Modern Stochastics: Theory and Applications

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.

Tests for proportional hazards assumption concerning specified covariates or groups of covariates are proposed. The class of alternatives is wide: log-hazard rates under different values of covariates may cross, approach, go away. The data may be right censored. The limit distribution of the test statistic is derived. Power of the test against approaching alternatives is given. Real data examples are considered.

We consider a multivariate functional measurement error model $AX\approx B$. The errors in $[A,B]$ are uncorrelated, row-wise independent, and have equal (unknown) variances. We study the total least squares estimator of X, which, in the case of normal errors, coincides with the maximum likelihood one. We give conditions for asymptotic normality of the estimator when the number of rows in A is increasing. Under mild assumptions, the covariance structure of the limit Gaussian random matrix is nonsingular. For normal errors, the results can be used to construct an asymptotic confidence interval for a linear functional of X.

We prove the asymptotic normality of the discretized maximum likelihood estimator for the drift parameter in the homogeneous ergodic diffusion model.

We present large sample properties and conditions for asymptotic normality of linear functionals of powers of the periodogram constructed with the use of tapered data.