Modern Stochastics: Theory and Applications

The mixed model with polynomial drift of the form $X(t)=\theta \mathcal{P}(t)+\alpha W(t)+\sigma {B_{H}^{n}}(t)$ is studied, where ${B_{H}^{n}}$ is the nth-order fractional Brownian motion with Hurst index $H\in (n-1,n)$ and $n\ge 2$, independent of the Wiener process W. The polynomial function $\mathcal{P}$ is known, with degree $d(\mathcal{P})\in [1,n)$. Based on discrete observations and using the ergodic theorem estimates of H, ${\alpha ^{2}}$ and ${\sigma ^{2}}$ are given. Finally, a continuous time maximum likelihood estimator of θ is provided. Both strong consistency and asymptotic normality of the proposed estimators are established.

We consider a mixture with varying concentrations in which each component is described by a nonlinear regression model. A modified least squares estimator is used to estimate the regressions parameters. Asymptotic normality of the derived estimators is demonstrated. This result is applied to confidence sets construction. Performance of the confidence sets is assessed by simulations.

Stationary processes have been extensively studied in the literature. Their applications include modeling and forecasting numerous real life phenomena such as natural disasters, sales and market movements. When stationary processes are considered, modeling is traditionally based on fitting an autoregressive moving average (ARMA) process. However, we challenge this conventional approach. Instead of fitting an ARMA model, we apply an AR(1) characterization in modeling any strictly stationary processes. Moreover, we derive consistent and asymptotically normal estimators of the corresponding model parameter.

We consider a finite mixture model with varying mixing probabilities. Linear regression models are assumed for observed variables with coefficients depending on the mixture component the observed subject belongs to. A modification of the least-squares estimator is proposed for estimation of the regression coefficients. Consistency and asymptotic normality of the estimates is demonstrated.

A mixture with varying concentrations is a modification of a finite mixture model in which the mixing probabilities (concentrations of mixture components) may be different for different observations. In the paper, we assume that the concentrations are known and the distributions of components are completely unknown. Nonparametric technique is proposed for testing hypotheses on functional moments of components.