Abstract This is a short survey about asymptotic properties of a supercritical branching process ( Z n ) (Z_{n}) with immigration in a stationary and ergodic or independent and identically distributed random environment. We first present basic properties of the fundamental submartingale ( W n ) (W_{n}) , about the a.s. convergence, the non-degeneracy of its limit 𝑊, the convergence in L p L^{p} for p ≥ 1 pgeq 1 , and the boundedness of the harmonic moments E W n - a mathbb{E}W_{n}^{-a} , a > 0 a>0 . We then present limit theorems and large deviation results on log Z n log Z_{n} , including the law of large numbers, large and moderate deviation principles, the central limit theorem with Berry–Esseen’s bound, and Cramér’s large deviation expansion. Some key ideas of the proofs are also presented.
摘要本文研究了平稳遍历或独立同分布随机环境下具有迁移的超临界分支过程(zn) {(Z＿n)}的渐近性质。我们首先给出了基本次鞅(wn) {(W＿n)}的基本性质，关于a.s.收敛性，极限的非简并性𝑊，当p≥1 p {}geq 1时，L^在L^p中的收敛性，以及谐波矩E ^ W n- a mathbb{E} W＿n{^ }a, a{> a>0的有界性。在此基础上，我们给出了log ln zn }log Z＿n{上的极限定理和大偏差结果，包括大数定律、大偏差和中等偏差原理、Berry-Esseen界的中心极限定理和cram大偏差展开式。给出了证明的一些关键思想。}
{"title":"Asymptotic Properties of a Supercritical Branching Process with Immigration in a Random Environment","authors":"Yanqing Wang, Quansheng Liu","doi":"10.1515/eqc-2021-0030","DOIUrl":"https://doi.org/10.1515/eqc-2021-0030","url":null,"abstract":"Abstract This is a short survey about asymptotic properties of a supercritical branching process ( Z n ) (Z_{n}) with immigration in a stationary and ergodic or independent and identically distributed random environment. We first present basic properties of the fundamental submartingale ( W n ) (W_{n}) , about the a.s. convergence, the non-degeneracy of its limit 𝑊, the convergence in L p L^{p} for p ≥ 1 pgeq 1 , and the boundedness of the harmonic moments E W n - a mathbb{E}W_{n}^{-a} , a > 0 a>0 . We then present limit theorems and large deviation results on log Z n log Z_{n} , including the law of large numbers, large and moderate deviation principles, the central limit theorem with Berry–Esseen’s bound, and Cramér’s large deviation expansion. Some key ideas of the proofs are also presented.","PeriodicalId":37499,"journal":{"name":"Stochastics and Quality Control","volume":"35 1","pages":"145 - 155"},"PeriodicalIF":0.0,"publicationDate":"2021-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77745581","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Linear fractional Galton–Watson branching processes in i.i.d. random environment are, on the quenched level, intimately connected to random difference equations by the evolution of the random parameters of their linear fractional marginals. On the other hand, any random difference equation defines an autoregressive Markov chain (a random affine recursion) which can be positive recurrent, null recurrent and transient and which, as the forward iterations of an iterated function system, has an a.s. convergent counterpart in the positive recurrent case given by the corresponding backward iterations. The present expository article aims to provide an explicit view at how these aspects of random difference equations and their stationary limits, called perpetuities, enter into the results and the analysis, especially in quenched regime. Although most of the results presented here are known, we hope that the offered perspective will be welcomed by some readers.
{"title":"Linear Fractional Galton–Watson Processes in Random Environment and Perpetuities","authors":"G. Alsmeyer","doi":"10.1515/eqc-2021-0037","DOIUrl":"https://doi.org/10.1515/eqc-2021-0037","url":null,"abstract":"Abstract Linear fractional Galton–Watson branching processes in i.i.d. random environment are, on the quenched level, intimately connected to random difference equations by the evolution of the random parameters of their linear fractional marginals. On the other hand, any random difference equation defines an autoregressive Markov chain (a random affine recursion) which can be positive recurrent, null recurrent and transient and which, as the forward iterations of an iterated function system, has an a.s. convergent counterpart in the positive recurrent case given by the corresponding backward iterations. The present expository article aims to provide an explicit view at how these aspects of random difference equations and their stationary limits, called perpetuities, enter into the results and the analysis, especially in quenched regime. Although most of the results presented here are known, we hope that the offered perspective will be welcomed by some readers.","PeriodicalId":37499,"journal":{"name":"Stochastics and Quality Control","volume":"77 1","pages":"111 - 127"},"PeriodicalIF":0.0,"publicationDate":"2021-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90077397","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract A nonparametric sequential probability ratio test control chart to monitor the process dispersion based on the sequential sign statistic is proposed. The statistical performance of this chart is evaluated by comparing it with that of the charts for dispersion based on sign statistic in the existing literature. It is found that the proposed chart outperforms all these charts uniformly in detecting a shift of any size over a wide range. An implementation of the chart is illustrated through an example.
{"title":"The SPRT Sign Chart for Process Dispersion","authors":"Dadasaheb G. Godase, Shashibhushan B. Mahadik","doi":"10.1515/eqc-2021-0026","DOIUrl":"https://doi.org/10.1515/eqc-2021-0026","url":null,"abstract":"Abstract A nonparametric sequential probability ratio test control chart to monitor the process dispersion based on the sequential sign statistic is proposed. The statistical performance of this chart is evaluated by comparing it with that of the charts for dispersion based on sign statistic in the existing literature. It is found that the proposed chart outperforms all these charts uniformly in detecting a shift of any size over a wide range. An implementation of the chart is illustrated through an example.","PeriodicalId":37499,"journal":{"name":"Stochastics and Quality Control","volume":"47 1","pages":"101 - 106"},"PeriodicalIF":0.0,"publicationDate":"2021-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85191078","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract A properly scaled critical Galton–Watson process converges to a continuous state critical branching process ξ ( ⋅ ) xi(,{cdot},) as the number of initial individuals tends to infinity. We extend this classical result by allowing for overlapping generations and considering a wide class of population counts. The main result of the paper establishes a convergence of the finite-dimensional distributions for a scaled vector of multiple population counts. The set of the limiting distributions is conveniently represented in terms of integrals ( ∫ 0 y ξ ( y - u ) d u γ int_{0}^{y}xi(y-u),du^{gamma} , y ≥ 0 ygeq 0 ) with a pertinent γ ≥ 0 gammageq 0 .
当初始个体数量趋于无穷时，适当缩放的临界高尔顿-沃森过程收敛为连续状态临界分支过程ξ∞(⋅)xi (， {cdot} ，)。我们扩展了这个经典的结果，允许重叠代和考虑一个广泛的人口计数类。本文的主要结果建立了多个种群数量的缩放向量的有限维分布的收敛性。极限分布的集合可以方便地用积分(∫0 y ξ∞(y-u)∑u γ int ＿0{^}y {}xi (y-u)，du^ {gamma}, y≥0 y geq 0)表示，其中相关的γ≥0 gammageq 0。
{"title":"Critical Galton–Watson Processes with Overlapping Generations","authors":"S. Sagitov","doi":"10.1515/eqc-2021-0027","DOIUrl":"https://doi.org/10.1515/eqc-2021-0027","url":null,"abstract":"Abstract A properly scaled critical Galton–Watson process converges to a continuous state critical branching process ξ ( ⋅ ) xi(,{cdot},) as the number of initial individuals tends to infinity. We extend this classical result by allowing for overlapping generations and considering a wide class of population counts. The main result of the paper establishes a convergence of the finite-dimensional distributions for a scaled vector of multiple population counts. The set of the limiting distributions is conveniently represented in terms of integrals ( ∫ 0 y ξ ( y - u ) d u γ int_{0}^{y}xi(y-u),du^{gamma} , y ≥ 0 ygeq 0 ) with a pertinent γ ≥ 0 gammageq 0 .","PeriodicalId":37499,"journal":{"name":"Stochastics and Quality Control","volume":"30 1","pages":"87 - 110"},"PeriodicalIF":0.0,"publicationDate":"2021-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74537142","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Recently, entropy and extropy-based tests for the uniform distribution have attracted the attention of some researchers. This paper proposes nonparametric entropy and extropy estimators based on progressive type-II censoring and investigates their properties and behavior. Performance of the proposed estimators is studied via simulations. Entropy and extropy-based goodness-of-fit tests for uniformity are developed by the well performed estimators. The powers of the proposed uniformity tests are compared also via simulations assuming various alternatives and censoring schemes.
{"title":"Kernel and CDF-Based Estimation of Extropy and Entropy from Progressively Type-II Censoring with Application for Goodness of Fit Problems","authors":"Raja Hazeb, H. A. Bayoud, M. Z. Raqab","doi":"10.1515/eqc-2020-0035","DOIUrl":"https://doi.org/10.1515/eqc-2020-0035","url":null,"abstract":"Abstract Recently, entropy and extropy-based tests for the uniform distribution have attracted the attention of some researchers. This paper proposes nonparametric entropy and extropy estimators based on progressive type-II censoring and investigates their properties and behavior. Performance of the proposed estimators is studied via simulations. Entropy and extropy-based goodness-of-fit tests for uniformity are developed by the well performed estimators. The powers of the proposed uniformity tests are compared also via simulations assuming various alternatives and censoring schemes.","PeriodicalId":37499,"journal":{"name":"Stochastics and Quality Control","volume":"61 1","pages":"73 - 83"},"PeriodicalIF":0.0,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80686114","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Extropy was introduced as a dual complement of the Shannon entropy. In this investigation, we consider failure extropy and its dynamic version. Various basic properties of these measures are presented. It is shown that the dynamic failure extropy characterizes the distribution function uniquely. We also consider weighted versions of these measures. Several virtues of the weighted measures are explored. Finally, nonparametric estimators are introduced based on the empirical distribution function.
{"title":"Failure Extropy, Dynamic Failure Extropy and Their Weighted Versions","authors":"S. Kayal","doi":"10.1515/eqc-2021-0008","DOIUrl":"https://doi.org/10.1515/eqc-2021-0008","url":null,"abstract":"Abstract Extropy was introduced as a dual complement of the Shannon entropy. In this investigation, we consider failure extropy and its dynamic version. Various basic properties of these measures are presented. It is shown that the dynamic failure extropy characterizes the distribution function uniquely. We also consider weighted versions of these measures. Several virtues of the weighted measures are explored. Finally, nonparametric estimators are introduced based on the empirical distribution function.","PeriodicalId":37499,"journal":{"name":"Stochastics and Quality Control","volume":"1 1","pages":"59 - 71"},"PeriodicalIF":0.0,"publicationDate":"2021-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89577926","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The present work aims to propose an approximation of the sample median distribution with a normal parent distribution. Although the mean is usually used as the central tendency measure for normal samples, the median has also been used in engineering, process control in particular. The proposed method approximates the normal sample median distribution only using the normal distribution function. It outperforms Castagliola’s method for small samples and serves as an alternative approximation for trading off accuracy against computational complexity for large samples.
{"title":"Approximating the Normal Sample Median Distribution in Process Control","authors":"Char Leung","doi":"10.1515/eqc-2021-0003","DOIUrl":"https://doi.org/10.1515/eqc-2021-0003","url":null,"abstract":"Abstract The present work aims to propose an approximation of the sample median distribution with a normal parent distribution. Although the mean is usually used as the central tendency measure for normal samples, the median has also been used in engineering, process control in particular. The proposed method approximates the normal sample median distribution only using the normal distribution function. It outperforms Castagliola’s method for small samples and serves as an alternative approximation for trading off accuracy against computational complexity for large samples.","PeriodicalId":37499,"journal":{"name":"Stochastics and Quality Control","volume":"10 1","pages":"21 - 25"},"PeriodicalIF":0.0,"publicationDate":"2021-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87783724","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Dynamic cumulative residual entropy is a new addition to the class of information measures. In the present paper, we study its relationship with excess wealth transform and derive some identities connecting the two using the quantile-based approach. Some theoretical results that have applications to infer ageing properties and risk measures are presented. These are used as tools to analyse real life data.
{"title":"Relation Between Cumulative Residual Entropy and Excess Wealth Transform with Applications to Reliability and Risk","authors":"N. Unnikrishnan Nair, B. Vineshkumar","doi":"10.1515/eqc-2020-0036","DOIUrl":"https://doi.org/10.1515/eqc-2020-0036","url":null,"abstract":"Abstract Dynamic cumulative residual entropy is a new addition to the class of information measures. In the present paper, we study its relationship with excess wealth transform and derive some identities connecting the two using the quantile-based approach. Some theoretical results that have applications to infer ageing properties and risk measures are presented. These are used as tools to analyse real life data.","PeriodicalId":37499,"journal":{"name":"Stochastics and Quality Control","volume":"8 1","pages":"43 - 57"},"PeriodicalIF":0.0,"publicationDate":"2021-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86682594","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper we advance a nonlinear optimization problem for hedging wind power variability by using a dispatchable energy source (DES) like gas. The model considers several important aspects such as modeling of wind power production, electricity price, nonlinear penalization scheme for energy underproduction and interrelations among the considered variables. Results are given in terms of optimal co-generation policy with DES. The optimal policy is interpreted and analyzed in different penalization scenarios and related to a 48 MW hypothetical wind park. The model is suitable for integration of wind energy especially for isolated grids. Some probabilistic results for special moments of a Log-Normal distribution are obtained; they are necessary for the evolution of the optimal policy.
{"title":"Hedging the Risk of Wind Power Production Using Dispatchable Energy Source","authors":"G. D’Amico, Bice Di Basilio, F. Petroni","doi":"10.1515/EQC-2020-0033","DOIUrl":"https://doi.org/10.1515/EQC-2020-0033","url":null,"abstract":"Abstract In this paper we advance a nonlinear optimization problem for hedging wind power variability by using a dispatchable energy source (DES) like gas. The model considers several important aspects such as modeling of wind power production, electricity price, nonlinear penalization scheme for energy underproduction and interrelations among the considered variables. Results are given in terms of optimal co-generation policy with DES. The optimal policy is interpreted and analyzed in different penalization scenarios and related to a 48 MW hypothetical wind park. The model is suitable for integration of wind energy especially for isolated grids. Some probabilistic results for special moments of a Log-Normal distribution are obtained; they are necessary for the evolution of the optimal policy.","PeriodicalId":37499,"journal":{"name":"Stochastics and Quality Control","volume":"59 1","pages":"1 - 20"},"PeriodicalIF":0.0,"publicationDate":"2021-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75147395","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, we study one-dimensional backward stochastic differential equations under logarithmic growth in the 𝑧-variable (|z||ln|z||)(lvert zrvertsqrt{lvertlnlvert zrvertrvert}). We show the existence and the uniqueness of the solution when the noise is driven by a Brownian motion and an independent Poisson random measure. In addition, we highlight the connection of such BSDEs with stochastic optimal control problem, where we show the existence of an optimal strategy for the control problem.
摘要本文研究了𝑧-variable (|z| zi |ln²|z|)(lvert z rvertsqrt{lvertlnlvert zrvertrvert})中对数增长下的一维倒向随机微分方程。我们证明了当噪声由布朗运动和独立泊松随机测量驱动时解的存在性和唯一性。此外，我们强调了这种BSDEs与随机最优控制问题的联系，在那里我们证明了控制问题的最优策略的存在性。
{"title":"BSDEs with Logarithmic Growth Driven by Brownian Motion and Poisson Random Measure and Connection to Stochastic Control Problem","authors":"Khalid Oufdil","doi":"10.1515/eqc-2021-0012","DOIUrl":"https://doi.org/10.1515/eqc-2021-0012","url":null,"abstract":"Abstract In this paper, we study one-dimensional backward stochastic differential equations under logarithmic growth in the 𝑧-variable (|z||ln|z||)(lvert zrvertsqrt{lvertlnlvert zrvertrvert}). We show the existence and the uniqueness of the solution when the noise is driven by a Brownian motion and an independent Poisson random measure. In addition, we highlight the connection of such BSDEs with stochastic optimal control problem, where we show the existence of an optimal strategy for the control problem.","PeriodicalId":37499,"journal":{"name":"Stochastics and Quality Control","volume":"1 1","pages":"27 - 42"},"PeriodicalIF":0.0,"publicationDate":"2020-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88758091","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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