Abstract
The paper focuses on the estimation of the force of mortality of living time distribution. We use a third-order B-spline function to construct the logarithm for force of mortality of living time. The number of the knots, their locations and B-spline coefficients based on a sample of observations are estimated by the maximum likelihood estimation method. Evaluation of B-spline parameters estimated by maximum likelihood estimation tested with criteria of the modified chi-squared goodness of the fit statistic. An algorithm developed to calculate Sequential Procedure for the modified chi-squared goodness of the fit testing. The Matlab code was written using the algorithm. Within this evaluation, the number of knots in the model has significantly reduced. The developed method was used to explain the mortality rate of women aged 0 to 69 among the Mongolian population in 2019 and estimate the life expectancy of Mongolians. The results of this experiment provided an excellent estimation of the force of mortality. Construction of a mortality rate estimation gives possibilities to determine mortality trends and force of mortality. Here, force of mortality is further used to construct a survival function, a lifetime distribution function, and a lifetime distribution probability density function. The method can also be used in financial market models and in models that estimate the useful life of equipment.
Highlights
F Mortality, B-splines, Chi-square Statistic, Forecasting life insurance, health, and demographic surveillance system
The number of the knots, their locations and B-spline coefficients based on a sample of observations are estimated by the maximum likelihood estimation method
We evaluate the logarithm of the force of mortality in the form of the third order B-spline, where its parameters have been estimated by the Maximum Likelihood Estimation
Summary
Let’s denote by X is the living time of people. by assuming X ≥ 0 to be continuous random variable, we define distribution function FX (x) as and FX (x) = P (X ≤ x). x ≥ 0. The times interval x + ∆x, for given X ≥ x is: g(x, θ, ck,n) ∈ Sk B-splines are defined on the set of knots. Survival function is Basis B-splines are defined on set of knots ck,n through the expressed as a function of the force of mortality and can be Mansfield-De Boor-Cox recurrence relation written in the following x. The cumulative distribution and the density functions are expressed via the force of mortality as: FX (x) = 1 − s(x) = 1 − exp − λ(t)dt (17). Let nx denotes the number of death of people aged in the interval x − 1, x. Empirical x=1 estimation for the function of living time distribution and empirical estimation for the force of mortality are defined as FX (x) n1
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