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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-MSTA94

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.

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-MSTA93

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

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

2010 Mathematics Subject Classification index

https://doi.org/10.15559/17-MSTA44MI

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

Subject index

https://doi.org/10.15559/17-MSTA44SI

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

Author index

https://doi.org/10.15559/17-MSTA44AI

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.

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