Consistent estimators of the baseline hazard rate and the regression parameter are constructed in the Cox proportional hazards model with heteroscedastic measurement errors, assuming that the baseline hazard function belongs to a certain class of functions with bounded Lipschitz constants.
We investigate the fractional Vasicek model described by the stochastic differential equation dXt=(α−βXt)dt+γdBHt, X0=x0, driven by the fractional Brownian motion BH with the known Hurst parameter H∈(1/2,1). We study the maximum likelihood estimators for unknown parameters α and β in the non-ergodic case (when β<0) for arbitrary x0∈R, generalizing the result of Tanaka, Xiao and Yu (2019) for particular x0=α/β, derive their asymptotic distributions and prove their asymptotic independence.
Cox proportional hazards model with measurement errors is considered. In Kukush and Chernova (2017), we elaborated a simultaneous estimator of the baseline hazard rate λ(⋅) and the regression parameter β, with the unbounded parameter set Θ=Θλ×Θβ, where Θλ is a closed convex subset of C[0,τ] and Θβ is a compact set in Rm. The estimator is consistent and asymptotically normal. In the present paper, we construct confidence intervals for integral functionals of λ(⋅) and a confidence region for β under restrictions on the error distribution. In particular, we handle the following cases: (a) the measurement error is bounded, (b) it is a normally distributed random vector, and (c) it has independent components which are shifted Poisson random variables.
Cox proportional hazards model is considered. In Kukush et al. (2011), Journal of Statistical Research, Vol. 45, No. 2, 77–94 simultaneous estimators λn(⋅) and βn of baseline hazard rate λ(⋅) and regression parameter β are studied. The estimators maximize the objective function that corrects the log-likelihood function for measurement errors and censoring. Parameter sets for λ(⋅) and β are convex compact sets in C[0,τ] and Rk, respectively. In present paper the asymptotic normality for βn and linear functionals of λn(⋅) is shown. The results are valid as well for a model without measurement errors. A way to compute the estimators is discussed based on the fact that λn(⋅) is a linear spline.