Pub Date : 2018-12-28DOI: 10.19195/0208-4147.38.2.9
Rafał M. Wieczorek, H. Podsędkowska
The entropic upper bound for Bayes risk in a general quantum case is presented. We obtained generalization of the entropic lower bound for probability of detection. Our result indicates upper bound for Bayes risk in a particular case of loss function – for probability of detection in a pretty general setting of an arbitrary finite von Neumann algebra. It is also shown under which condition the indicated upper bound is achieved.
{"title":"Entropic upper bound for Bayes risk in the quantum case","authors":"Rafał M. Wieczorek, H. Podsędkowska","doi":"10.19195/0208-4147.38.2.9","DOIUrl":"https://doi.org/10.19195/0208-4147.38.2.9","url":null,"abstract":"The entropic upper bound for Bayes risk in a general quantum case is presented. We obtained generalization of the entropic lower bound for probability of detection. Our result indicates upper bound for Bayes risk in a particular case of loss function – for probability of detection in a pretty general setting of an arbitrary finite von Neumann algebra. It is also shown under which condition the indicated upper bound is achieved.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48461076","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-12-28DOI: 10.19195/0208-4147.38.2.3
Dejian Zhou, Weiwei Li, Y. Jiao
In this paper, we give an equivalent characterization of weak BMO martingale spaces due to Ferenc Weisz 1998.
本文给出了Ferenc-Weisz 1998给出的弱BMO鞅空间的等价刻画。
{"title":"An equivalent characterization of weak BMO martingale spaces","authors":"Dejian Zhou, Weiwei Li, Y. Jiao","doi":"10.19195/0208-4147.38.2.3","DOIUrl":"https://doi.org/10.19195/0208-4147.38.2.3","url":null,"abstract":"In this paper, we give an equivalent characterization of weak BMO martingale spaces due to Ferenc Weisz 1998.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48363675","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-12-28DOI: 10.19195/0208-4147.38.2.8
Zhenxia Liu, Xiangfeng Yang
In the first n steps of a two-state success and failure Markov chain, the longest success run Ln has been attracting considerable attention due to its various applications. In this paper, we study Ln in terms of its two closely connected properties: moment generating function and large deviations. This study generalizes several existing results in the literature, and also finds an application in statistical inference. Our method on the moment generating function is based on a global estimate of the cumulative distribution function of Ln proposed in this paper, and the proofs of the large deviations include the Gärtner–Ellis theorem and the moment generating function.
{"title":"On the longest runs in Markov chains","authors":"Zhenxia Liu, Xiangfeng Yang","doi":"10.19195/0208-4147.38.2.8","DOIUrl":"https://doi.org/10.19195/0208-4147.38.2.8","url":null,"abstract":"In the first n steps of a two-state success and failure Markov chain, the longest success run Ln has been attracting considerable attention due to its various applications. In this paper, we study Ln in terms of its two closely connected properties: moment generating function and large deviations. This study generalizes several existing results in the literature, and also finds an application in statistical inference. Our method on the moment generating function is based on a global estimate of the cumulative distribution function of Ln proposed in this paper, and the proofs of the large deviations include the Gärtner–Ellis theorem and the moment generating function.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42368931","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-12-28DOI: 10.19195/0208-4147.38.2.7
Bünyamin Kızıldemir, Nicolas Privault
We construct a dependence structure for binomial, Poisson and Gaussian random vectors, based on partially ordered binary trees and sums of independent random variables. Using this construction, we characterize the supermodular ordering of such random vectors via the componentwise ordering of their covariance matrices. For this, we apply Möbius inversion techniques on partially ordered trees, which allow us to connect the Lévy measures of Poisson random vectors on the discrete d-dimensional hypercube to their covariance matrices.
{"title":"Supermodular ordering of Poisson and binomial random vectors by tree-based correlations","authors":"Bünyamin Kızıldemir, Nicolas Privault","doi":"10.19195/0208-4147.38.2.7","DOIUrl":"https://doi.org/10.19195/0208-4147.38.2.7","url":null,"abstract":"We construct a dependence structure for binomial, Poisson and Gaussian random vectors, based on partially ordered binary trees and sums of independent random variables. Using this construction, we characterize the supermodular ordering of such random vectors via the componentwise ordering of their covariance matrices. For this, we apply Möbius inversion techniques on partially ordered trees, which allow us to connect the Lévy measures of Poisson random vectors on the discrete d-dimensional hypercube to their covariance matrices.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45368088","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-12-28DOI: 10.19195/0208-4147.38.2.10
S. Abbasi, M. H. Alamatsaz
Stochastic comparisons of lifetime characteristics of reliability systems and their components are of common use in lifetime analysis. In this paper, using Harris family distributions, we compare lifetimes of two series systems with random number of components, with respect to several types of stochastic orders. Our results happen to enfold several previous findings in this connection. We shall also show that several stochastic orders and ageing characteristics, such as IHRA, DHRA, NBU, and NWU, are inherited by transformation to Harris family. Finally, some refinements are made concerning related existing results in the literature.
{"title":"Preservation properties of stochastic orders by transformation to Harris family","authors":"S. Abbasi, M. H. Alamatsaz","doi":"10.19195/0208-4147.38.2.10","DOIUrl":"https://doi.org/10.19195/0208-4147.38.2.10","url":null,"abstract":"Stochastic comparisons of lifetime characteristics of reliability systems and their components are of common use in lifetime analysis. In this paper, using Harris family distributions, we compare lifetimes of two series systems with random number of components, with respect to several types of stochastic orders. Our results happen to enfold several previous findings in this connection. We shall also show that several stochastic orders and ageing characteristics, such as IHRA, DHRA, NBU, and NWU, are inherited by transformation to Harris family. Finally, some refinements are made concerning related existing results in the literature.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43021728","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-12-28DOI: 10.19195/0208-4147.38.2.5
Maciej Kawecki, R. Różański, Grzegorz Chłapiński, M. Hławka, Krzysztof Jamróz, A. Zagdanski
In the paper, the construction of unconditional bootstrap prediction intervals and regions for some class of second order stationary multivariate linear time series models is considered. Our approach uses the sieve bootstrap procedure introduced by Kreiss 1992 and Bühlmann 1997. Basic theoretical results concerning consistency of the bootstrap replications and the bootstrap prediction regions are proved. We present a simulation study comparing the proposed bootstrap methods with the Box–Jenkins approach.
{"title":"Prediction intervals and regions for multivariate time series models with sieve bootstrap","authors":"Maciej Kawecki, R. Różański, Grzegorz Chłapiński, M. Hławka, Krzysztof Jamróz, A. Zagdanski","doi":"10.19195/0208-4147.38.2.5","DOIUrl":"https://doi.org/10.19195/0208-4147.38.2.5","url":null,"abstract":"In the paper, the construction of unconditional bootstrap prediction intervals and regions for some class of second order stationary multivariate linear time series models is considered. Our approach uses the sieve bootstrap procedure introduced by Kreiss 1992 and Bühlmann 1997. Basic theoretical results concerning consistency of the bootstrap replications and the bootstrap prediction regions are proved. We present a simulation study comparing the proposed bootstrap methods with the Box–Jenkins approach.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47671696","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-12-28DOI: 10.19195/0208-4147.38.2.1
M. Ben Chrouda, Khalifa El Mabrouk, Kods Hassine
Let Δk be the Dunkl Laplacian on ℜd associated with a reflection group W and a multiplicity function k. The purpose of this paper is to establish the existence and the uniqueness of a positive solution on the unit ball B of ℜd to the following boundary value problem:Δku = φu in B and u = ƒ on ∂BWe distinguish two cases of nonnegative perturbation φ: trivial and nontrivial.
{"title":"Boundary value problems for the Dunkl Laplacian","authors":"M. Ben Chrouda, Khalifa El Mabrouk, Kods Hassine","doi":"10.19195/0208-4147.38.2.1","DOIUrl":"https://doi.org/10.19195/0208-4147.38.2.1","url":null,"abstract":"Let Δk be the Dunkl Laplacian on ℜd associated with a reflection group W and a multiplicity function k. The purpose of this paper is to establish the existence and the uniqueness of a positive solution on the unit ball B of ℜd to the following boundary value problem:Δku = φu in B and u = ƒ on ∂BWe distinguish two cases of nonnegative perturbation φ: trivial and nontrivial. ","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49261267","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-12-19DOI: 10.37190/0208-4147.41.1.10
I. Yaroslavtsev
In the present paper we introduce the notion of strongly orthogonal martingales. Moreover, we show that for any UMD Banach space $X$ and for any $X$-valued strongly orthogonal martingales $M$ and $N$ such that $N$ is weakly differentially subordinate to $M$ one has that for any $1
{"title":"On strongly orthogonal martingales in UMD Banach spaces","authors":"I. Yaroslavtsev","doi":"10.37190/0208-4147.41.1.10","DOIUrl":"https://doi.org/10.37190/0208-4147.41.1.10","url":null,"abstract":"In the present paper we introduce the notion of strongly orthogonal martingales. Moreover, we show that for any UMD Banach space $X$ and for any $X$-valued strongly orthogonal martingales $M$ and $N$ such that $N$ is weakly differentially subordinate to $M$ one has that for any $1<p<infty$ [ mathbb E |N_t|^p leq chi_{p, X}^p mathbb E |M_t|^p,;;; tgeq 0, ] with the sharp constant $chi_{p, X}$ being the norm of a decoupling-type martingale transform and being within the range [ maxBigl{sqrt{beta_{p, X}}, sqrt{hbar_{p,X}}Bigr} leq max{beta_{p, X}^{gamma,+}, beta_{p, X}^{gamma, -}} leq chi_{p, X} leq min{beta_{p, X}, hbar_{p,X}}, ] where $beta_{p, X}$ is the UMD$_p$ constant of $X$, $hbar_{p, X}$ is the norm of the Hilbert transform on $L^p(mathbb R; X)$, and $beta_{p, X}^{gamma,+}$ and $ beta_{p, X}^{gamma, -}$ are the Gaussian decoupling constants.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43043742","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-07-30DOI: 10.19195/0208-4147.38.1.11
Ying Dong, Lixin Song, Mingqiu Wang, Muhammad Amin
To investigate the features of the individual from the mixedtype model, a novel model, named the mixed-type generalized linear model, is proposed firstly in this work, which is verified to be realistic and useful. We consider the robustness of M-estimation to estimate the unknown parameters of the mixed-type generalized linear model. By applying the law of large numbers and the central limit theorem, the consistency and asymptotic normality of the M-estimation for the mixed-type generalized linear model are proved with regularity assumptions. At last, in order to evaluate the finite sample performance of the estimator for the new model, several applied instances are presented, which show the good performance of the estimator.
{"title":"M-estimation of the mixed-type generalized linear model","authors":"Ying Dong, Lixin Song, Mingqiu Wang, Muhammad Amin","doi":"10.19195/0208-4147.38.1.11","DOIUrl":"https://doi.org/10.19195/0208-4147.38.1.11","url":null,"abstract":"To investigate the features of the individual from the mixedtype model, a novel model, named the mixed-type generalized linear model, is proposed firstly in this work, which is verified to be realistic and useful. We consider the robustness of M-estimation to estimate the unknown parameters of the mixed-type generalized linear model. By applying the law of large numbers and the central limit theorem, the consistency and asymptotic normality of the M-estimation for the mixed-type generalized linear model are proved with regularity assumptions. At last, in order to evaluate the finite sample performance of the estimator for the new model, several applied instances are presented, which show the good performance of the estimator.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44144023","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-07-30DOI: 10.19195/0208-4147.38.1.6
A. Adler, P. Matuła
We study the almost sure convergence of weighted sums of dependent random variables to a positive and finite constant, in the case when the random variables have either mean zero or no mean at all. These are not typical strong laws and they are called exact strong laws of large numbers. We do not assume any particular type of dependence and furthermore consider sequences which are not necessarily identically distributed. The obtained results may be applied to sequences of negatively associated random variables.
{"title":"On exact strong laws of large numbers under general dependence conditions","authors":"A. Adler, P. Matuła","doi":"10.19195/0208-4147.38.1.6","DOIUrl":"https://doi.org/10.19195/0208-4147.38.1.6","url":null,"abstract":"We study the almost sure convergence of weighted sums of dependent random variables to a positive and finite constant, in the case when the random variables have either mean zero or no mean at all. These are not typical strong laws and they are called exact strong laws of large numbers. We do not assume any particular type of dependence and furthermore consider sequences which are not necessarily identically distributed. The obtained results may be applied to sequences of negatively associated random variables.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49506051","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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