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**The natural variability of vital rates and associated statistics (with discussion).**
*(English)*
Zbl 0611.62136

The author develops approximations to the distributions of crude mortality rates, age-specific mortality rates, age-standardized rates, standardized mortality ratios, and the like for the case of a closed population or period study.

Assuming Poisson birth times and independent lifetimes, it is found that the number of deaths and the corresponding midyear population have a bivariate Poisson distribution. It is suggested that it will be satisfactory to approximate the distribution of the number of deaths given the population size, by a Poisson with mean proportional to the population size. It is further suggested that situations in which explanatory variables are present may be modelled via a doubly stochastic Poisson distribution for the number of deaths, with mean proportional to the population size and an exponential function of a linear combination of the explanatories. Such models are found to fit to mortality data for Canadian females. The models with extra-Poisson variation are found to lead to substantially improved fits.

Assuming Poisson birth times and independent lifetimes, it is found that the number of deaths and the corresponding midyear population have a bivariate Poisson distribution. It is suggested that it will be satisfactory to approximate the distribution of the number of deaths given the population size, by a Poisson with mean proportional to the population size. It is further suggested that situations in which explanatory variables are present may be modelled via a doubly stochastic Poisson distribution for the number of deaths, with mean proportional to the population size and an exponential function of a linear combination of the explanatories. Such models are found to fit to mortality data for Canadian females. The models with extra-Poisson variation are found to lead to substantially improved fits.

Reviewer: I.Křivý

### MSC:

62P10 | Applications of statistics to biology and medical sciences; meta analysis |