Points of T-axis, determination of the representative point in age category
If the mortality rate is strongly age dependent, then the arithmetic mean of the extreme points of one age category
is not a representative point of the age category. It coudl not ba used in regression analysis.
Generally, mortality rate R(t) can be defined as a continuous function:
R(t) = [-dS(t)/dt]/S(t) , where S(t) is the percentage of living individuals at the age t.
In other words, it is necessary for the analysis of mortality decline along with age to assign
one representative point Ti to all dead individuals within one age category [ti-1; ti].
The number of the living individuals within one age category (the size of whole population) is almost constant.
Thus the decrease of deaths d(t) corresponds to -dS(t)/dt and it represents
the decisive factor for the dependence of the mortality upon the age within one age category.
The representative point Ti can be calculated as expectated value E(t) of age t at the moment of death .
Population of death individuals in age category should be describe by the same density function
for each age category if the type of function d(t) is known.
As the d(t) function is also valid within the age category,
the number of dead individuals within the age category [ti-1; ti] equals to:
Expectation value E(Xi) of interval Ti does not depend of the parameter C.
The alternative description
The problem with the representative points could be particulary solved using the cumulatice number of deaths up to age limit T
(T could be the upper limit of demographic age interval).
It is covenient for the first day of life if the function C/t is not defined.
The projection does not satisfied to standard regression model (x values are not independent).