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Quantifying and estimating additive measures of interaction from case-control data

Ola Hössjer Lars Alfredsson Anna Karin Hedström Magnus Lekman Ingrid Kockum Tomas Olsson

https://doi.org/10.15559/17-VMSTA77

Pub. online: 26 Apr 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 2 (2017), pp. 109–125

Abstract

In this paper we develop a general framework for quantifying how binary risk factors jointly influence a binary outcome. Our key result is an additive expansion of odds ratios as a sum of marginal effects and interaction terms of varying order. These odds ratio expansions are used for estimating the excess odds ratio, attributable proportion and synergy index for a case-control dataset by means of maximum likelihood from a logistic regression model. The confidence intervals associated with these estimates of joint effects and interaction of risk factors rely on the delta method. Our methodology is illustrated with a large Nordic meta dataset for multiple sclerosis. It combines four studies, with a total of 6265 cases and 8401 controls. It has three risk factors (smoking and two genetic factors) and a number of other confounding variables.

Quantifying and estimating additive measures of interaction from case-control data

Ola Hössjer Lars Alfredsson Anna Karin Hedström Magnus Lekman Ingrid Kockum Tomas Olsson

https://doi.org/10.15559/17-MSTA77

Pub. online: 26 Apr 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 4, Issue 2 (2017), pp. 109–125

Abstract

A functional limit theorem for random processes with immigration in the case of heavy tails

Alexander Marynych Glib Verovkin

https://doi.org/10.15559/17-VMSTA76

Pub. online: 12 Apr 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 2 (2017), pp. 93–108

Abstract

Let $(X_{k},\xi _{k})_{k\in \mathbb{N}}$ be a sequence of independent copies of a pair $(X,\xi )$ where X is a random process with paths in the Skorokhod space $D[0,\infty )$ and ξ is a positive random variable. The random process with immigration $(Y(u))_{u\in \mathbb{R}}$ is defined as the a.s. finite sum $Y(u)=\sum _{k\ge 0}X_{k+1}(u-\xi _{1}-\cdots -\xi _{k})\mathbb{1}_{\{\xi _{1}+\cdots +\xi _{k}\le u\}}$. We obtain a functional limit theorem for the process $(Y(ut))_{u\ge 0}$, as $t\to \infty $, when the law of ξ belongs to the domain of attraction of an α-stable law with $\alpha \in (0,1)$, and the process X oscillates moderately around its mean $\mathbb{E}[X(t)]$. In this situation the process $(Y(ut))_{u\ge 0}$, when scaled appropriately, converges weakly in the Skorokhod space $D(0,\infty )$ to a fractionally integrated inverse stable subordinator.

A functional limit theorem for random processes with immigration in the case of heavy tails

Alexander Marynych Glib Verovkin

https://doi.org/10.15559/17-MSTA76

Pub. online: 12 Apr 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 4, Issue 2 (2017), pp. 93–108

Abstract

Asymptotic behaviour of non-isotropic random walks with heavy tails

Mark Kelbert Enzo Orsingher

https://doi.org/10.15559/17-VMSTA75

Pub. online: 6 Apr 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 1 (2017), pp. 79–89

Abstract

A random flight on a plane with non-isotropic displacements at the moments of direction changes is considered. In the case of exponentially distributed flight lengths a Gaussian limit theorem is proved for the position of a particle in the scheme of series when jump lengths and non-isotropic displacements tend to zero. If the flight lengths have a folded Cauchy distribution the limiting distribution of the particle position is a convolution of the circular bivariate Cauchy distribution with a Gaussian law.

Asymptotic behaviour of non-isotropic random walks with heavy tails

Mark Kelbert Enzo Orsingher

https://doi.org/10.15559/17-MSTA75

Pub. online: 6 Apr 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 4, Issue 1 (2017), pp. 79–89

Abstract

Randomly stopped maximum and maximum of sums with consistently varying distributions

Ieva Marija Andrulytė Martynas Manstavičius Jonas Šiaulys

https://doi.org/10.15559/17-VMSTA74

Pub. online: 6 Mar 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 1 (2017), pp. 65–78

Abstract

Let $\{\xi _{1},\xi _{2},\dots \}$ be a sequence of independent random variables, and η be a counting random variable independent of this sequence. In addition, let $S_{0}:=0$ and $S_{n}:=\xi _{1}+\xi _{2}+\cdots +\xi _{n}$ for $n\geqslant 1$. We consider conditions for random variables $\{\xi _{1},\xi _{2},\dots \}$ and η under which the distribution functions of the random maximum $\xi _{(\eta )}:=\max \{0,\xi _{1},\xi _{2},\dots ,\xi _{\eta }\}$ and of the random maximum of sums $S_{(\eta )}:=\max \{S_{0},S_{1},S_{2},\dots ,S_{\eta }\}$ belong to the class of consistently varying distributions. In our consideration the random variables $\{\xi _{1},\xi _{2},\dots \}$ are not necessarily identically distributed.

Randomly stopped maximum and maximum of sums with consistently varying distributions

Ieva Marija Andrulytė Martynas Manstavičius Jonas Šiaulys

https://doi.org/10.15559/17-MSTA74

Pub. online: 6 Mar 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 4, Issue 1 (2017), pp. 65–78

Abstract

L^{p} (p \geq 2)

-solutions of generalized BSDEs with jumps and monotone generator in a general filtration

M’hamed Eddahbi Imade Fakhouri Youssef Ouknine

https://doi.org/10.15559/17-VMSTA73

Pub. online: 7 Feb 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 1 (2017), pp. 25–63

Abstract

In this paper, we study multidimensional generalized BSDEs that have a monotone generator in a general filtration supporting a Brownian motion and an independent Poisson random measure. First, we prove the existence and uniqueness of ${\mathbb{L}}^{p}(p\ge 2)$-solutions in the case of a fixed terminal time under suitable p-integrability conditions on the data. Then, we extend these results to the case of a random terminal time. Furthermore, we provide a comparison result in dimension 1.

L^{p} (p \geq 2)

-solutions of generalized BSDEs with jumps and monotone generator in a general filtration

M’hamed Eddahbi Imade Fakhouri Youssef Ouknine

https://doi.org/10.15559/17-MSTA73

Pub. online: 7 Feb 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications* Volume 4, Issue 1 (2017), pp. 25–63

Abstract

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