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The Actuary The magazine of the Institute & Faculty of Actuaries

Book review: Systems of Frequency Curves

Andrew Smith revisits Systems of Frequency Curves by W P Elderton & N L Johnson 


TITLE: Systems of Frequency Curves

PUBLISHER: Cambridge University Press

ISBN-10: 0521093368

RRP: £17.99


Developments in risk modelling have sent actuaries trawling for heavy-tailed and asymmetric probability distributions. The 2009 reprinting of Systems of Frequency Curves, by William Palin Elderton and Norman Lloyd Johnson, has come at a good time.

The actuarial search for exotic distributions goes back a long way. The book's first edition was published in 1906. Both authors were British actuaries - with Elderton President of the Institute from 1932-1934. The book incorporates frequent references to actuarial work. Most of the examples relate to life insurance statistics.

Those familiar with modern computing techniques can only be impressed by the ingenuity of previous generations to reduce calculation burdens. Any statistical work starts by sorting the data into discrete groups. Moments are estimated based on the midpoint of each group, with elaborate (but optional) corrections for the distortions created by the grouping. Another section details the interpolation of arctangent functions from printed tables of the tangent function, discussing the relevant merits of tables in degrees or radians. With modern computing power, these considerations are obsolete as we calculate moments directly and have arctangent functions wired into computer processors.

The book gives a fascinating view into statistical thinking in the early 20th century, prior to the pioneering work of Ronald Fisher that dominates most other statistical textbooks. Some of Elderton and Johnson's terms are preserved in today's actuarial conventions despite having fallen out of general statistical use.

For example, distributions are 'graduated' rather than 'estimated'; the objective is the smoothing of data rather than statistically efficient parameter error. Modern actuaries might talk in this way about the graduation of mortality curves but would probably not use this language for fitting a distribution. The book pre-dates the modern convention that probability density functions integrate to 1. Instead, the proposed 'frequency curves' integrate to the number of data points - an approach that lives on in the actuarial tradition of mortality tables starting at 100,000 rather than one.

Yet this is not a purely actuarial text, but rather part of the mainstream statistical canon. For the past century, this book has been prized for its clear exposition of Pearson's system of distributions. Analysts following developments in Value-at-Risk techniques will welcome the material on series expansions and Johnson's unbounded distributions. Actuaries debating the relative merits of maximum likelihood estimation against the method of moments can find some of today's technical arguments in the pages of Elderton & Johnson's book. Once again, there is an interesting historical context.

The place of this book in the statistical literature is a credit to its authors but also reflects well on actuaries generally. Likewise, the Institute's reputation is surely enhanced by having such an eminent statistician serve as President. Extrapolating from this experience, actuaries writing tomorrowÕs learned texts should be encouraged by the contribution they are making to the profession's standing in 2012.

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