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Consistency of the total least squares estimator in the linear errors-in-variables regression

Sergiy Shklyar

https://doi.org/10.15559/18-MSTA104

Pub. online: 30 May 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 5, Issue 3 (2018), pp. 247–295

Abstract

This paper deals with a homoskedastic errors-in-variables linear regression model and properties of the total least squares (TLS) estimator. We partly revise the consistency results for the TLS estimator previously obtained by the author [18]. We present complete and comprehensive proofs of consistency theorems. A theoretical foundation for construction of the TLS estimator and its relation to the generalized eigenvalue problem is explained. Particularly, the uniqueness of the estimate is proved. The Frobenius norm in the definition of the estimator can be substituted by the spectral norm, or by any other unitarily invariant norm; then the consistency results are still valid.

On closeness of two discrete weighted sums

Vydas Čekanavičius Palaniappan Vellaisamy

https://doi.org/10.15559/18-VMSTA103

Pub. online: 21 May 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 2 (2018), pp. 207–224

Abstract

The effect that weighted summands have on each other in approximations of $S={w_{1}}{S_{1}}+{w_{2}}{S_{2}}+\cdots +{w_{N}}{S_{N}}$ is investigated. Here, ${S_{i}}$’s are sums of integer-valued random variables, and ${w_{i}}$ denote weights, $i=1,\dots ,N$. Two cases are considered: the general case of independent random variables when their closeness is ensured by the matching of factorial moments and the case when the ${S_{i}}$ has the Markov Binomial distribution. The Kolmogorov metric is used to estimate the accuracy of approximation.

On closeness of two discrete weighted sums

Vydas Čekanavičius Palaniappan Vellaisamy

https://doi.org/10.15559/18-MSTA103

Pub. online: 21 May 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 5, Issue 2 (2018), pp. 207–224

Abstract

Large deviations of regression parameter estimator in continuous-time models with sub-Gaussian noise

Alexander V. Ivanov

Igor V. Orlovskyi

https://doi.org/10.15559/18-VMSTA102

Pub. online: 7 May 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 2 (2018), pp. 191–206

Abstract

A continuous-time regression model with a jointly strictly sub-Gaussian random noise is considered in the paper. Upper exponential bounds for probabilities of large deviations of the least squares estimator for the regression parameter are obtained.

Large deviations of regression parameter estimator in continuous-time models with sub-Gaussian noise

Alexander V. Ivanov

Igor V. Orlovskyi

https://doi.org/10.15559/18-MSTA102

Pub. online: 7 May 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 5, Issue 2 (2018), pp. 191–206

Abstract

Properties of Poisson processes directed by compound Poisson-Gamma subordinators

Khrystyna Buchak Lyudmyla Sakhno

https://doi.org/10.15559/18-VMSTA101

Pub. online: 2 May 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 2 (2018), pp. 167–189

Abstract

In the paper we consider time-changed Poisson processes where the time is expressed by compound Poisson-Gamma subordinators $G(N(t))$ and derive the expressions for their hitting times. We also study the time-changed Poisson processes where the role of time is played by the processes of the form $G(N(t)+at)$ and by the iteration of such processes.

Properties of Poisson processes directed by compound Poisson-Gamma subordinators

Khrystyna Buchak Lyudmyla Sakhno

https://doi.org/10.15559/18-MSTA101

Pub. online: 2 May 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 5, Issue 2 (2018), pp. 167–189

Abstract

Computation of option greeks under hybrid stochastic volatility models via Malliavin calculus

Bilgi Yilmaz

https://doi.org/10.15559/18-VMSTA100

Pub. online: 24 Apr 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 2 (2018), pp. 145–165

Abstract

This study introduces computation of option sensitivities (Greeks) using the Malliavin calculus under the assumption that the underlying asset and interest rate both evolve from a stochastic volatility model and a stochastic interest rate model, respectively. Therefore, it integrates the recent developments in the Malliavin calculus for the computation of Greeks: Delta, Vega, and Rho and it extends the method slightly. The main results show that Malliavin calculus allows a running Monte Carlo (MC) algorithm to present numerical implementations and to illustrate its effectiveness. The main advantage of this method is that once the algorithms are constructed, they can be used for numerous types of option, even if their payoff functions are not differentiable.

Computation of option greeks under hybrid stochastic volatility models via Malliavin calculus

Bilgi Yilmaz

https://doi.org/10.15559/18-MSTA100

Pub. online: 24 Apr 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 5, Issue 2 (2018), pp. 145–165

Abstract

Exponential bounds for the tail probability of the supremum of an inhomogeneous random walk

Dominyka Kievinaitė Jonas Šiaulys

https://doi.org/10.15559/18-VMSTA99

Pub. online: 15 Mar 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 2 (2018), pp. 129–143

Abstract

Let $\{{\xi _{1}},{\xi _{2}},\dots \}$ be a sequence of independent but not necessarily identically distributed random variables. In this paper, the sufficient conditions are found under which the tail probability $\mathbb{P}(\,{\sup _{n\geqslant 0}}\,{\sum _{i=1}^{n}}{\xi _{i}}>x)$ can be bounded above by ${\varrho _{1}}\exp \{-{\varrho _{2}}x\}$ with some positive constants ${\varrho _{1}}$ and ${\varrho _{2}}$. A way to calculate these two constants is presented. The application of the derived bound is discussed and a Lundberg-type inequality is obtained for the ultimate ruin probability in the inhomogeneous renewal risk model satisfying the net profit condition on average.

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