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Construction of a Graduated Table of Mortality from a Limited Experience.
ILLUSTRATION. (2) (3) (4) (5) (6)
(8) (9) (10) (11) (12) (13)
1 Ez a'z
mz (log q=-04) (1-2)
Construction of a Graduated Table of Mortality, &c.—(continued).
Similarly, when t=46,
a=.00713, b=.0000749. For determining the weights we shall adopt
1 from which we compute the factor
in column (6) to be
mx(1 -mx) multiplied into Ex to obtain the weight wx in column (7). We can now form the normal equations (6). For the purpose of this illustration, five places of logarithms have been deemed sufficient
in calculating the auxiliaries wxq*+t, wxm'æ, &c. Making the necessary summations we get Ew==6,707,974, Ewrq*+1=278,341,720, Ewqm'c = 67,952,
20+1=23,249,674,696, Ewrqu+km's=3,580,417. Introducing these in (6) and solving, we get the most probable values,
a=.0074319, b=.000065025. Column (8) shows the rates of mortality calculated from these by means of (2), and column (9) the differences (computed minus observed) from the actual ratios.
To ascertain the probable errors of the determination we have (n being the number of equations of condition) the mean error of an observation of the weight unity, Ewx(m.z-m'x) 80.495
= +1.1304; n-2
63 also the weights of the determinations of a and b,
From these the mean errors of a and ó are found to be
and, from the constant relation between the mean and probable errors, r=-6745€,
ra= +0004150, ry=+.00007049. Recapitulating, then, the results of this determination, a=.0074319 with the probable error of +.0004150. b=000065025
+.000007049. To illustrate the use of the formulas for finding the most probable value of q at the same time with those of a and b, let us assume the approximate valuesa'= .00726,
b'=.000074, a=1.096,478 or log q'=:04, With the value m'' of the mortality rates corresponding to these assumptions we find the values of u from (7). Then, performing the requisite multiplications and summations, we deduce the following normal equations, according to formulas (10),
6,707,974X+ 278,341,720Y+ 826,858 Z- 1,267 =0, 278,341,720X+23,249,674,696Y+81,926,500 Z-159,723 =0,
826,858X+ 81,926,500Y+ 298,367-62- 460:6=0, whence
X= +.0001829, Y=-000005229, 2=-·0012554, are the corrections to be added to a', 6', and q'.
The correction Z comes out with the large mean error + .001070. Also, the mean error of an observation of the weight unity is Ewr(mr-mx)3 81-597
62 In the solution previously made, when log q was taken = '04, € was +1.1304. These results would lead us to infer (if no mistake has been made in the computation) that the assumption log q=:04 introduced no error appreciable by the observations, and that we may safely adopt that value as the best attainable in this
It has before been remarked that the experience of even the oldest American companies is insufficient to supply trustworthy information as to the mortality prevailing in extreme old age. Nevertheless, it may sometimes be desirable, for certain purposes, to extend the scale from the point where the observations fail, to the limit of human life, according to our general knowledge, from other sources, of the mortality for this period. As to the precise mode of connecting the observed rates with those according to a known table, adopted as a standard, without rendering the transition too violent, no general rules can be laid down. A process which will, I think, answer the purpose sufficiently well in most cases, is the following:
Comparing the rates of mortality deduced from the observations with those of the standard table at the older ages, where, on account of the paucity of the data, we begin to lose confidence in the former, and incline to the latter as the properer index of the probabilities of dying; we can conceive of a point where we shall be unable to assign a preference for either. Denote the observed rates by single accents and those of the standard table by double accents. Put, also, for the age x at which these two measures of the probability of dying may be presumed of equal weight,
Then, taking convenient equidistant intervals on either side of the age x, we can obtain for any age within the limits x-nw and *+nw the interpolated value mox+t from the system of differences derived from the values m'x-no, mʻx-(n-1)0, ...., mʻx-20,
m" In general, it will not be worth while to employ more than five terms, in which case n=2.
Denote by a and c the means of the first and of the third differences immediately above and below the line of mox; and by b and d the second and fourth differences on this line. Then for the age x+t we shall have, by the theory of interpolation, 2 1
1 + at
184.108.40.206 which may be directly employed. I have found it more convenient, however, to use this series developed according to the powers of t, putting
1 1 D
W220.127.116.11 whence we have
mor+t=mo+At+Bto+Ct + Dt. .. (13) t being taken positive for older, and negative for younger, ages
A check on the whole computation is that at ages x + 2w, x+w, the interpolation should reproduce the fundamental values.
For illustration, let us in the previous example suppose that it is desired to extend the series in column (8) to the end of life, hypothetically, making use of the Actuaries' (old) table for this purpose, according to the foregoing method. Take x=70, w=5. Then