Probability in the Engineering and Informational Sciences最新文献

{"title":"On the probability of a Pareto record","authors":"James Allen Fill, Ao Sun","doi":"10.1017/s0269964824000081","DOIUrl":"https://doi.org/10.1017/s0269964824000081","url":null,"abstract":"<p>Given a sequence of independent random vectors taking values in <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240603131552515-0032:S0269964824000081:S0269964824000081_inline1.png\"><span data-mathjax-type=\"texmath\"><span>${mathbb R}^d$</span></span></img></span></span> and having common continuous distribution function <span>F</span>, say that the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240603131552515-0032:S0269964824000081:S0269964824000081_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$n^{rm scriptsize}$</span></span></img></span></span>th observation <span>sets a (Pareto) record</span> if it is not dominated (in every coordinate) by any preceding observation. Let <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240603131552515-0032:S0269964824000081:S0269964824000081_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$p_n(F) equiv p_{n, d}(F)$</span></span></img></span></span> denote the probability that the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240603131552515-0032:S0269964824000081:S0269964824000081_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$n^{rm scriptsize}$</span></span></img></span></span>th observation sets a record. There are many interesting questions to address concerning <span>p<span>n</span></span> and multivariate records more generally, but this short paper focuses on how <span>p<span>n</span></span> varies with <span>F</span>, particularly if, under <span>F</span>, the coordinates exhibit negative dependence or positive dependence (rather than independence, a more-studied case). We introduce new notions of negative and positive dependence ideally suited for such a study, called <span>negative record-setting probability dependence</span> (NRPD) and <span>positive record-setting probability dependence</span> (PRPD), relate these notions to existing notions of dependence, and for fixed <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240603131552515-0032:S0269964824000081:S0269964824000081_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$d geq 2$</span></span></img></span></span> and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240603131552515-0032:S0269964824000081:S0269964824000081_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$n geq 1$</span></span></img></span></span> prove that the image of the mapping <span>p<span>n</span></span> on the domain of NRPD (respectively, PRPD) distributions is <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://stati","PeriodicalId":54582,"journal":{"name":"Probability in the Engineering and Informational Sciences","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141254853","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Quantile-based information generating functions and their properties and uses","authors":"Suchandan Kayal, N. Balakrishnan","doi":"10.1017/s0269964824000068","DOIUrl":"https://doi.org/10.1017/s0269964824000068","url":null,"abstract":"\u0000 Information generating functions (IGFs) have been of great interest to researchers due to their ability to generate various information measures. The IGF of an absolutely continuous random variable (see Golomb, S. (1966). The information generating function of a probability distribution. IEEE Transactions in Information Theory, 12(1), 75–77) depends on its density function. But, there are several models with intractable cumulative distribution functions, but do have explicit quantile functions. For this reason, in this work, we propose quantile version of the IGF, and then explore some of its properties. Effect of increasing transformations on it is then studied. Bounds are also obtained. The proposed generating function is studied especially for escort and generalized escort distributions. Some connections between the quantile-based IGF (Q-IGF) order and well-known stochastic orders are established. Finally, the proposed Q-IGF is extended for residual and past lifetimes as well. Several examples are presented through out to illustrate the theoretical results established here. An inferential application of the proposed methodology is also discussed","PeriodicalId":54582,"journal":{"name":"Probability in the Engineering and Informational Sciences","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141111804","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Orderings of extremes among dependent extended Weibull random variables","authors":"Ramkrishna Jyoti Samanta, Sangita Das, N. Balakrishnan","doi":"10.1017/s026996482400007x","DOIUrl":"https://doi.org/10.1017/s026996482400007x","url":null,"abstract":"<p>In this work, we consider two sets of dependent variables <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240507063908403-0649:S026996482400007X:S026996482400007X_inline1.png\"><span data-mathjax-type=\"texmath\"><span>${X_{1},ldots,X_{n}}$</span></span></img></span></span> and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240507063908403-0649:S026996482400007X:S026996482400007X_inline2.png\"><span data-mathjax-type=\"texmath\"><span>${Y_{1},ldots,Y_{n}}$</span></span></img></span></span>, where <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240507063908403-0649:S026996482400007X:S026996482400007X_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$X_{i}sim EW(alpha_{i},lambda_{i},k_{i})$</span></span></img></span></span> and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240507063908403-0649:S026996482400007X:S026996482400007X_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$Y_{i}sim EW(beta_{i},mu_{i},l_{i})$</span></span></img></span></span>, for <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240507063908403-0649:S026996482400007X:S026996482400007X_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$i=1,ldots, n$</span></span></img></span></span>, which are coupled by Archimedean copulas having different generators. We then establish different inequalities between two extremes, namely, <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240507063908403-0649:S026996482400007X:S026996482400007X_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$X_{1:n}$</span></span></img></span></span> and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240507063908403-0649:S026996482400007X:S026996482400007X_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$Y_{1:n}$</span></span></img></span></span> and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240507063908403-0649:S026996482400007X:S026996482400007X_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$X_{n:n}$</span></span></img></span></span> and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240507063908403-0649:S026996482400007X:S026996482400007X_inline9.png\"><span data-mathjax-type=\"texmath\"><span>$Y_{n:n}$</span></span></img></span></span>, in terms of the usual stochastic, star, Lorenz, hazard rate, reversed hazar","PeriodicalId":54582,"journal":{"name":"Probability in the Engineering and Informational Sciences","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140884164","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An inequality for log-concave functions and its use in the study of failure rates","authors":"Mahdi Alimohammadi, N. Balakrishnan, T. Simon","doi":"10.1017/s0269964824000056","DOIUrl":"https://doi.org/10.1017/s0269964824000056","url":null,"abstract":"\u0000 We establish here an integral inequality for real log-concave functions, which can be viewed as an average monotone likelihood property. This inequality is then applied to examine the monotonicity of failure rates.","PeriodicalId":54582,"journal":{"name":"Probability in the Engineering and Informational Sciences","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140670395","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Discounted cost exponential semi-Markov decision processes with unbounded transition rates: a service rate control problem with impatient customers","authors":"Bora Çekyay","doi":"10.1017/s0269964824000044","DOIUrl":"https://doi.org/10.1017/s0269964824000044","url":null,"abstract":"<p>We focus on exponential semi-Markov decision processes with unbounded transition rates. We first provide several sufficient conditions under which the value iteration procedure converges to the optimal value function and optimal deterministic stationary policies exist. These conditions are also valid for general semi-Markov decision processes possibly with accumulation points. Then, we apply our results to a service rate control problem with impatient customers. The resulting exponential semi-Markov decision process has unbounded transition rates, which makes the well-known uniformization technique inapplicable. We analyze the structure of the optimal policy and the monotonicity of the optimal value function by using the customization technique that was introduced by the author in prior work.</p>","PeriodicalId":54582,"journal":{"name":"Probability in the Engineering and Informational Sciences","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140609514","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Discounted densities of overshoot and undershoot for Lévy processes with applications in finance","authors":"Hui Gao, Chuancun Yin","doi":"10.1017/s0269964824000032","DOIUrl":"https://doi.org/10.1017/s0269964824000032","url":null,"abstract":"This paper considers the first passage times to constant boundaries and the two-sided exit problem for Lévy process with a characteristic exponent in which at least one of the two jumps having rational Laplace transforms. The joint distribution of the first passage times and undershoot/overshoot are obtained. The processes recover many models that have appeared in the literature such as the compound Poisson risk models, the perturbed compound Poisson risk models, and their dual ones. As applications, we obtain the solutions for popular path-dependent options such as lookback and barrier options in terms of Laplace transforms.","PeriodicalId":54582,"journal":{"name":"Probability in the Engineering and Informational Sciences","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140165478","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Some properties of convex and increasing convex orders under Archimedean copula","authors":"Qingyuan Guan, Bing Xing Wang","doi":"10.1017/s0269964824000020","DOIUrl":"https://doi.org/10.1017/s0269964824000020","url":null,"abstract":"<p>In this paper, the ordering properties of convex and increasing convex orders of the dependent random variables are studied. Some closure properties of the convex and increasing convex orders under independent random variables are extended to the dependent random variables under the Archimedean copula. Two applications are provided to illustrate our results.</p>","PeriodicalId":54582,"journal":{"name":"Probability in the Engineering and Informational Sciences","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140034523","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Analyzing the multi-state system under a run shock model","authors":"Murat Ozkut, Cihangir Kan, Ceki Franko","doi":"10.1017/s0269964824000019","DOIUrl":"https://doi.org/10.1017/s0269964824000019","url":null,"abstract":"A system experiences random shocks over time, with two critical levels, <jats:italic>d</jats:italic><jats:sub>1</jats:sub> and <jats:italic>d</jats:italic><jats:sub>2</jats:sub>, where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0269964824000019_inline1.png\" /> <jats:tex-math>$d_{1} lt d_{2}$</jats:tex-math> </jats:alternatives> </jats:inline-formula>. <jats:italic>k</jats:italic> consecutive shocks with magnitudes between <jats:italic>d</jats:italic><jats:sub>1</jats:sub> and <jats:italic>d</jats:italic><jats:sub>2</jats:sub> partially damaging the system, causing it to transition to a lower, partially working state. Shocks with magnitudes above <jats:italic>d</jats:italic><jats:sub>2</jats:sub> have a catastrophic effect, resulting in complete failure. This theoretical framework gives rise to a multi-state system characterized by an indeterminate quantity of states. When the time between successive shocks follows a phase-type distribution, a detailed analysis of the system’s dynamic reliability properties such as the lifetime of the system, the time it spends in perfect functioning, as well as the total time it spends in partially working states are discussed.","PeriodicalId":54582,"journal":{"name":"Probability in the Engineering and Informational Sciences","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139769316","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"General distributions of number representation elements","authors":"Félix Balado, Guénolé C. M. Silvestre","doi":"10.1017/s0269964823000207","DOIUrl":"https://doi.org/10.1017/s0269964823000207","url":null,"abstract":"We provide general expressions for the joint distributions of the <jats:italic>k</jats:italic> most significant <jats:italic>b</jats:italic>-ary digits and of the <jats:italic>k</jats:italic> leading continued fraction (CF) coefficients of outcomes of arbitrary continuous random variables. Our analysis highlights the connections between the two problems. In particular, we give the general convergence law of the distribution of the <jats:italic>j</jats:italic>th significant digit, which is the counterpart of the general convergence law of the distribution of the <jats:italic>j</jats:italic>th CF coefficient (Gauss-Kuz’min law). We also particularise our general results for Benford and Pareto random variables. The former particularisation allows us to show the central role played by Benford variables in the asymptotics of the general expressions, among several other results, including the analogue of Benford’s law for CFs. The particularisation for Pareto variables—which include Benford variables as a special case—is especially relevant in the context of pervasive scale-invariant phenomena, where Pareto variables occur much more frequently than Benford variables. This suggests that the Pareto expressions that we produce have wider applicability than their Benford counterparts in modelling most significant digits and leading CF coefficients of real data. Our results may find practical application in all areas where Benford’s law has been previously used.","PeriodicalId":54582,"journal":{"name":"Probability in the Engineering and Informational Sciences","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139768952","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Log-concavity and relative log-concave ordering of compound distributions","authors":"Wanwan Xia, Wenhua Lv","doi":"10.1017/s0269964823000293","DOIUrl":"https://doi.org/10.1017/s0269964823000293","url":null,"abstract":"In this paper, we compare the entropy of the original distribution and its corresponding compound distribution. Several results are established based on convex order and relative log-concave order. The necessary and sufficient condition for a compound distribution to be log-concave is also discussed, including compound geometric distribution, compound negative binomial distribution and compound binomial distribution.","PeriodicalId":54582,"journal":{"name":"Probability in the Engineering and Informational Sciences","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139579048","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}