[Statistical models of longevity, aging and degradation in demography, gerontology and oncology].

Several interesting papers related with the problems of mathematical modelling in connection with a study of aging and longevity have been published recently in "Advances of Gerontology"; see, for example, [1-4, 10, 11]. Following the main idea of these papers we consider here some approaches for construction of statistical models used today in survival analysis and reliability, and which can be used in demography, gerontology and carcinogenesis studies. We expose the so called dynamic regression models, which are well adapted for studies of survival in dynamic environments.

Flexible regression models for dynamic prediction of survival in elderly in presence of cancer or other chronic diseases.

We consider degradation and failure time models with multiple failure modes adapted for the statistical analysis of survival in the elderly in presence of chronic diseases such as cancer, but also other conditions like dementia, Alzheimer, diabetes, etc... These models can be applied in gerontology, general medicine, biology and demography for the analysis of longevity and survival of highly complex biological systems as any human is.

Estimation in degradation models with explanatory variables.

Influence of covariates on degradation is modelled. Models which include dependence of the intensity of the process of traumatic events on the degradation level are also discussed. Estimation of reliability and degradation characteristics from data with covariates is considered in dynamic environments.

A probabilistic description of radioactive contamination: a multivariate model.

A multivariate discrete probability model is used to facilitate the description of gamma-ray spectroscopic data obtained from radioactively contaminated territory, east of the former Semipalatinsk nuclear test site in Kazakhstan. Two possible estimators of probabilities of interest have been considered: maximum likelihood and unbiased estimators. We show that unbiased estimators are much easier to compute. The model was used in two variants: (i) several radionuclides in spatially independent measurements, (ii) a single radionuclide in spatially dependent measurements. We show that, in both cases, it is important to take into account the correlation for the accurate evaluation of probabilities of interest.

Generalized proportional hazards model based on modified partial likelihood.

The proportional hazards (Cox) model is generalized by assuming that at any moment the ratio of hazard rates is depending not only on values of covariates but also on resources used until this moment. Relations with generalized multiplicative, frailty and linear transformation models are considered. A modified partial likelihood function is proposed, and properties of the estimators are investigated.