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REVIEWS

The Fundamental Principles of Mathematical Statistics. By HUGH H. WOLFENDEN.

[Pp. 379+xv. The Macmillan Co. of Canada, Toronto, 1942. $5.00]

THE author of a book on statistics must be faced with a number of major decisions to make.

The first decision will concern the mathematical ability to be assumed in the reader. Wolfenden, specifically writing ‘with reference to the requirements of actuaries and vital statisticians ’, has commendably kept to his brief. The ordinary actuarial student needs little more than an occasional refresher to find the whole of this book within his mathematical competence.

The second decision will concern the scope of the book. The territory is immense and cannot be covered in one volume. Wolfenden’s decision here is to skirmish over a large part of the field, occasionally attacking some actuarial strong-point in force (for example, the point binomial and its significance and developments) but leaving other regions almost-entirely untouched (e.g. analysis of variance, correlation, regression). This decision has its merits: it ensures continuity and reasonable comprehensiveness, and it illuminates many of the interrelations of different aspects of statistical theory-for example, the links are drawn between the Lexis theory, the x2 distribution, the t- and z-distributions and the least-square theory. Naturally, however, there are demerits; the tempo of the development tends to be too rapid so that the beginner may find it difficult to focus attention on the salient features.

These aspects, both merits and demerits, are accentuated by the unusual arrangement the author has adopted. The book is divided into five sections: a condensed presentation of the whole subject (148 pages); history (28 pages); mathematics and interpretations (83 pages); applications (86 pages); two bibliographies of historical and of current value, and an index (34 pages), There are considerable advantages in this arrangement since the first section, uninter- rupted by detail, moves along in one orderly and impressive sweep, while the second, third and fourth sections (which are well supported by a highly efficient index) provide useful illustrations and elaborations of the main theme. These later sections also form a valuable book of reference in themselves.

It is clear, however, that the author. must have been in occasional doubt as to which of the various sections were the proper homes for many parts of the subject, and as a consequence there is a certain amount of unevenness in the book. For example, it may be doubted whether the new student could under- stand the condensed development of the Lexis theory in the first section; reference to the third section should resolve his difficulties but the rhythm of the first section is lost.

One of the most satisfactory features is the treatment of the philosophy of the subject. The words ‘Fundamental Principles’ in the title are justified. Although the book, partly because of its wide range, is necessarily brief in its discussion of controversial matters, yet the fundamental issues at stake are well signposted. The treatment, though elementary, is not superficial.

Each reader will of course find his points of divergence from the author. The reviewer, for example, objects to the liberal use of an assumed ‘true underlying probability’. The assumption of such a probability will usually be found to be

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unnecessary and therefore dangerous. If it is true that the proof of the scientific pudding is in the eating, then it is also true that the main cause of scientific indigestion has lain, not in the natural ingredients, but in the cooking utensils which have been inadvertently left in the pudding. One may mention the assumption of the reality of Kelvin’s ether, the assumption of the Euclidean straight line-and the assumption of a true underlying probability. Perhaps Wolfenden saves himself by referring to ‘true’ in inverted commas, but he occasionally forgets the significance of the inverted commas and draws the plausible but meaningless distinction between the postulated value which is under test and some ‘true ’ value.

An attraction of the book to the actuary is the emphasis placed on actuarial matters, particularly in the fourth section on applications. The author is to be congratulated on his useful marshalling of much widely scattered information on such varied matters as graduation, risk-theory, population growth, and so on. A noticeable omission, however, is the simple formula pq for the variance of claims-the fundamental formula which has npq as a special case and which seems to be the subject of some conspiracy of silence to judge by its absence from current actuarial literature in the English language. This omission is the more remarkable since it is implicit in several more elaborate formulae which are set out at length (e.g. Cody’s formulae in groups of sums assured). Perhaps it is different in America but few English actuaries carry this simple formula as part of their instinctive equipment.

Among a few minor points for criticism may be mentioned the author’s. emphasis on the necessity that, for a valid application of ‘Student’s’ t-test, the value of s should not be unusual. Surely the issue is simply that as a test of

the t-test is unobjectionable, but that as a test of ( it must be interpreted with caution.

In conclusion, Wolfenden has done us a distinct service in writing a book specifically for actuaries which will make a valuable addition to any actuary’s bookshelf. F. M. R.

Surrender and Paid-up Policy Values. By H. N. FREEMAN, G. F. MENZIES and M. E. OGBORN.

[Pp. 25. T. and A. Constable, Ltd., 1946. 4s.]

THIS booklet has been written for the use of students studying for the examina- tions of the Institute of Actuaries and the Faculty of Actuaries. The authors have fulfilled a useful purpose and students will find the booklet a valuable aid to their studies. It is, however, only with difficulty that such a complex subject can be condensed into twenty-five pages and the result cannot be expected to cover every aspect.

The early sections discuss the general bases and basic formulae. The practical and theoretical points of view are closely interwoven. A clearer distinction might have been preserved between the prospective or ‘sale value’ aspect and the retrospective or ‘amount in hand’ aspect, which are of importance, varying with duration in force, as bases for testing scales of surrender values and paid-up policies (not even excepting the generally accepted proportionate paid-up policy). Thus, for endowment assurances approaching maturity (say within two years) it is not unusual to calculate surrender values solely from the

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prospective approach by discounting the sum assured and bonuses anticipated to be payable at maturity and deducting any unpaid premiums.

The necessity for an office to recuperate itself on surrender for the unliquidated portion of initial expenses is rightly underlined, but it should, at the same time, be remembered that an office taxed on ‘interest less expenses ’ will have already recovered tax at 7s. 6d. in the £ on initial expenses.

The difficult problem of allowance for profits is well treated. Altered-class policies, double endowment assurances, pure endowments and children’s deferred assurances are fully discussed in the later sections, where mention is also made of family income policies, extra premiums and non-forfeiture schemes. w. E. H. H.

Tables of Fractional Powers. Prepared by the MATHEMATICAL TABLES PROJECT under the sponsorship of the NATIONAL BUREAU OF STANDARDS.

[Pp. 489. Columbia University Press, New York, 1946. $7.50]

THESE tables are described in a foreword to the book as being in a sense a generalization of Barlow’s well-known tables. Approximately half the book is taken up by tables of Ax for values of A from .01 to .99 by intervals of .01 and also between .100 and 1.000 where 1000 A is a prime, the values of x in both cases running from .001 to .01 by intervals of .001 and then to .99 by intervals of .01. There are also tables of Aa where A is a whole number between 2 and IO inclusive and where A= , and there are tables of square roots, cube roots, etc. In nearly all the tables the entries are tabulated to 15 places of decimals.

An introduction explains how the tables were constructed, and shows how values of Ax not tabulated may be found by multiplication or interpolation. A bibliography is included.

Versicherungsmathematik, By E. ZWINGGI.

[Pp. 199. Birkhäuser, Basle, 1945. 34.s]

THIS book can be recommended to all actuaries interested in the mathematical basis of Life Contingencies. It is extraordinarily compact considering the ground covered. The size of the book is about three-fifths that of Spurgeon’s text-book, but the scope is appreciably wider. In fact the scope is fairly close to that which will be required under the new Institute syllabus.

For British actuaries, unfamiliar with Continental text-books, the most interesting feature of the book will be Prof. Zwinggi’s logical approach to the subject. This approach is very similar to that adopted by W. G. Bailey and the reviewer in the booklet Some theoretical aspects, of multiple decrement tables, and hence it can be inferred that the reviewer has found himself in almost complete sympathy with it.

In an interesting introduction Prof. Zwinggi explains his attitude towards ‘ life assurance mathematics ’. He points out that, although life assurance has been practised for many years, the logical foundations have not yet been laid to the satisfaction of all. Assured events do not occur in a completely ‘planless’ manner; there is some regularity in the amount and time structure of claims

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when a large number of persons are insured. The ‘law’ of these events can be found only from past experience, and if we apply this ‘law’ to the future we anticipate that all relevant conditions will not vary very much. The mathematical ‘set-up ’ can be regarded as a model of the assured events through time, and although we are certain that there will not be an exact repetition of the past we are still prepared to use the model.

Life Contingencies is, then, the working-out of the implications of the model. It follows from this assumption that the theory of probability as a basis of scientific inference is not part of the subject-matter, and for this reason Prof. Zwinggi would have preferred to drop the word ‘probability’ and develop in effect a calculus of frequency. However, on account of long usage among actuaries he has retained the terminology of the calculus of probability, but warns his readers that it is misleading and that in the development of the section on assurances involving more than one life the rules of the calculus are used only for convenience, because without them the mathematics would become very unwieldy.

The book is divided into seven parts, and in Parts I and II the model in its most general form is developed. The foundation of the model is a set of forces of decrement (i= 1, 2, . . ., h), and thus Prof. Zwinggi commences with the notion of data subject to more than one cause of decrement ; unlike Spurgeon he does not consider it necessary that students should be introduced in the first place to ideas based on one cause of decrement only. British actuaries will, no doubt, contest this view with the argument that, whilst Prof. Zwinggi’s method is suitable for graduates in mathematics, it is not the best way of teaching the subject to boys who have had only a secondary school education. In the reviewer’s opinion much would depend on the teacher. A good teacher could afford to try the more general approach. He would then give his students a more unified picture than he could probably give by the other method. In Prof. Zwinggi’s hands the unified method gives much insight into the basic assump- tions underlying actuarial technique.

In order to define Prof. Zwinggi first defines a function as the number of individuals leaving the group l[x] from cause (i) during time t ; thus

The dependent probability is then defined as

and

From this definition and the usual definition for independent rates Prof. Zwinggi demonstrates the mathematical relationships between these various functions and obtains approximate practical formulae for the relationships between dependent and independent rates.

A peculiar characteristic of the book is that the infinitesimal calculus appears only in these demonstrations. Once it has been shown that the different rates depend on the basic postulate of a set of forces, the remainder of the mathematics in the book is based on rates only and the values of contributions and benefits are shown as discrete multiple summations. The reason for this procedure is the desirability of obtaining practical formulae. Even so, the reviewer is not in

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complete sympathy with Prof. Zwinggi on this point. In many cases the integral is the more natural symbol to use, and practical approximations can usually be obtained.

In Chapter I of Part II Prof. Zwinggi sets out the generalized equality between the value of premiums and the value of benefits. The equation is written

where E[x] = single payment due at outset, P[x]+t = varying annual payment for n years, varying benefit if the assured withdraws from cause (i), m ( n) = term of assurance, T[x]+m = benefit if the assured survives m years. This approach, based on a multiple decrement table, enables Prof. Zwinggi to deal with all types of assurances, whether on one life or more than one life, as special cases of the above general type. From a consideration of this method of com- mencing with a postulate of a set óf forces of decrement and deriving therefrom a generalized equality between the values of benefits and premiums it is easy to appreciate the unity of Prof. Zwinggi’s book and the reason why he has been able to cover so much ground in relatively few pages.

In Part III (about 83 pages) the subject of assurances on single lives is fully developed. In addition to the usual treatment of net premiums, office premiums, and policy values, there are sections on invalidity assurances, methods of group and approximate valuation, and analysis and distribution of surplus. In the case of the latter two subjects the treatment is,much fuller than that of Spurgeon and will be of interest to those responsible for the new Institute text-book.

Part IV is a very short section of eight pages which describes a form of group life assurance well known on the Continent of Europe.

Part v deals with assurances involving more than one life. Only seven pages are given to joint-life annuities and assurances, on the grounds that the principles are exactly the same as those for single-life assurances. The remainder of the part deals with group and pension assurances.

It is doubtful whether Parts VI and VII should be included in a book which eschews probability and deals only with the mathematics of the multiple decrement table. Part VI, entitled Variations in the calculation bases, considers ‘period’ and ‘generation’ mortality and the extrapolation of mortality rates, and Part VII deals with graduation. A full treatment of these subjects would require the methods of modern statistics which is outside the sdope of the book. Prof. Zwinggi can, therefore, do no more than describe the various well-known mathematical methods that have been used to fit curves to mortality rates. He has made use of the very fine paper of Cramér and Wold entitled Mortality variations in Sweden-a study in graduation and forecasting which appeared in Skandinavisk Aktuarietidskrift, 1935—a paper which should be in the Institute Part III reading. It gives one of the most elegant and simple accounts known to the reviewer of the fitting of Makeham’s Law to ‘period’ and ‘generation’ mortality. An outline of Cramér and Weld’s investigation is well worth in- cluding in any text-book.

The book would not, by itself, be suitable for Institute students ; in addition a book on problems would be required. The Institute examination up to 1942 was tending to become more and more an examination in relatively difficult algebraic problems dressed up in the garb of Life Contingencies. A student, therefore, is more interested in learning the tricks for solving such problems than in gaining insight into the methodological principles of his technique.

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Prof. Zwinggi’s book does not even contain any simple examples, except in so far as the mathematical treatment of a particular type of assurance serves as an illustration of the method.

While this review was being written, a review of Prof. Zwinggi’s book by Dr S. Vajda appeared in the Journal of the Students’ Society (Vol. VI, p. 149). The latter contains some interesting comments on parts of Prof. Zwinggi’s book that are not dealt with here. H. w. H.

Mathematical Methods of Statistics. By HARALD CRAMÉR.

[Pp. 575 + xvi. Almqvist & Wiksells Boktryckeri AB., Uppsala, 1945. Kr. 45]

ALL those who have read Prof. Cramer’s Cambridge Tract, published in 1937, on Random variables and probability distributions must have hoped that he would provide a fuller treatise on similar lines, and it is a pleasure to his many friends and admirers that, in spite of the many demands on his time, he undertook the work on his new book, completed it in May 1945, and arranged for its publication in the Princeton Mathematical Series as well as in a Scandi- navian edition.

The book, as would be expected by those familiar with its predecessor, is mathematical and will appal especially to mathematicians interested in the theoretical side of statistics, but it will also be of interest to those statisticians who, though more interested in practical statistical applications, want to appre- ciate the mathematical background of the subject. It is no doubt for the sake of the latter class of reader that the author has devoted the first 136 pages to a mathematical introduction dealing primarily with sets and the Lebesgue and Lebesgue-Stieltje integrals but including chapters on Fourier integrals, on matrices, determinants, etc., and on the Euler-Maclaurin formula, I’ and B functions, Stirling’s formula, and orthogonal polynomials. This part, in fact, describes the mathematical tools that are used in the rest of the book. The actuarial reader who is unfamiliar with the idea of sets might start by reading what is said about them in Courant and Robbins’s What is Mathematics?, where the connexion with probability is mentioned-he will enjoy the whole book-, and, if he wants further information about matrices and determinants, he will find A. C. Aitken’s book very helpful.

The second part of Cramer’s book bears the same title as his Cambridge Tract but is by no means a repetition., It starts with a chapter on ‘ Statistics and Probability’ which may be said to give the author’s philosophic reflections on those subjects and their relationship. It is beautifully clear and even actuaries who are fearful of Cramer’s mathematics should give themselves the pleasure of reading this chapter though they may skip a bit about sets in § 13, 3. After a general talk about random experiments he says: ‘it does not seem possible to give a precise definition of what is meant by the word “random”. The sense of the word is best conveyed by some examples ‘-and by means of them he succeeds. He then describes and explains what is meant by statistical regularity and this leads him to the object of a mathematical theory; he says that ‘ when. . . we find evidence of a confirmed regularity, we may try to form a mathematical theory of the subject’ and that we ‘ choose as our starting-point some of the most essential and most elementary features’ and build up various propositions by purely logical deduction from the axioms laid down. ‘ Every proposition of such

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a system is true, in the mathematical sense of the word, as soon as it is correctly deduced from the axioms. On the other hand. . . no proposition of any mathe- matical theory proves anything about the events that will, in fact, happen.’ We rely on the future agreement between theory and experience, bearing in mind that any theory which does not fit the facts must be modified. Cramér then sets out in general terms a description of mathematical probability and concludes the chapter with notes on the different opinions that have been expressed about the foundation of the theory of probability, i.e. the principle of equally possible cases, the definition as a limit of frequency (von Mises), the Kolmogoroff view from certain axioms, and the conception of probability as a theory of degrees of reasonable belief (Keynes and Jeffreys). On this last theory, after remarking that in its extreme form it might assign a numerical measure to such a statement as ‘ The “ Masque de Fer ” was the brother of Louis XIV ’, he writes, ’ probabilities of this type have no direct connexion with random experiments, and thus no obvious frequency interpretation. . . ; we shall not attempt to discuss the question whether such probabilities are numerically measurable and, if the question could be answered in the affirmative, whether such measurement would serve any useful purpose’. All this summary of the views held about probability is printed in small type and Cramér’s comments so printed should never be skipped-they are, perhaps, even more important to his readers than the small print in a prospectus of a new issue is to the investor!

The author, as is well known from his earlier work, adopts the Kolmogoroff point of view, and proceeds in Chapter 14 to give in mathematical form three axioms, almost the same as Kolmogoroff’s ; as set out they will appear ultra- mathematical to an ordinary reader, but they are not really hard to follow and in the conclusion to the chapter, the author may be said to have translated them into ordinary language.

Chapters 15-20 are concerned mainly with variables and distributions in one dimension. Starting with frequency distributions, mean values, moments, semi- invariants and measures of location, of skewness and of excess, we come to the first use of characteristic functions which.were explained in Part I, Chapter IO, and which play so important a part in all the subsequent mathematical work. We proceed to a chapter on various discrete distributions beginning, of course, with the binomial and its connexion with the.normal curve and the Poisson distribu- tion, and this is followed by a chapter on the normal distribution and the addition of independent normal variables, so that we reach the central limit theorem which, beginning with independent variables; has been shown to be capable of extension to various cases when the variables are not independent-on this subject much original work has been done by Cramér himself. After a brief reference to the logarithmic-normal curve, there is a discussion of the orthogonal distribution of the normal with warnings about convergency and then of Edgeworth’s asymptotic expansion which is not open to the same objections. The summary at the end of Chapter 17 on the role of the normal distribution in statistics is excellent-just the sort of thing that is wanted. Chapter 18 gives the x2 distribution, transformations of it that are needed in applications, ‘ Student’s’ distribution and Fisher’s z-distribution. The next chapter considers, among further continuous distributions, the rectangular, Cauchy’s, a special case of ‘ Student’s ’ in which ‘ no moment of positive order, not even the mean, is finite ’, and the Pearson system-for which the author gives the differential equation and explains that it covers many of the distributions already discussed, but he does not deal with the system further except to mention the actual types of which

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the distributions he has described previously are special cases. After a chapter giving some convergence theorems, there are four chapters on cases of two or more dimensions in which he discusses the line of closest fit to the regression curve, which in the case of a straight. line is identical with the mean-square regression line. The parabolic mean-square regression of order n>1, which generalizes the linear, is dealt with by considering the regression polynomial as a linear aggregate of the orthogonal polynomials associated with the marginal distribution, so that coefficients are obtained directly without first having to solve a system of linear equations; hence, if we know the regression poly- nomial of degree n and require it for degree n + 1, it is only necessary to add an additional term. He then points out that it is not essential that the functions, py (x) in his notation, are polynomials, but ‘any sequence of functions satisfying the orthogonality conditions may be used to form a mean-square regression curve y =g (x)= cypy (x)’ and the relations will hold good irrespective of the form of py (x). How far would this cover cases where the regression curve is of some form other than a polynomial? In common with practically all writings on the subject, other possible curves for the regression curve are not mentioned.

From the regression curves the author takes us to the correlation coefficient and says that ‘two uncorrelated variables are not necessarily independent’; he gives a mathematical example in which p , previously defined as is zero and he then writes, ‘thus two variables with this distribution are uncorrelated’, but, apart from any question of a particular example, is not this going perilously near to saying that µ11/ is the only correct measure of correlation? This does not seem to be wholly logical unless and until we can prove that no other measure of correlation, in the dictionary sense of that word, is possible. All it seems he has really done is to say (1) this is a measure of correlation, (2) it is zero so there is no correlation, (3) here is a case where the measure is zero but the variables are not independent. Moreover, a few pages later other measures that have been devised to characterize the ‘degree of dependence’, e.g. the mean-square contingency, are discussed.

In Chapter 23 the author deals with regression and correlation with n variables, and a small-type paragraph at the end is on orthogonal mean-square regression —yet one more example of small print not to be missed. The final chapter in Part 11 relates to the normal distribution with n variables and the central limit theorem in such cases—again a subject on which his own work may be recalled.

At the end of certain chapters in Part 11 are some examples which he leaves the reader to work out; they are all mathematical examples and many of them have appeared in some form or another in published material—a point that may be borne in mind in considering the total content of the volume.

The third part of the book is entitled ‘Statistical Inference’ and provides evi- dence that the theory dealt with in Part II is in line with the experience which must be its ultimate justification. This part starts with a chapter on preliminary notions on sampling, and the author then devotes a chapter to general considerations of statistical inference which are divided into description, analysis, and prediction, and so link up with Chapter 13. All this is set out clearly and is good reading. The next chapter deals mathematically with the characteristics of sampling distributions and connects with the corresponding work in the second part which, as already explained, deals with the theory—the characteristics being the mean, moments, certain functions of moments, correlation coefficients, etc.—; the corrections for grouping (Sheppard) are also given. We next have a

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chapter dealing with the asymptotic properties of sampling distributions, and the wide range within which various sample characteristics are asymptotically normal for increasing size of sample is brought out. In Chapter 29 we come to exact sampling distributions, and the author introduces it by pointing out that though we have studied in the preceding chapters the conditions that arise when the size of the sample tends to infinity we are, in many cases, more concerned with the form of the distribution when the sample is small, but though we may be able to deal with the problems involved numerically with tables based on approximate formulae the solution explicitly in terms of known functions can only be reached, at present, in comparatively few cases. Some isolated results had been obtained earlier, but for many of the exact forms already discovered we are indebted to R. A. Fisher, and the author gives proofs of some of Fisher’s results using the method of characteristic functions. One of Fisher’s discoveries was the distribution of the coefficient of correlation and the transformation

which turns the skew distributions into normal distributions for

moderate-sized samples: the author’s diagrams 29 and 30 are helpfully explana- tory. There is one rather difficult place in this chapter where a formula (29, 5, 1) appears but it is not till some paragraphs later that after a nice piece of work its origin seems clear. In Chapter 30 we have tests for goodness of fit, etc., and Cramér proves the basic x2 theorem by means of the characteristic function and paves the way to tests of significance and descends (or is it ascends?) to some well-chosen numerical examples. The author gives on pp. 449-450 a test of whether the death-rates from two sets of data differ significantly. This test has been used before and an example is given where an attempt is made to decide whether there are any significant differences in mortality between the two sexes during the first year after finding T.B. + , and the author concludes that the deviation is ‘almost significant’, i.e. the probability lies between the 1% and 5% limits. I confess that I have never felt completely happy about the test, and in the particular example the old-fashioned method of comparing the results for men and women separately with the figures obtained from the combined exposed and deaths would have told all that we want in practice-perhaps this is another way of saying that the test seems to answer a problem different from the one that I should set!

Chapters 32-34 deal with the theory of estimation, and, as the author says in a footnote to p. 473, the topics treated are highly controversial and there are divided opinions about the relative merits of the concepts and methods discussed. But, whatever view may be taken, the reader will find that the author has a valuable mathematical contribution on the estimation of parameters, has pro- duced something solid in connexion with Fisher’s concept of efficiency, and has proved that Fisher’s maximum likelihood estimate for one parameter is asymp- totically normally distributed with the least possible variance and so on. It may be mentioned that in Chapter 32, §§6-8, the author deals with the case of two or more unknown parameters, and he has subsequently given the proof that he merely indicates in the book (see Skandinavisk Aktuarietidskrift, 1946, pp. 85-94).

In Chapter 34 (confidence regions) the author compares, for the case of a single unknown parameter, the old treatment by Bayes’s theorem with the modern method of confidence intervals due to Neyman and brings out clearly the distinction between the two methods; some of what appears might almost be taken as a comment on Perks’s recent paper had it not been written more than a year earlier.

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Finally we come to chapters on the general theory of testing statistical hypotheses, in which he explains and discusses the Neyman-Pearson theory, on the simpler case of the analysis of variance which has proved valuable in many branches of statistical work, and on some regression problems involving non- random variables.

The book is excellently printed and the proofs seem to have been read with great care. The errors we have noticed are unimportant, such as when in one place ‘variance’is used for instead of or when a line does not quite read as at the bottom of p. 376, or when the singular ‘term’ has been printed instead of the plural ‘terms’ (p. 435), or when on p. 444 the last sentence just above formula (30, 5, 2) seems to want re-wording because the use of the phrase ‘on the other hand ’implies a difference from what precedes where no difference apparently exists; and Maclaurin is spelt throughout with a capital L.

So, having come to the end of this interesting and valuable work, we look back and ask ourselves what else Cramer might have given us and, remembering his attractive course of lectures in London before the war, we may hope that some day in the near future he may find time to provide us with something on statistical time-series and periodograms. We should have liked to know his views on many questions on the arithmetical side of statistical research but that would have meant a lot of numerical examples and, as he says in his preface, the book had to be kept within reasonable limits. If therefore a reader wants something else added he should, to be fair, say what he would exclude and that would indeed be a puzzle. The difficulty of that puzzle is a measure of our gratitude for the book.

On the page following the title-page is a dedication in two words to his wife: this will give pleasure to the many actuaries in Great Britain and all over the world who hold both Harald and Marta Cramér in affectionate regard.

W. P. E.

Actuarial Statistics, Volume 1, Statistics and Graduation. By H. TETLEY, M.A., F.I.A. (Co-ordinating Editor: HARRY FREEMAN, M.A., F.I.A.)

[Pp. xvi + 285. Cambridge University Press, 1946. 21s.]

THIS book, issued under the auspices of the Institute and the Faculty, represents the first half of what is to be the Part III text-book. Volume 11, on the con- struction of mortality and other tables, is in course of preparation.

The book is primarily intended for the actuarial student, and in a preface the author states that no attempt has been made to produce a statistics text-book for general use. As the author explains, mathematics has been given rather more prominence than is usual in elementary statistical works because it has been found that actuarial students find this form of treatment interesting and stimu- lating. The book has grown out of lesson notes prepared by the tutors for the Institute and Faculty examinations. It was commissioned before the war and its production has been held up by the war; it does not therefore reflect the changed order and emphasis contemplated in the training of students in statistics under the proposed new syllabus of the Institute’s examinations.

There is a short introduction and ten chapters, of which the first four are concerned with statistics (variables and grouped data, the binomial and normal distributions, correlation, sampling) and the remaining six with graduation

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(general considerations and tests, graphic method, reference to standard table, summation formulae, mathematical formulae, osculatory interpolation). The book is copiously illustrated with tables and diagrams and contains many worked-out examples, At the end of each chapter there is a set of examples for the exercise of the student as well as a short bibliography. An index is included.

The task of the author of a text-book such as this is unenviable. On the one hand is the need, in order to make the subject-matter intelligible and interesting to the student, to give at each stage the ‘why’ and ‘wherefore’ in terms not too erudite or abstruse; on the other hand is the danger of framing the explanations in a manner which would not stand up to critical examination by experienced statisticians. Mr Tetley has clearly had these considerations in mind. It may be thought that the introduction of the normal curve has been made too abrupt, insufficient exposition being given of the reasons for its existence and importance; but against this it can be argued that it is better for the student to learn the reasons gradually as he works his way through the book. Among minor printer’s errors—which will no doubt be corrected in subsequent editions—are noticed the use of the Greek a instead of the italic a in ‘a (f) and a (m) tables’ on p. 248, and the omission of the apostrophe in ‘Actuaries’ Investment Index’ on p. 25; the expression ‘Female Government Annuitants Table’ on pp. 153, etc., is not a very happy choice. It would be easier to find one’s way about the book if the right-hand pages were headed with the titles of the chapters rather than of the, sections.

There has been a long-felt need in the course of reading for Part III of the Institute examinations for a suitable text-book on statistics and graduation. It is to be hoped that Mr Tetley’s book will fill the gap.

Some Theoretical Aspects of Multiple Decrement Tables. By W. G. BAILEY, B.A., F.I.A., and H. W. HAYCOCKS, B.Sc. (Econ.), F.I.A.

[Pp. 40. T. and A. Constable Ltd., 1946. 5s.]

THE data available for the study of multiple decrement tables do not usually form a sufficient basis for rigorous mathematical treatment, but the application of the theory to unusual problems during the war necessitated a more thorough treatment than the subject had been given in the past. The mathematical theory is set out in a booklet published by the Institute of Actuaries and the Faculty of Actuaries.

Though prepared for students the booklet should be studied by actuaries generally.

Suppose that forces of decrement (denoted severally by the indices etc.) are in operation, and suppose that in the multiple decrement table the corre- sponding forces are denoted by the same symbols prefixed with a letter (a, say) to define the multiple decrement table. The total force, aµ, within the multiple decrement table is the sum of the separate forces within that table, i.e.

The conditions under which the elements of the right-hand side of the equation can be put identically equal to the several separate forces (i.e. aµ = µ , etc.) go to the roots of the practical problems involved in dealing with data subject to more than one force of decrement. The actuary—as distinct from the student—

Reviews 175

could, we think, turn with profit to §§ 10 and 11 where these practical problems are considered before he studies the mathematical exposition.

Given the identity referred to, we may write aµ = µ + µ +µ + . . . . Each force is assumed to give rise to functions defined by. the special index or prefix on the basis of relationships similar to those obtaining between µ and the ordinary life- table functions. Remembering the exponential relationship between µ and p, it follows at once that the p’s are compounded by multiplying them together. The simple rule that µ’s are added but that p’s are multiplied forms the basis. of the mathematical work. An interesting corollary is the identity of the process between tables based on several forces of decrement and tables based on several lives. M. E. 0.

National Insurance Bill, 1946. Report by the Government Actuary on the Financial Provisions of the Bill. (Cmd. 6730.)

[pp. 20 +Appendix, 12. H.M. Stationery Office, 1946. 6d.]

THIS Report contains an interesting and valuable Appendix on the actuarial basis of the estimates made. There is, of course, much of interest in the methods by which the estimates of contributions and emerging cost have been built up, but these methods follow the lines of earlier reports on social insurance, and this note is confined to a brief review of the mortality and other rates adopted.

The Government Actuary states that he has made use of the most recent available data relating to mortality, marriage and other vital statistics, but that no allowance has been made for possible future changes except as regards sickness. Since the cost of retirement pensions is the largest element of the whole National Insurance Scheme, any marked improvement in vitality in the future will mean that the estimates of emerging cost are too low. As is pointed out in the Ap- pendix, however, the future position is obscure, and ‘on balance it was thought preferable, for the purpose of financial estimates, to adopt an up-to-date ex- perience in respect of a favourable period without modification for conjectural future improvement ‘.

Mortality. The mortality basis adopted was derived from the death-rates (excluding war deaths) experienced by civilians in Great Britain during the three years 1942–44. These rates were, in fact, light, and it would appear (though no reference is made to the point in the Appendix) that the data have been adjusted at the younger male ages, since there is no sign of the heavy rates at these ages shown by W. S. Hocking (J.I.A. Vol. LXXII, pp. 122, 522) to have occurred during the war amongst civilians owing to the powerful selective force of con- scription for the armed forces.

The actual rates adopted are shown at quinquennial ages, and it is interesting to compare these rates with those calculated by Sir Alfred Watson from the 1921 census and the deaths in Great Britain in the years 1920–22 (J.I.A. Vol. LIX, p. 310), and with the English Life No. 10 tables relative to deaths in England and Wales in 1930–32. The tables overleaf give the comparison for all men and all women respectively.

It will be noted that the relative improvement in female mortality, except at advanced ages, is materially greater than that in male mortality.

Other mortality rates shown in the Appendix relate to married men and widows. In the case of married men, it was necessary to have recourse to the

176 Reviews

ratios of the rates for married men to those for all men in 1930–32 in Scotland, as this was the most recent source from which complete information could be obtained.

Age X

17 22 27 32 37 42 47 52 57 62 67 72 77 82 87

68 66 64 60 64 71

81 81 84 84 88 88 88 88 88

Females

17 22 27 32 37 42 47 52 57 62 67 72 77 82 87

1920–22 (Watson)

qx

.00280

.00377

.00408

.00485

.00605

.00762

.00975

.01403

.02032

.03107

.04690

.07273

.11127

.16418

.22733

.00265

.00332

.00375

.00428

.00495

.00589

.00762

.01082

.01521

.2326

.0556

.05737

.09140

.14079

.19896

Males

E.L. No. 10 qx

.00259

.00330

.00328

.00361

.00474

.00639

.00925

.01295

.01890

.02875

.04568

.07246

.11325

.16927

.24078

.00235

.00282

.00306

.00332

.00392

.00486

.00668

.06941

.01377

.02110

.03321

.05435

.09025

.14065

.20844

1942–44 qx

.0019

.0025

.0026

.0029

.0039

.0054

.0079

.0113

.0171

.0260

.0414

.0640

.0978

.1444

.1998

.0015 .0018 .0020 .0023 .0028 .0036 .0050 .0072 .0109 .0171 .0275 .0450 .0745 .1161 .1720

1942-44 1920-22

%

57 54 53 54 57 61 66 67 72 74 77 78 82 82 86

1942-44 E.L. No. 10

%

73 76 79 80 82 85 85 87 90 90 91 88 86 85 83

64 64 65 69 71 74 75 77 79 81 83 83 83 83 83

Marital status. The Appendix gives a table showing, for quinquennial age- groups, the proportions of men who are (a) married, (b) bachelors and widowers; and of women who are (a) married, (b) spinsters, (c) widows. These proportions are based on the National Registration taken on 29 September 1939, though the Report states that ‘for certain purposes, the proportion of widows thus obtained has been somewhat reduced and the proportion of married women correspond- ingly increased, in order to eliminate the abnormal proportion of widows at some ages remaining from the 1914–18 war‘.

A table is also given of the percentage age-distribution of wives for various ages of husbands (in both cases in quinquennial age-groups). These statistics are not new, but relate to the 1931 census of England and Wales. Comparable information was not obtained in the Scottish census of 1931.

Fertility. For building up the estimates of the future population, recourse was had to the rates of issue for all women in Great Britain in 1938 made available as a result of the Population (Statistics) Act of 1938. For the purpose of estimating

Reviews 177

the cost of various benefits, however, it was necessary to have further information as to the incidence of births in respect of (a) insured men, (b) gainfully occupied married women, and (c) insured unmarried women, and the rates adopted were obtained by rating down the issue rates incorporated in the 1938 N.H.I. Valua- tion Regulations in the light of the most recent N.H.I. experience so as to obtain agreement in total with the births resulting from the 1938 fertility rates. The over- all ratios adopted were 60% for men and 75% for employed married women, while married women not gainfully occupied were assumed to have issue rates 50% higher than those for gainfully occupied married women. The N.H.I. rates are shown in the Appendix for quinquennial ages. The Appendix also contains an interesting table derived from the experience of the existing Contributory Pensions Scheme, showing at quinquennial ages for (a) married men, (b) women newly widowed, (c) widows, the proportions who have one or more children, under age 16. The table also shows for widows the average age of the youngest child at widowhood.

Remarriage of widows. Remarriage rates of widows, according to age at widow- hood and duration since widowhood, are also given. The rates adopted are those which were compiled in the course of the first Actuarial Review of the’ Contributory Pensions Scheme based on the experience of the years 1926–32. The Government Actuary states that even if more recent data were available, the experience of war-time rates of remarriage would not provide a proper guide for the future.

Proportions employed. A table is given showing the proportions of (a) men, (b) spinsters, (c) married women, (d) widows, assumed to be gainfully occupied. This table is based on the 1931 census enumeration for Great Britain, but the census data were adjusted in the light of National Insurance and other data to give effect to the trend towards earlier retirement and the more extensive em- ployment of women, especially married women. No account was taken of war- time changes, nor was it thought possible to forecast post-war changes.

Sickness. As regards sickness rates, the Government Actuary decided, on a balance of considerations, to increase the rates which form the financial basis of the present N.H.I. Scheme (and which are approximately equivalent in the aggregate to the experience of the immediate pre-war period) by about 20% for men, 30% for unmarried women, and 5% for married women. It may be of interest to note that these loadings are materially heavier than the 12½% loading adopted by Sir George Epps in his Appendix to the Beveridge Report.

The sickness rates resulting from these loadings were then adjusted to allow for the following factors:

(1) the change from 12 to 3 months in the off period; (2) the payment of benefit in respect of the 3 days’ waiting period if there is

a further 9 days of ‘interruption of employment’; (3) the exclusion of incapacity within the 13 weeks’ confinement period, for

which separate provision is made; (4) the payment in respect of a certain amount of incapacity attributable to

industrial injuries or to war disability for which benefit may be payable in part under the National Insurance Scheme;

(5) the non-payment of benefit during the first 24 days of sickness in the case of self-employed persons (though this difference between the benefits of self-employed and employed persons was removed before the Bill became an Act).

178 Reviews

The adjusted rates of sickness for employed and self-employed persons are set out in the Appendix for quinquennial age points, and in the following table the rates for employed persons are compared, as a matter of intrest, with the Manchester Unity, 1893–97, Whole Society rates.

Age

32

17 22 27

37 42 47 52

M.U. 1893–97

980 .890 .954

1.003 1.262 1.582 1.979 2.745 4.019 6.360

Men

1.1 1.1 1.2 1.3 1.5 1.9 2.4 3.2 4.7 7.5

Ratesadopted

Unmarried women

1.3 1.9 2.3 2.6 2.0 2.9 3.4 4.0 5.0 6.6 —

Married women

2.4 2.9 3.3 3.6 3.9 4.4 5.1 6.2 7.9 —

F. J. C. H.

57 62