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Confidence regions in Cox proportional hazards model with measurement errors and unbounded parameter set

Oksana Chernova Alexander Kukush

https://doi.org/10.15559/18-VMSTA94

Pub. online: 31 Jan 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 1 (2018), pp. 37–52

Abstract

Cox proportional hazards model with measurement errors is considered. In Kukush and Chernova (2017), we elaborated a simultaneous estimator of the baseline hazard rate $\lambda (\cdot )$ and the regression parameter β, with the unbounded parameter set $\varTheta =\varTheta _{\lambda }\times \varTheta _{\beta }$, where $\varTheta _{\lambda }$ is a closed convex subset of $C[0,\tau ]$ and $\varTheta _{\beta }$ is a compact set in ${\mathbb{R}}^{m}$. The estimator is consistent and asymptotically normal. In the present paper, we construct confidence intervals for integral functionals of $\lambda (\cdot )$ and a confidence region for β under restrictions on the error distribution. In particular, we handle the following cases: (a) the measurement error is bounded, (b) it is a normally distributed random vector, and (c) it has independent components which are shifted Poisson random variables.

A moment-distance hybrid method for estimating a mixture of two symmetric densities

David Källberg Yuri Belyaev Patrik Rydén

https://doi.org/10.15559/17-VMSTA93

Pub. online: 18 Jan 2018 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 5, Issue 1 (2018), pp. 1–36

Abstract

In clustering of high-dimensional data a variable selection is commonly applied to obtain an accurate grouping of the samples. For two-class problems this selection may be carried out by fitting a mixture distribution to each variable. We propose a hybrid method for estimating a parametric mixture of two symmetric densities. The estimator combines the method of moments with the minimum distance approach. An evaluation study including both extensive simulations and gene expression data from acute leukemia patients shows that the hybrid method outperforms a maximum-likelihood estimator in model-based clustering. The hybrid estimator is flexible and performs well also under imprecise model assumptions, suggesting that it is robust and suited for real problems.

2010 Mathematics Subject Classification index

https://doi.org/10.15559/17-VMSTA44MI

Pub. online: 29 Dec 2017 Type: 2010 Mathematics Subject Classification Index

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 4 (2017), pp. 431–434

Subject index

https://doi.org/10.15559/17-VMSTA44SI

Pub. online: 29 Dec 2017 Type: Subject Index

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 4 (2017), pp. 429–430

Author index

https://doi.org/10.15559/17-VMSTA44AI

Pub. online: 29 Dec 2017 Type: Author Index

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 4 (2017), pp. 427–428

On the size of the block of 1 for Ξ-coalescents with dust

Fabian Freund Martin Möhle

https://doi.org/10.15559/17-VMSTA92

Pub. online: 27 Dec 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 4 (2017), pp. 407–425

Abstract

We study the frequency process $f_{1}$ of the block of 1 for a Ξ-coalescent Π with dust. If Π stays infinite, $f_{1}$ is a jump-hold process which can be expressed as a sum of broken parts from a stick-breaking procedure with uncorrelated, but in general non-independent, stick lengths with common mean. For Dirac-Λ-coalescents with $\varLambda =\delta _{p}$, $p\in [\frac{1}{2},1)$, $f_{1}$ is not Markovian, whereas its jump chain is Markovian. For simple Λ-coalescents the distribution of $f_{1}$ at its first jump, the asymptotic frequency of the minimal clade of 1, is expressed via conditionally independent shifted geometric distributions.

On model fitting and estimation of strictly stationary processes

Marko Voutilainen

Lauri Viitasaari Pauliina Ilmonen

https://doi.org/10.15559/17-VMSTA91

Pub. online: 22 Dec 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 4 (2017), pp. 381–406

Abstract

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.

Double barrier reflected BSDEs with stochastic Lipschitz coefficient

Mohamed Marzougue Mohamed El Otmani

https://doi.org/10.15559/17-VMSTA90

Pub. online: 8 Dec 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 4 (2017), pp. 353–379

Abstract

This paper proves the existence and uniqueness of a solution to doubly reflected backward stochastic differential equations where the coefficient is stochastic Lipschitz, by means of the penalization method.

The risk model with stochastic premiums, dependence and a threshold dividend strategy

Olena Ragulina

https://doi.org/10.15559/17-VMSTA89

Pub. online: 8 Dec 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 4 (2017), pp. 315–351

Abstract

The paper deals with a generalization of the risk model with stochastic premiums where dependence structures between claim sizes and inter-claim times as well as premium sizes and inter-premium times are modeled by Farlie–Gumbel–Morgenstern copulas. In addition, dividends are paid to its shareholders according to a threshold dividend strategy. We derive integral and integro-differential equations for the Gerber–Shiu function and the expected discounted dividend payments until ruin. Next, we concentrate on the detailed investigation of the model in the case of exponentially distributed claim and premium sizes. In particular, we find explicit formulas for the ruin probability in the model without either dividend payments or dependence as well as for the expected discounted dividend payments in the model without dependence. Finally, numerical illustrations are presented.

Integrated quantile functions: properties and applications

Alexander A. Gushchin

Dmitriy A. Borzykh

https://doi.org/10.15559/17-VMSTA88

Pub. online: 8 Dec 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 4 (2017), pp. 285–314

Abstract

In this paper we provide a systematic exposition of basic properties of integrated distribution and quantile functions. We define these transforms in such a way that they characterize any probability distribution on the real line and are Fenchel conjugates of each other. We show that uniform integrability, weak convergence and tightness admit a convenient characterization in terms of integrated quantile functions. As an application we demonstrate how some basic results of the theory of comparison of binary statistical experiments can be deduced using integrated quantile functions. Finally, we extend the area of application of the Chacon–Walsh construction in the Skorokhod embedding problem.

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Online ISSN: 2351-6054
Print ISSN: 2351-6046

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