) is defined as an potentially uncountable or infinite set of such events or individuals, while statistics correspond to the finite set of events or individuals that correspond to actual data.
In order to bridge this gap between the idealized world of parent populations and the statistical data that they beget, Kendall and Student introduce the reader to a variety of mathematical tools, some of which are used to characterize parent populations; while others belong to the realm of 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 assume the value of zero--thus greatly simplifying mathematical expressions that correspond to a sum of such moments. 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. Augmented symmetric functions are statistics that are merely the sum of products. Each such product can be broken in to simple moments that are referred to as power sum statistics.
One therefore proceeds as follows: express the quantity you wish to estimate in terms of the sum of products of parent moments. Express each parent moment as a sum of cumulant products. Now that your quantity corresponding to you parent population is expressed as a sum of cumulant products, you are ready to determine that statistic that is its unbiased estimator. The unbiased estimator of each cumulant product is equal to a k-statistic. Each k-statistic is expressible as a sum of augmented symmetric functions. Each augmented symmetric function is expressible as a sum of products of power sums. The final result is the statistic that is the unbiased estimator of your parent quantity expresses as a sum of products of power sums.
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 discussed in this book.