A comparison of methods of approximations for probabilities of death for fractions of a year (Q2759387)

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: NEWLINENEWLINENEWLINE(1) \(\widehat F(x+u) = \overline F(x)+u(\overline F(x+1)-\overline F(x))\);NEWLINENEWLINENEWLINE(2) \(\widehat F(x+u) = 1-(1-\overline F(x+1))^{u}(1-\overline F(x))^{1-u}\);NEWLINENEWLINENEWLINE(3) \(\widehat F(x+u)= 1-{(1-\overline F(x))(1-\overline F(x+1))\over u(1-\overline F(x))+(1-u)(1-\overline F(x+1))}\); NEWLINENEWLINENEWLINE(4) \(\widehat F(x+u)= (\overline F(x+2)-\overline F(x+1))(u^3-u^2)+(\overline F(x+1)-\overline F(x))(u+u^2-u^3)+\overline F(x))\), NEWLINENEWLINENEWLINE\(u\in [0,1]\), \(x=0,1,\ldots,\omega-1\). NEWLINENEWLINENEWLINETwo 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.