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Construction of maximum likelihood estimator in the mixed fractional–fractional Brownian motion model with double long-range dependence

Yuliya Mishura Ivan Voronov

https://doi.org/10.15559/15-VMSTA28

Pub. online: 20 Jul 2015 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 2, Issue 2 (2015), pp. 147–164

Abstract

We construct an estimator of the unknown drift parameter $\theta \in \mathbb{R}$ in the linear model

\[X_{t}=\theta t+\sigma _{1}{B}^{H_{1}}(t)+\sigma _{2}{B}^{H_{2}}(t),\hspace{0.2778em}t\in [0,T],\]

where ${B}^{H_{1}}$ and ${B}^{H_{2}}$ are two independent fractional Brownian motions with Hurst indices $H_{1}$ and $H_{2}$ satisfying the condition $\frac{1}{2}\le H_{1}<H_{2}<1$. Actually, we reduce the problem to the solution of the integral Fredholm equation of the 2nd kind with a specific weakly singular kernel depending on two power exponents. It is proved that the kernel can be presented as the product of a bounded continuous multiplier and weak singular one, and this representation allows us to prove the compactness of the corresponding integral operator. This, in turn, allows us to establish an existence–uniqueness result for the sequence of the equations on the increasing intervals, to construct accordingly a sequence of statistical estimators, and to establish asymptotic consistency.

Identifiability of logistic regression with homoscedastic error: Berkson model

Sergiy Shklyar

https://doi.org/10.15559/15-VMSTA27

Pub. online: 7 Jul 2015 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 2, Issue 2 (2015), pp. 131–146

Abstract

We consider the Berkson model of logistic regression with Gaussian and homoscedastic error in regressor. The measurement error variance can be either known or unknown. We deal with both functional and structural cases. Sufficient conditions for identifiability of regression coefficients are presented.

Conditions for identifiability of the model are studied. In the case where the error variance is known, the regression parameters are identifiable if the distribution of the observed regressor is not concentrated at a single point. In the case where the error variance is not known, the regression parameters are identifiable if the distribution of the observed regressor is not concentrated at three (or less) points.

The key analytic tools are relations between the smoothed logistic distribution function and its derivatives.

On the Feynman–Kac semigroup for some Markov processes

Victoria Knopova

https://doi.org/10.15559/15-VMSTA26

Pub. online: 18 Jun 2015 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 2, Issue 2 (2015), pp. 107–129

Abstract

For a (non-symmetric) strong Markov process X, consider the Feynman–Kac semigroup

\[{T_{t}^{A}}f(x):={\mathbb{E}}^{x}\big[{e}^{A_{t}}f(X_{t})\big],\hspace{1em}x\in {\mathbb{R}}^{n},\hspace{2.5pt}t>0,\]

where A is a continuous additive functional of X associated with some signed measure. Under the assumption that X admits a transition probability density that possesses upper and lower bounds of certain type, we show that the kernel corresponding to ${T_{t}^{A}}$ possesses the density ${p_{t}^{A}}(x,y)$ with respect to the Lebesgue measure and construct upper and lower bounds for ${p_{t}^{A}}(x,y)$. Some examples are provided.

Autoregressive approaches to import–export time series II: a concrete case study

Luca Di Persio Chiara Segala

https://doi.org/10.15559/15-VMSTA25

Pub. online: 1 Jun 2015 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 2, Issue 1 (2015), pp. 67–93

Abstract

The present work constitutes the second part of a two-paper project that, in particular, deals with an in-depth study of effective techniques used in econometrics in order to make accurate forecasts in the concrete framework of one of the major economies of the most productive Italian area, namely the province of Verona. It is worth mentioning that this region is indubitably recognized as the core of the commercial engine of the whole Italian country. This is why our analysis has a concrete impact; it is based on real data, and this is also the reason why particular attention has been taken in treating the relevant economical data and in choosing the right methods to manage them to obtain good forecasts. In particular, we develop an approach mainly based on vector autoregression where lagged values of two or more variables are considered, Granger causality, and the stochastic trend approach useful to work with the cointegration phenomenon.

Asymptotic normality of randomized periodogram for estimating quadratic variation in mixed Brownian–fractional Brownian model

Ehsan Azmoodeh Tommi Sottinen Lauri Viitasaari

https://doi.org/10.15559/15-VMSTA24

Pub. online: 11 May 2015 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 2, Issue 1 (2015), pp. 29–49

Abstract

We study asymptotic normality of the randomized periodogram estimator of quadratic variation in the mixed Brownian–fractional Brownian model. In the semimartingale case, that is, where the Hurst parameter H of the fractional part satisfies $H\in (3/4,1)$, the central limit theorem holds. In the nonsemimartingale case, that is, where $H\in (1/2,3/4]$, the convergence toward the normal distribution with a nonzero mean still holds if $H=3/4$, whereas for the other values, that is, $H\in (1/2,3/4)$, the central convergence does not take place. We also provide Berry–Esseen estimates for the estimator.

The rate of convergence to the normal law in terms of pseudomoments

Yuliya Mishura Yevheniya Munchak Petro Slyusarchuk

https://doi.org/10.15559/15-VMSTA23

Pub. online: 21 Apr 2015 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 2, Issue 2 (2015), pp. 95–106

Abstract

We establish the rate of convergence of distributions of sums of independent identically distributed random variables to the Gaussian distribution in terms of truncated pseudomoments by implementing the idea of Yu. Studnyev for getting estimates of the rate of convergence of the order higher than ${n}^{-1/2}$.

Autoregressive approaches to import–export time series I: basic techniques

Luca Di Persio

https://doi.org/10.15559/15-VMSTA22

Pub. online: 20 Apr 2015 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 2, Issue 1 (2015), pp. 51–65

Abstract

This work is the first part of a project dealing with an in-depth study of effective techniques used in econometrics in order to make accurate forecasts in the concrete framework of one of the major economies of the most productive Italian area, namely the province of Verona. In particular, we develop an approach mainly based on vector autoregressions, where lagged values of two or more variables are considered, Granger causality, and the stochastic trend approach useful to work with the cointegration phenomenon. Latter techniques constitute the core of the present paper, whereas in the second part of the project, we present how these approaches can be applied to economic data at our disposal in order to obtain concrete analysis of import–export behavior for the considered productive area of Verona.

Asymptotic normality of discretized maximum likelihood estimator for drift parameter in homogeneous diffusion model

Kostiantyn Ralchenko

https://doi.org/10.15559/15-VMSTA21

Pub. online: 13 Apr 2015 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 2, Issue 1 (2015), pp. 17–28

Abstract

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

Nonparametric Bayesian inference for multidimensional compound Poisson processes

Shota Gugushvili Frank van der Meulen Peter Spreij

https://doi.org/10.15559/15-VMSTA20

Pub. online: 13 Mar 2015 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 2, Issue 1 (2015), pp. 1–15

Abstract

Given a sample from a discretely observed multidimensional compound Poisson process, we study the problem of nonparametric estimation of its jump size density $r_{0}$ and intensity $\lambda _{0}$. We take a nonparametric Bayesian approach to the problem and determine posterior contraction rates in this context, which, under some assumptions, we argue to be optimal posterior contraction rates. In particular, our results imply the existence of Bayesian point estimates that converge to the true parameter pair $(r_{0},\lambda _{0})$ at these rates. To the best of our knowledge, construction of nonparametric density estimators for inference in the class of discretely observed multidimensional Lévy processes, and the study of their rates of convergence is a new contribution to the literature.

2010 Mathematics Subject Classification index

https://doi.org/10.15559/15-VMSTA12MI

Pub. online: 12 Feb 2015 Type: 2010 Mathematics Subject Classification Index

Journal: Modern Stochastics: Theory and Applications Volume 1, Issue 2 (2014), pp. 215–217

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Online ISSN: 2351-6054
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