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Jia 84 (1958) 0125-0165 INSTITUTE OF ACTUARIES A MEASURE OF SMOOTHNESS AND SOME REMARKS ON A NEW PRINCIPLE OF GRADUATION BY M. T. L. BIZLEY, F.I.A., F.S.S., F.I.S. [Submitted to the Institute, 27 January 19581 INTRODUCTION THE achievement of smoothness is one of the main purposes of graduation. Smoothness, however, has never been defined except in terms of concepts which themselves defy definition, and there is no accepted way of measuring it. This paper presents an attempt to supply a definition by constructing a quantitative measure of smoothness, and suggests that such a measure may enable us in the future to graduate without prejudicing the process by an arbitrary choice of the form of the relationship between the variables. 2. The customary method of testing smoothness in the case of a series or of a function, by examining successive orders of differences (called hereafter the classical method) is generally recognized as unsatisfactory. Barnett (J.1.A. 77, 18-19) has summarized its shortcomings by pointing out that if we require successive differences to become small, the function must approximate to a polynomial, while if we require that they are to be smooth instead of small we have to judge their smoothness by that of their own differences, and so on ad infinitum. Barnett’s own definition, which he recognizes as being rather vague, is as follows : a series is smooth if it displays a tendency to follow a course similar to that of a simple mathematical function. Although Barnett indicates broadly how the word ‘simple’ is to be interpreted, there is no way of judging decisively between two functions to ascertain which is the simpler, and hence no quantitative or qualitative measure of smoothness emerges; there is a further vagueness inherent in the term ‘similar to’ which would prevent the definition from being satisfactory even if we could decide whether any given function were simple or not. 3. In their textbook Mortality and other Investigations, Haycocks and Perks write ‘Smoothness is a concept that has eluded a precise and generally accepted mathematical definition’. This quotation from a recent official text- book is sufficient evidence that the whole question is in need of clarification at the theoretical level. There may be no practical need to study smoothness, because the failure of the classical method is seldom a serious embarrassment; when its verdicts are obviously wrong we can generally use graphical methods to decide if a series is smooth in the sense in which we intuitively understand the term. Nevertheless, it is unlikely that practice has nothing to gain from a better understanding of the foundation upon which it rests, and there is reason to believe that the invention-or discovery-of a method of measuring smoothness may one day revolutionize the processes of graduation which have all perforce been designed in the absence of any such method. 4. Although in graduation we generally deal with discrete series of values, our interest in smoothness has its origin in the representation of such series by continuous curves or functions. Indeed, almost all of the variables with which 9 AJ 126 Measure of Smoothmess our work is concerned are apparently of a continuous nature, and it is only for practical convenience that we use discrete steps. We shall therefore concen- trate our study upon continuous variables. We begin with a brief survey of the classical method to see just where and why it fails. THE CLASSICAL METHOD 5. In the discrete case, the classical method consists essentially of the examination of successive orders of differences of a function for equal intervals of the argument. The function is accepted as smooth if the absolute values of a fairly low order of differences (say the third order) are small, which implies that the previous order must be approximately constant. In the continuous case, the method demands analogously that a low order of differential coefficients with respect to the independent variable shall be numerically small for all relevant values of this variable. 6. The criterion becomes useless if the function (y) is increasing or decreas- ing rapidly compared with the argument (x). Consider the following examples: (i) y=A+Bcx; (c>e; x>0). Here successive orders of differential coeffi- cients with respect to x get numerically larger, and no order can be numerically small. Yet if c=3, say, the function is entirely free from any feature which would be objectionable from the standpoint of smoothness. (ii) y= 1OOO/(1OO-x); (x>0). As x increases towards 1OO, all orders of differential coefficients become large in absolute value and ultimately increase over all bounds. On the other hand, so far from becoming less smooth as x increases towards 1OO, the graph of y is asymptotic to a straight line—surely the smoothest of all curves. Thus, the classical criterion rejects the function over just the range of values of x where we should expect it to be most accept- able, and indeed does so with progressive vigour as it tends more and more nearly to ideal smoothness. (The fact that y becomes infinite when x = 1OO does not invalidate these remarks, for they still apply if we confine attention to the range x = 0 to x = 99.9.) These examples, which could be multiplied indefinitely, suffice to show that the tendency of successive differential coefficients to become small in absolute value does not constitute a necessary condition for smoothness, although it may be a sufficient one. The next example illustrates a less obvious weakness of the classical method. (iii) An actuary (A) wishes to represent the relation between two variables x and y by a smooth curve. He has plotted the data and has drawn the curve y = a + bx + cxa (for certain values of a, b, c) which gives a satisfactory fit. He claims that this is an ideally smooth curve because the third differential coefficient of y with respect to x vanishes everywhere. However, A has omitted to label his axes x and y and his colleague (B) looks at the diagram sideways by mistake, treating the original right-hand edge of the paper as if it were the top. From B’s point of view the curve drawn by A has the equation or Since B also abides by the classical criterion, he objects that this is not an ideally smooth curve, and points out that, from his point of view, no order of differen- tial coefficients of y with respect to x vanishes. Yet A and B are both looking at the same curve ! Clearly a proper criterion of smoothness should give the same Measure of Smoothness 127 result whether we look at a curve sideways or upside down, or even if we erase the axes altogether so that position and orientation are arbitrary. 7. It will be seen from these examples that the classical method is funda- mentally unsuccessful in specifying the conditions of smoothness, and that a complete breakdown occurs when y increases rapidly with respect to x. It might be thought that we could improve it by requiring instead that either an order of differential coefficients of y with respect to x, or else an order of differential coefficients of x with respect to y, shall be small. Example (iii) above shows, however, that between these two extremes there is an infinite number of intermediate standpoints, all having an equal claim to validity. Both in the discrete and in the continuous case, the classical method insists that one of the variables shall be regarded as the ‘independent variable’; thus in the discrete case we have to consider successive differences of the dependent variable for predetermined intervals of the independent variable, while in the continuous case all differential coefficients are taken with respect to the independent variable. This position has probably arisen because, as actuaries, we are accustomed to work with time (in the form of age or duration) and to consider mortality rates, premium rates and so on as depending upon time; thus in nearly all our work there is, in fact, one variable which it is natural to take as the independent variable. This, however, is a feature of the material with which we happen to be mainly concerned, and has nothing to do with smooth- ness itself. Thus, we have always asked the question: ‘Is y a smooth function of x?', while a more profitable enquiry would be : ‘Is the relationship between x and y a smooth one ?'. There is, after all, no reason why this relationship should be presented to us other than in the form f(x, y) = 0, and how are we to proceed to examine smoothness without making an arbitrary choice as to whether x is to be expressed in terms of y or vice versa? 8. The classical method makes use of the differential coefficients dy/dx, d2y/dx2, d3y/dx3, and so on. If we write yn = dny/dxn and xn = dnx/dyn, xn and yn are related by the familiar equations of elementary differential calculus and so on. Since y2 measures the rate of change of y1 with respect to x, y2 will be large in absolute value if y1 is changing rapidly as x changes, although y1 may be changing only slowly relative to its own value. Thus if y1 is large, y2 may also be large without there being any departure from smoothness in the intuitive sense. We should reach very different results if we measured the rate of change of y1 with respect to y, or if we worked throughout with y as independent variable and examined x2, the rate of change of x1 with respect to y.
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