The opinion is expressed sometimes that of recent years less interest has been taken by the profession generally in Sessional meetings of the Institute, and in the preparation of original contributions for the Journal. It is urged that actuarial subjects have been fully covered by earlier writers, and that in consequence a narrowing field for active interest exists.

The object of this paper is to examine the statistics available and, if possible, to decide whether the views stated above are justified. For this purpose two main ‘yard-sticks’ will be used.

f(x), f′(x),…, f(n+1)(x) are one-valued, finite and continuous (v≤x≤w).

Define

Then*

u (x) is one-valued, finite and continuous (v ≤ x ≤ w).

by writting

During the past 25 years much attention has been given by prominent statistical writers, especially J. Neyman and E. S. Pearson, to building up the theory of the Testing of Statistical Hypotheses. This theory and the practical tests which have been developed from it are of great interest to actuaries, those employed in normal actuarial spheres as well as those employed in purely statistical roles. The leading exponents of the theory, however, are mathematicians of no mean standing and the subject has been approached from a mathematical angle. From the point of view of the actuary or student whose study of mathematics has not proceeded beyond that of the Institute's examination syllabus this is unfortunate, as the consequent complexity of the approach masks from him the essential simplicity of the basic ideas underlying the theory.

In the discussion on the paper ‘The Variance Ratio test in Statistics’, given to the Students' Society on 13 December 1946, and printed in J.S.S. Vol. VI (1947), p. 172, the desire was expressed for a proof by relatively elementary methods of the distribution of the variance ratio. As such a proof must first derive the distribution of X2 modified into that of a variance estimate, three distributions are involved (in addition to that of the sample mean). The various steps are given in this paper. A note is added on the probability integral of the variance ratio distribution.

The values of the returns of past and future contributions on death before retirement, such contributions being accumulated to the date of death at rate j, are given respectively by

(TPC)xPCFx and (AC)xFCFx

where (TPC)x=total past contributions at age x accumulated with (compound) interest at rate j to the valuation date,

(AC)x = actual annual contributions payable at valuation age x.