ADJUSTMENT OF MORTALITY TABLES BY MEANS OF SMOOTHING SPLINES. L. D'HOOGE*, J. DE KERF** and M.J. GOOVAERTS* (Belgium)

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1 Sixth nternational Conference of Social Security Actuaries and Statisticians Helsinki) 1-6 September 1975 ADJUSTMENT OF MORTALTY TABLES BY MEANS OF SMOOTHNG SPLNES by L. D'HOOGE*, J. DE KERF** and M.J. GOOVAERTS* (Belgium) Onderzoeksrapport N 7511 * Katholieke Universiteit te Leuven nstituut voor Aktuariele Wetenschappen Dekenstraat 2, B-3000 Leuven Computer Centre Agfa-Gevaert N. V., B ~1ortsel

2 J ABSTRACT n the present contribution the b-ruto mortality data, taken from the last census of population hold in Belgium, are adjusted by smoothing splines, a relative recent development in approximating given data. Some research possibilities still to be performed in this framework are suggested.

3 2.. NTRODUCTON The aim of the present paper consists in deriving an adjustment of the mortality tables based on spline functions. Due to the structure of the spline function the method seems to be very promising in handling mortality tables. The spline function gives much more facilities than the analytical adjustment based on the function of Makeham. t also contains in it the approximation by means of orthogenal polynomials. The present study is intended as being a start in examining the problem of the adjustment of ~ortality tables by means of a rather recent.technique. Still a lot of work has to be done in detecting the statistical properties of this kind of inference technique, as well as in determining the consequence of the adjustment for the other actuarial quantities. However as will be shown by the results of our numerical example the adjustment seems to be rather promising. The following data, taken from the last census of the Belgian population (1), are adjusted.

4 3. X w X \ ~ l ll ~ o.oott8o o.o.n3n \ S ~ O.l2S O.l4C ! 0.153ld3 35! o ~8 25 o.oo o.l '?:820 26!~ ll ' l ' so l -~~ o_o_7_9_o_o,.;. z_14-.l..j---._ '---_j x = age q "" clement~~ of: the bruto mortality X = the number of:.'rersons of age x under: consideration divided by 1000

5 4...smoothing and ap:eroodmatin~ spline fun_c_~_iol_!.s n ref (2) an excellent survey the theory and applications of spline functions given. The work also contains a paper concerning the approximatio~ for the computation of interpretating functions. by splines, as well as some algo:ri thms approximating spline For an extensive study of the technique of spline functions the interested reader is referred to ref (2). For ref (3) P. Dierckx gives an excellent algorithm for smoothing, differentiating and integrating experimental data using spline functions. This algorithm has been applied in order to perform the adjustment of the mortality tables. Given a strictly increasing sequence of real numbers., t k n- a spline function s(x) of degree k with knots t., j=k+ 1,, n-k J is a function defiued on the range (tk+l' tn-k) having the following two properties i) n each lnterva 1 ( tj" tj+l ), J=k+l,,n-k-1,. s(x) is given by some polynomial of degree k or less; ii) s (x) and its dex.ivates of orders, 2,.., k-l are continuous everywhere in the :range ( tk+ 1 ~ tn-k). The class nk(tk+j', tn-k) of spline functions of degree k with knots tk+l,, tn-k is a (n-k-1) dimensional vector space. Let { { { k (t-x) 0 if t ~X if t <X or where H is the usual notation for the Heaviside function defined as : H(x) = { {o if X ~ 0 if X < 0

6 5. then the B-spline M. k(x) given as the ) th divided J.., difference of gk(t,x) on ti ti+l. ~ for fixed x, 1 i.e. : while the so called normalised B-spline N. k(x) is :t., N. k(x) = (t.lk~' - t.) M. k(x) 1, LT ~l l. J.., The B-spline Mi,k(x.) and Ni,k(x) are positive for ti < x < ti+k+l and zero otherwise~ i.e. N. k(x) "" 0 l., N. k(x) > 0 l., X~ t. l or n order to obtain a basis for nk(tk+l~, tn-k)2k additional knots t 1, t 2,, t are introduced n Now every s (x} E: nk ( tk+ 1,, tn-k) has a unique representation. as a linear combination of the normalised B-splines N. k(x), J, j = 1, 2,, n -k. n ref (3) P. Dierckx then gives an algorithm for solving the following problem : Given the set of data points (x.,y.), i=l,2,...,m in the range l. l. (a,b) and a set of pc;sitive numbers w., i=t,2,..,m one determines l. a spline function s(x) of degree k with knots t., j=k+l,k+2,,n-k J from the condition that n-k-1 : r=k+2 is minimal for all s(x) satisfying the constraint

7 6. 'where d represents the discontinuity jump of s(k)(x) at t, i.e. r r d = s(k)(t +0) - s(k)(t -o) r r r and where S is a given constant. n the present case of the adjustment of the mortality data we take : yi "" ln qi w. = the number of persons of age i under consideration ]. divided by OQP x..., i i = l,2,.~.,n l. Hence the mortality table. n-k-1 exp { E 1 adjusted by The algorithm of ref (3) provides us with the set {c.}, as well J as with the set of knots {t. }. J Another important feature of the algorithm consists of the fact that s(v)(x), as well as~ s(x) dx can be evaluated. This provides us with an important instrument in deriving other actuarial quantities from the deduced approximation for qx. However in the present contribution we will restrict ourselves to the case of the adjustment of the q mortality table. X Numerical results n what follows we give the position of the knots, the coeffi- cients of the spline expansion, the values ln q, ln q *, ln q */q, * X X :X X w;,e qx and qx. n fig. 1, 2, 3 the bruto data and the smoothed qx * curve is given for some values of S, namely S = 500, 250, 2. all the results are acceptable statistically. Following the x 2 -test t is clear that our results are superior to those that can be obtained from Makeham' s formula. This becomes clear if fig. l~ 'i.s compared to the correspoading results obtained in the present contribution.

8 a) s = soo 7. The following set of knots is obtained : -o. 4 goooooe E QOE 02 -o. 1 9oooooE E 01 O.OOOOOOE lioooooe lOOOOOE E JOOOOOE E E E E E looooooe J OOOOOE E i300000e 03 O. J400000E 03 O.l500000E 03 The following set of coefficients is obtained : E 03 -D E 01 -o. 69W768E E J059E O E E E E E O E 01 -o E E 00 -D lE Ol E 02 Hence the following table is obtained : X.ln qx ln q* ln q*/q w qx q* X X X X X o ! o o O.OOOS o. t44o n o o ! o J o ! O.OO!lr o ! ! o !

9 {j l L ! L7l07 t n M l!2 i ! o.o o ! , ~ ! ~ J ! l ! l Oi 2209 o.oo o.oot~ ( l ! ! ' ! ! ! o ! ~ ll ! o o. i i5i

10 ~2 21 o l ' o l o l o l o

11 10. o~~~~~~~~~~~~~-~~~ :.7 -a = Fig. l

12 b) s = 250 The following set of knots is obtained : E iOOOOOE E SOOOOOE E OOOOOOE Ol E l700000e E E E E E lOOOOOE E E E E E 03 O.lOSOOOOE E i240000e 03 O.l320000E E 03 The following set of coefficients is obtained : !8E E E Ol E Ol E E E E Ol E E E E E 01 -O.l793ll4E Ol -O.l381547E E E E 02. Hence the following table is obtained X ln q ln q* ln q */q w qx c.. X X X X X l ~7834 O.i O.OO i o ! lo ! ll o ! o lt o o ! o. to ! ! ' o. 207/./r ! ! l 221 o ! lf.t !_ t L

13 J io l : ] L ! ! o.oo ' : \) o.oo9t69 O.OOt o.006l o ' ,_ * o.o699s l o i i o !28535 o o o l o.1so14s o l

14 ' t o ! o l o.1467 l f

15 o~~~~~~~~~~~~~~~~ -1-2 " a S= Fig. 2

16 c) s ""' 2 The following set of knots is obtained : 15. -o. 1 goooooe SOOOOOE l OOOOOE E E 0% 0. looooooe OJ O.SOOOOOOE E 01 O.l300000E 02 O. l700000e E E E E E 02 o.4loooooe E E E E E E, E E E E E 02 O E 03 O E ll20000e E E E E 02 O.lOOOOOOE E 03 The following set of coefficients is obtained : E E 01 -: E 01 -o JE E OJ -o. 11 J3234E E OJ E Ol E o e E E E Ol Ol E E 01 -o.34585e 01.-o E E E E 01 -o E E OJ -O.l27213SE 01 -o.tot 1461E 01 -o. s E S380E E 02 Hence we get the following result : X ln q ln q* ln q */q w X X X X X qx ~ ! ! ! t l !2 295 o. 00! i ! ! o ! l!j o oo jl, ! ,6il3l M ! !44.,...,,...,._,.,...,.., ~----'--

17 t !~ s.zno llllo ~ ! ? o.oo ! l l O.Olt3l ! ! l / o.! o o. 12! o o o o.! ~~~6st85 8 L

18 o.oot~s 21 o o : o ) JO l l J ] l l o !

19 18. o----~--~~----~~~~--~ a c S= Fig. 3

20 Makeham bruto data stnothed value 19. : 0,5. (), 1 -. ' 0,05 0,010 0,005 0,001 O,OOOb!'!-!- ~, ~_\ ~ ~ ~!--'"",~ --~ ' ~ \! - A l(/,v,/ -" 1/ - t/,r J ll \ /fl - kf lf / v ' v ~1.;J v v ', [\! j 0, j ~-.~..-~...:...._ GO X

21 20. CONCLUSONS n the present contribution it is shown that the approximation of brute mortality tables by means of spline functions provides no difficulties, even not for ages x < 30. A set of approximating curves corresponding to a set of s values can be accepted statistically. A lot of work still has to be performed in determining the "best" approximating curve. To solve this problem one has to examine the consequences of these approximations to the numerical results of the actuarial quantities.

22 REFERENCES (1) Y. BALLEGEER, Bruto en afgeronde sterftetafels , tische Studien, 35 (1975). (2) T.N.E. GREVLLE, Theory and applications of Spline functions, (eds.) Academic Press, London (!969). (3) P. DERCKX, "An algorithm for smoothing, differentiation and integration of experimental data using spline functions", Journ. Comp. Appl. Math.v.x, Nr. 3 (1975). See also references quoted there.