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Summation and Some Other Methods of Graduation – the Foundations of Theory

Published online by Cambridge University Press:  07 November 2014

M. D. W. Elphinstone
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Synopsis

Several methods of graduation depend on linear operators–that is, operators of the type …+λ−nE−n+‥+λ0+…+λnEn+… Few attempts have been made to compare these methods, and there is no general theory. There is need for such a theory, for it is easy to show that unless some relation is assumed to hold between neighbouring graduated rates, there is no justification for graduating at all. Any such theory must explain this relation, which cannot be of the simple type that leads to curve fitting.

The paper has two objects : to compare and judge these methods, and to devise a general theory applying to them all.

The first approach is by the classical method–separate examination of fidelity and smoothness.

Type
Research Article
Information
Transactions of the Faculty of Actuaries , Volume 20 , 1951 , pp. 15 - 77
Copyright
Copyright © Institute and Faculty of Actuaries 1951

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References

page 16 note * The Calculus of Observations by Whittaker and Robinson, 4th Edition, p. 289. Actuarial Statistics, vol. I, “Statistics and Graduation,” by H. Tetley, p. 179. My general indebtedness to these two books is evident, and I do not acknowledge them as sources except in respect of certain special particulars.

page 17 note * J.I.A., 26, p. 114.

page 18 note * J.I.A., 26, p. 110.

page 18 note † This word means a sine curve of any amplitude, period or phase, but no other periodic curve.

page 20 note * E{f(x)} is called the “expectation of f(x)” and is defined for the purposes of this paper as

where f(x) may be an ordinary function of x or a function of a random or autoregressive variable, such as εx 2. The limit, however, is not exactly the usual limit of elementary mathematics, for this limit is defined to be l if, given ε (any positive number, however small), there is an N, depending on ε, such that

whenever

Here this need not be true, but even here, given ε, η (any two positive numbers, however small), there is an N, depending on both ε and η, such that the probability that

will be greater than 1—η. I do not seek mathematical rigour in this paper, and many parts might benefit by subsequent examination in detail.

page 21 note * References in this form refer to the corresponding paragraph in Appendix C.

page 26 note * Kendall: The Advanced Theory of Statistics, vol. 2, p. 404.

page 27 note * The “intensity” of a sinusoid is the square of its amplitude. The intensity of A cos ax+B sin ax is A 2 + B 2.

page 27 note † Econometrica, 5, p. 105.

page 33 note * Annals of Mathematical Statistics, 10, p. 254, and 12, p. 127.

page 33 note † The second term does not, however, conform to the particular set of sufficient conditions given by Slutzky.

page 34 note * See Wolfenden, T.A.S.A., 26, p. 81.

page 34 note † J.I.A., 48, pp. 171–390, J.I.A., 49, p. 148.

page 35 note * J.I.A., 41, p. 355.

page 35 note † Tracts for Computers, No. vi.

page 36 note * J.I.A., 53, p. 92. Whittaker and Robinson, pp. 300–302.

page 37 note * Whittaker and Robinson, pp. 303–312 and the references given in those pages.

page 37 note † T.F.A., 11, p.31

page 42 note * J.I.A., 71, pp. 21–27.

page 43 note * J.I.A., 26, p. 114.