Modern Stochastics: Theory and Applications

We investigate the fractional Vasicek model described by the stochastic differential equation $d{X_{t}}=(\alpha -\beta {X_{t}})\hspace{0.1667em}dt+\gamma \hspace{0.1667em}d{B_{t}^{H}}$, ${X_{0}}={x_{0}}$, driven by the fractional Brownian motion ${B^{H}}$ with the known Hurst parameter $H\in (1/2,1)$. We study the maximum likelihood estimators for unknown parameters α and β in the non-ergodic case (when $\beta <0$) for arbitrary ${x_{0}}\in \mathbb{R}$, generalizing the result of Tanaka, Xiao and Yu (2019) for particular ${x_{0}}=\alpha /\beta $, derive their asymptotic distributions and prove their asymptotic independence.

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.