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A comparison of methods of approximations for probabilities of death for fractions of a year. (English) Zbl 0979.91043
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.
MSC:
91B30 Risk theory, insurance (MSC2010)
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References:
[1] Life Insurance Mathematics. Springer: Berlin, 1990. · doi:10.1007/978-3-662-02655-7
[2] Jasiulewicz, Insurance: Mathematics and Economics 19 pp 237– (1997)
[3] Probabilities of death for fractions of a year. Technical Report #40/1996, Institute of Mathematics, Wroc?aw University of Technology, Wroc?aw. 1996.
[4] The influence of the method of lifetime approximation on the premium in life insurance. Master’s Thesis, Institute of Mathematics, Wroc?aw University of Technology, Wroc?aw, 1999 (in Polish).
[5] Polish Life Tables 1990-1991. GUS: Warsaw, 1993 (in Polish).
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