Modelling modal age by various smoothing methods with regard to the asymmetry in mortality: the case of the Czech Republic 1950–2014

Martina Miskolczi, University of Economics, Prague (VSE)
Jitka Langhamrova, University of Economics, Prague (VSE)
Jana Vrabcova Langhamrova, University of Economics, Prague (VSE)

The objective of the article is to introduce and compare several methods of smoothing mortality curve, estimating mode and describe statistically age distribution of deaths in the population, separately for males and females. Among considered methods there are statistical models such as Gompertz–Makeham model, logistic models (Himes-Preston-Condran model, Heligman-Pollard model, Thatcher model, Kannisto model), cubic model, polynomial model, Coale-Kisker model, Denuit-Goderniaux model, P-spline-smoothing method based on penalised Poisson likelihood or models with term representing asymmetry. Second, the trend in the period 1950–2014 is analyzed for the case of the Czech Republic, how parameters of parametric distributions change over a time and whether this approach can be used for well-founded and justified projection to the future. Finally, the focus is concentrated on the asymmetry of the age distribution of deaths. Based on the estimated age distribution moments can be expressed, besides the first moment (expected value) also the second moment (variability), the third moment (skewness) and the fourth moment (kurtosis). This is the foundation for the estimate of maximal age that is reached by an individual in the population and, especially, how this maximum is evolving over a time and for each gender, i.e. separately for males and for females. Estimate of age distribution of deaths and its characteristics leads to the assessment of measurement of longevity, estimation of modal age at death for males, females and its convergence and in time, estimation of maximal age, comparison of various methods of mortality smoothing, modal age estimates among each other and comparison of these with other measures, such as life expectancy, median age and life expectancy in health.