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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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