an:01681793
Zbl 0979.91043
Jasiulewicz, Helena
A comparison of methods of approximations for probabilities of death for fractions of a year
EN
Appl. Stoch. Models Bus. Ind. 17, No. 3, 245-260 (2001).
00081790
2001
j
91B30
methods of approximations; probabilities of death; lifetime distribution; Kolmogorov statistic
This paper deals with the comparison of methods of approximations for probabilities of death for fractions of a year. Let human lifetime \(X\) be a continuous random variable with distribution function \(F(x)\). Let \(\overline F(x)\) denote the empirical lifetime distribution and \(\widehat F(x)\) denote the interpolating function of \(\overline F(x)\) such that \(\widehat F(x)=\overline F(x)\) at integer points \(x=1,\ldots, \omega-1\), where \(\omega\) is the maximum age for human being. The author considers four methods of approximation:
(1) \(\widehat F(x+u) = \overline F(x)+u(\overline F(x+1)-\overline F(x))\);
(2) \(\widehat F(x+u) = 1-(1-\overline F(x+1))^{u}(1-\overline F(x))^{1-u}\);
(3) \(\widehat F(x+u)= 1-{(1-\overline F(x))(1-\overline F(x+1))\over u(1-\overline F(x))+(1-u)(1-\overline F(x+1))}\);
(4) \(\widehat F(x+u)= (\overline F(x+2)-\overline F(x+1))(u^3-u^2)+(\overline F(x+1)-\overline F(x))(u+u^2-u^3)+\overline F(x))\),
\(u\in [0,1]\), \(x=0,1,\ldots,\omega-1\).
Two criteria based on the Kolmogorov statistic and the measure of distance \(L^2(x)\) are used. The author shows that none of the four methods are better than the other three.
A.D.Borisenko (Ky??v)