Abstract The estimation of stress-strength reliability in a time-dependent context deals with either the stress or strength or both dynamic. The repeated occurrence of stress in random intervals of time induces a change in the distribution of strength over time. In this paper, we study the stress-strength reliability of a system whose strength reduces by a constant over each run and the stress is considered as either fixed over time or as increasing by a constant over each run. The number of runs in any interval of time is assumed to be random. The stress-strength reliability of the system is obtained, assuming continuous phase-type distribution for the duration of time taken for completion of each run in any interval of time and Weibull or gamma distribution for initial stress and strength. We obtain matrix-based expressions for the stress-strength reliability and numerical illustrations are also discussed.
{"title":"Time-Dependent Stress-Strength Reliability Models with Phase-Type Cycle Times","authors":"M. Drisya, Joby K. Jose","doi":"10.1515/eqc-2020-0023","DOIUrl":"https://doi.org/10.1515/eqc-2020-0023","url":null,"abstract":"Abstract The estimation of stress-strength reliability in a time-dependent context deals with either the stress or strength or both dynamic. The repeated occurrence of stress in random intervals of time induces a change in the distribution of strength over time. In this paper, we study the stress-strength reliability of a system whose strength reduces by a constant over each run and the stress is considered as either fixed over time or as increasing by a constant over each run. The number of runs in any interval of time is assumed to be random. The stress-strength reliability of the system is obtained, assuming continuous phase-type distribution for the duration of time taken for completion of each run in any interval of time and Weibull or gamma distribution for initial stress and strength. We obtain matrix-based expressions for the stress-strength reliability and numerical illustrations are also discussed.","PeriodicalId":37499,"journal":{"name":"Stochastics and Quality Control","volume":"47 1","pages":"97 - 112"},"PeriodicalIF":0.0,"publicationDate":"2020-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91349843","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 This study considers a stochastic control model in which an economic unit has productive capital and liabilities in the form of debt. The worth of capital changes over time through investment and random Brownian fluctuations in the unit price of capital. Income from production is also subject to the random Brownian fluctuations. The existence of the solutions to the associated Hamilton Jacobi Bellman equation for this model is established and the optimal policies are characterized. The optimal advertising rate as a function of the market share, the optimal consumption rate and the fraction of the wealth invested in stock at any time are obtained. The worth of the capital and the optimal consumption policy are derived for the stochastic optimal investment consumption model associated with the Hamilton Jacobi Bellman equation. Analysis and numerical simulations are then presented.
{"title":"A Stochastic Control Model of Investment and Consumption with Applications to Financial Economics","authors":"Md. Azizul Baten, Ruzelan Khalid","doi":"10.1515/eqc-2020-0017","DOIUrl":"https://doi.org/10.1515/eqc-2020-0017","url":null,"abstract":"Abstract This study considers a stochastic control model in which an economic unit has productive capital and liabilities in the form of debt. The worth of capital changes over time through investment and random Brownian fluctuations in the unit price of capital. Income from production is also subject to the random Brownian fluctuations. The existence of the solutions to the associated Hamilton Jacobi Bellman equation for this model is established and the optimal policies are characterized. The optimal advertising rate as a function of the market share, the optimal consumption rate and the fraction of the wealth invested in stock at any time are obtained. The worth of the capital and the optimal consumption policy are derived for the stochastic optimal investment consumption model associated with the Hamilton Jacobi Bellman equation. Analysis and numerical simulations are then presented.","PeriodicalId":37499,"journal":{"name":"Stochastics and Quality Control","volume":"50 1","pages":"43 - 55"},"PeriodicalIF":0.0,"publicationDate":"2020-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78136392","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, a new probability density function with bounded domain is presented. The new distribution arises from the Lindley distribution proposed in 1958. It presents the advantage of not including any special function in its formulation. The new transformed model, called the reflected-shifted-truncated Lindley distribution can be used to model left-skewed data. We provide a comprehensive treatment of general mathematical and statistical properties of this distribution. We estimate the model parameters by maximum likelihood methods based on complete and right-censored data. To assess the performance and consistency of the maximum likelihood estimators, we conduct a simulation study with varying sample sizes. Finally, we use the distribution to model left-skewed survival and failure data from two real data sets. For the real data sets containing complete data and right-censored data, this distribution is superior in its ability to sufficiently model the data as compared to the power Lindley, exponentiated power Lindley, generalized inverse Lindley, generalized weighted Lindley and the well-known Gompertz distributions.
{"title":"The Reflected-Shifted-Truncated Lindley Distribution with Applications","authors":"S. Dey, Sophia D. Waymyers, D. Kumar","doi":"10.1515/eqc-2020-0008","DOIUrl":"https://doi.org/10.1515/eqc-2020-0008","url":null,"abstract":"Abstract In this paper, a new probability density function with bounded domain is presented. The new distribution arises from the Lindley distribution proposed in 1958. It presents the advantage of not including any special function in its formulation. The new transformed model, called the reflected-shifted-truncated Lindley distribution can be used to model left-skewed data. We provide a comprehensive treatment of general mathematical and statistical properties of this distribution. We estimate the model parameters by maximum likelihood methods based on complete and right-censored data. To assess the performance and consistency of the maximum likelihood estimators, we conduct a simulation study with varying sample sizes. Finally, we use the distribution to model left-skewed survival and failure data from two real data sets. For the real data sets containing complete data and right-censored data, this distribution is superior in its ability to sufficiently model the data as compared to the power Lindley, exponentiated power Lindley, generalized inverse Lindley, generalized weighted Lindley and the well-known Gompertz distributions.","PeriodicalId":37499,"journal":{"name":"Stochastics and Quality Control","volume":"28 1","pages":"67 - 77"},"PeriodicalIF":0.0,"publicationDate":"2020-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88269967","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 consider the Bayesian inference of M/M/𝑅 queue with 𝑅 heterogeneous servers with service rates μ 1 , μ 2 , … , μ R mu_{1},mu_{2},ldots,mu_{R} , where μ 1 > μ 2 > ⋯ > μ R mu_{1}>mu_{2}>cdots>mu_{R} . Assuming multivariate gamma prior distribution for service rates and gamma prior distribution for arrival rate 𝜆, we derive the conditional posterior densities of mean arrival rate and mean service rates. We apply the Markov chain Monte Carlo method and compute the Bayes estimates and credible interval for the M/M/3 queue, as a particular case of the M/M/𝑅 queue under squared error loss function, entropy loss function and linex loss function corresponding to a different set of hyperparameters.
在本文中，我们考虑了具有𝑅异构服务器的M/M/𝑅队列的贝叶斯推理，其服务速率为μ 1， μ 2，…，μ R mu_{1}，mu_{2}，ldots，mu_{R}，其中μ 1 > μ R mu_{1}>mu_{2}>cdots>mu_{R}。假设服务率的多元先验分布和到达率的多元先验分布，我们推导了平均到达率和平均服务率的条件后验密度。本文应用马尔可夫链蒙特卡罗方法，以M/M/𝑅队列为例，计算了M/M/3队列在不同超参数集对应的平方误差损失函数、熵损失函数和线性损失函数下的Bayes估计和可信区间。
{"title":"Bayesian Estimation of an M/M/𝑅 Queue with Heterogeneous Servers Using Markov Chain Monte Carlo Method","authors":"V. Deepthi, Joby K. Jose","doi":"10.1515/eqc-2020-0010","DOIUrl":"https://doi.org/10.1515/eqc-2020-0010","url":null,"abstract":"Abstract In this paper, we consider the Bayesian inference of M/M/𝑅 queue with 𝑅 heterogeneous servers with service rates μ 1 , μ 2 , … , μ R mu_{1},mu_{2},ldots,mu_{R} , where μ 1 > μ 2 > ⋯ > μ R mu_{1}>mu_{2}>cdots>mu_{R} . Assuming multivariate gamma prior distribution for service rates and gamma prior distribution for arrival rate 𝜆, we derive the conditional posterior densities of mean arrival rate and mean service rates. We apply the Markov chain Monte Carlo method and compute the Bayes estimates and credible interval for the M/M/3 queue, as a particular case of the M/M/𝑅 queue under squared error loss function, entropy loss function and linex loss function corresponding to a different set of hyperparameters.","PeriodicalId":37499,"journal":{"name":"Stochastics and Quality Control","volume":"24 1","pages":"57 - 66"},"PeriodicalIF":0.0,"publicationDate":"2020-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85105127","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 Multivariate regression models and ANOVA are probably the most frequently applied methods of all statistical analyses. We study the case where the predictors are qualitative variables and the response variable is quantitative. In this case, we propose an alternative to the classic approaches that does not assume homoscedasticity but assumes that a Markov chain can describe the covariates’ correlations. This approach transforms the dependent covariates using a change of measure to independent covariates. The transformed estimates allow a pairwise comparison of the mean and variance of the contribution of different values of the covariates. We show that, under standard moment conditions, the estimators are asymptotically normally distributed. We test our method with data from simulations and apply it to several classic data sets.
{"title":"Estimations of Means and Variances in a Markov Linear Model","authors":"Abraham Gutierrez, S. Müller","doi":"10.1515/eqc-2022-0004","DOIUrl":"https://doi.org/10.1515/eqc-2022-0004","url":null,"abstract":"Abstract Multivariate regression models and ANOVA are probably the most frequently applied methods of all statistical analyses. We study the case where the predictors are qualitative variables and the response variable is quantitative. In this case, we propose an alternative to the classic approaches that does not assume homoscedasticity but assumes that a Markov chain can describe the covariates’ correlations. This approach transforms the dependent covariates using a change of measure to independent covariates. The transformed estimates allow a pairwise comparison of the mean and variance of the contribution of different values of the covariates. We show that, under standard moment conditions, the estimators are asymptotically normally distributed. We test our method with data from simulations and apply it to several classic data sets.","PeriodicalId":37499,"journal":{"name":"Stochastics and Quality Control","volume":"20 3 1","pages":"21 - 43"},"PeriodicalIF":0.0,"publicationDate":"2020-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79695449","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}
A. Afify, H. Yousof, M. Alizadeh, I. Ghosh, Samik Ray, G. Ozel
Abstract We introduce a new family of univariate continuous distributions called the Marshall–Olkin transmuted-G family which extends the transmuted-G family pioneered by Shaw and Buckley (2007). Special models for the new family are provided. Some of its mathematical properties including quantile measure, explicit expressions for the ordinary and incomplete moments, generating function, Rényi and Shannon entropies, order statistics and probability weighted moments are derived. The estimation of the model parameters is performed by maximum likelihood. The flexibility of the proposed family is illustrated by means of two applications to real data sets.
{"title":"The Marshall–Olkin Transmuted-G Family of Distributions","authors":"A. Afify, H. Yousof, M. Alizadeh, I. Ghosh, Samik Ray, G. Ozel","doi":"10.1515/eqc-2020-0009","DOIUrl":"https://doi.org/10.1515/eqc-2020-0009","url":null,"abstract":"Abstract We introduce a new family of univariate continuous distributions called the Marshall–Olkin transmuted-G family which extends the transmuted-G family pioneered by Shaw and Buckley (2007). Special models for the new family are provided. Some of its mathematical properties including quantile measure, explicit expressions for the ordinary and incomplete moments, generating function, Rényi and Shannon entropies, order statistics and probability weighted moments are derived. The estimation of the model parameters is performed by maximum likelihood. The flexibility of the proposed family is illustrated by means of two applications to real data sets.","PeriodicalId":37499,"journal":{"name":"Stochastics and Quality Control","volume":"19 1","pages":"79 - 96"},"PeriodicalIF":0.0,"publicationDate":"2020-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85320988","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 We extend the concept of maximum coefficient of determination Max R 2 operatorname{Max}R^{2} caused by repeat runs to ideas about a maximum test statistic F 0 F_{0} and a minimum p-value Min P for regression
{"title":"A Note on Pure Error and Its Effect on Regression Model Significance","authors":"J. Chimka, Salma Boudhoum, Katelyn Burrows","doi":"10.1515/eqc-2020-0004","DOIUrl":"https://doi.org/10.1515/eqc-2020-0004","url":null,"abstract":"Abstract We extend the concept of maximum coefficient of determination Max R 2 operatorname{Max}R^{2} caused by repeat runs to ideas about a maximum test statistic F 0 F_{0} and a minimum p-value Min P for regression","PeriodicalId":37499,"journal":{"name":"Stochastics and Quality Control","volume":"8 1","pages":"39 - 42"},"PeriodicalIF":0.0,"publicationDate":"2020-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82222725","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 extreme value theory (EVT) is used to assess the risk of extreme events caused by natural calamities or untoward circumstances in the social and economic sectors. The theory can be used to study the frequency of rare events and to build up a predictive model so that one can attempt to forecast the frequency of such future extreme events such as a financial collapse and the amount of damage from such a collapse. Even though many statistical techniques have been used to analyze the manner in which the Federal Reserve determines the level of the Federal Fund Rates, no known study has used EVT to analyze and predict the extreme fund rates. In this study, the US Federal Funds Rate, one of the most publicized and important economic indicators in the financial world, from 1954–2019 has been analyzed. The contributions of this study are: (1) to provide an appropriate model for the normalized Federal Funds Rate data; (2) to compare several estimation techniques in estimating parameters for two possible models; (3) to predict the maximum economic return rate from a Federal Funds Rate in the future by using the concept of the return period; and (4) to investigate the bias of estimated parameters applying a simulation study. Simulated data and real financial data are used for the study, and the outcome satisfies the efficiency of its application.
{"title":"Predicting Federal Funds Rate Using Extreme Value Theory","authors":"Ashim Kumar Dey, K. Das","doi":"10.1515/eqc-2020-0003","DOIUrl":"https://doi.org/10.1515/eqc-2020-0003","url":null,"abstract":"Abstract The extreme value theory (EVT) is used to assess the risk of extreme events caused by natural calamities or untoward circumstances in the social and economic sectors. The theory can be used to study the frequency of rare events and to build up a predictive model so that one can attempt to forecast the frequency of such future extreme events such as a financial collapse and the amount of damage from such a collapse. Even though many statistical techniques have been used to analyze the manner in which the Federal Reserve determines the level of the Federal Fund Rates, no known study has used EVT to analyze and predict the extreme fund rates. In this study, the US Federal Funds Rate, one of the most publicized and important economic indicators in the financial world, from 1954–2019 has been analyzed. The contributions of this study are: (1) to provide an appropriate model for the normalized Federal Funds Rate data; (2) to compare several estimation techniques in estimating parameters for two possible models; (3) to predict the maximum economic return rate from a Federal Funds Rate in the future by using the concept of the return period; and (4) to investigate the bias of estimated parameters applying a simulation study. Simulated data and real financial data are used for the study, and the outcome satisfies the efficiency of its application.","PeriodicalId":37499,"journal":{"name":"Stochastics and Quality Control","volume":"62 1","pages":"1 - 15"},"PeriodicalIF":0.0,"publicationDate":"2020-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72816976","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 major problem in analyzing control charts is to work with autocorrelated data. This problem can be solved by fitting a suitable model to the data and using the control chart for the residuals. The problem becomes very important, when the distribution of observation is nonnormal, in addition to being autocorrelated. Much recent research has focused on the development of appropriate statistical process control techniques for the autocorrelated data or nonnormal distribution, but few studies have considered monitoring the process mean of both nonnormal and autocorrelated data. In this paper, a simulation study is conducted to compare the performances of the control chart based on the median absolute deviation method (MAD) with those of existing control charts for the skew normal distribution. Simulation results indicate considerable improvement over existing control charts for nonnormal data can be achieved when the control charts with control limits based on the MAD method are used to monitor the process mean of nonnormal autocorrelated data.
{"title":"MAD Control Chart for Autoregressive Models with Skew-Normal Distribution","authors":"Vahideh Gorgin, B. Sadeghpour Gildeh","doi":"10.1515/eqc-2019-0006","DOIUrl":"https://doi.org/10.1515/eqc-2019-0006","url":null,"abstract":"Abstract The major problem in analyzing control charts is to work with autocorrelated data. This problem can be solved by fitting a suitable model to the data and using the control chart for the residuals. The problem becomes very important, when the distribution of observation is nonnormal, in addition to being autocorrelated. Much recent research has focused on the development of appropriate statistical process control techniques for the autocorrelated data or nonnormal distribution, but few studies have considered monitoring the process mean of both nonnormal and autocorrelated data. In this paper, a simulation study is conducted to compare the performances of the control chart based on the median absolute deviation method (MAD) with those of existing control charts for the skew normal distribution. Simulation results indicate considerable improvement over existing control charts for nonnormal data can be achieved when the control charts with control limits based on the MAD method are used to monitor the process mean of nonnormal autocorrelated data.","PeriodicalId":37499,"journal":{"name":"Stochastics and Quality Control","volume":"15 2 1","pages":"17 - 23"},"PeriodicalIF":0.0,"publicationDate":"2020-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83889247","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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