Chernova O. Estimation and goodness-of-fit test in Cox proportional hazards model with measurement errors.

The thesis is devoted to the investigation of asymptotic properties of the simultaneous estimator of a baseline hazard rate and a regression parameter in the Cox proportional hazards model when parameter set is unbounded. The regression model introduced by D. Cox in 1972 and its modifications are of vital importance in survival analysis. Such models are commonly used in oncology and other branches of medicine to characterize the relation between survival time and covariates (e.g., type of treatment). In the thesis a lifetime is modeled using absolutely continuous positive random variable. The Cox proportional hazards model assumes that the effect of covariates is linear and additive on log-rate scale. In the classical Cox model, one estimates the time-independent regression parameter, while the baseline hazard rate that describes dependence of time is assumed to be a nuisance parameter and is not to be estimated. In this thesis, baseline hazard rate is estimated assuming that it belongs to the set of nonnegative Lipsñhitz functions on a fixed interval with known Lipsñhitz constant, and regression parameter is from a compact set. It is a semiparametric model whose parameters are the baseline hazard rate and the regression parameter.