H. Saulo, Jeremias Leão, V. Leiva, R. Vila, V. Tomazella
The Birnbaum–Saunders distribution has been widely used to model reliability and fatigue data. In this paper, we propose a regression of generalized linear models type based on a new bivariate Birnbaum–Saunders distribution. This is parameterized in terms of its means and allows data to be described in their original scale. We estimate the model parameters and carry out inference with the maximum likelihood method. A case study with real-world reliability data is conducted for motivating our investigation, illustrating the potential applications of the proposed results. We obtain a predictive model which can be a useful addition to the tool-kit of diverse practitioners, reliability engineers, applied statisticians, and data scientists.
{"title":"A bivariate fatigue-life regression model and its application to fracture of metallic tools","authors":"H. Saulo, Jeremias Leão, V. Leiva, R. Vila, V. Tomazella","doi":"10.1214/20-bjps490","DOIUrl":"https://doi.org/10.1214/20-bjps490","url":null,"abstract":"The Birnbaum–Saunders distribution has been widely used to model reliability and fatigue data. In this paper, we propose a regression of generalized linear models type based on a new bivariate Birnbaum–Saunders distribution. This is parameterized in terms of its means and allows data to be described in their original scale. We estimate the model parameters and carry out inference with the maximum likelihood method. A case study with real-world reliability data is conducted for motivating our investigation, illustrating the potential applications of the proposed results. We obtain a predictive model which can be a useful addition to the tool-kit of diverse practitioners, reliability engineers, applied statisticians, and data scientists.","PeriodicalId":51242,"journal":{"name":"Brazilian Journal of Probability and Statistics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48604845","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}
For modelling unbounded count data, Poisson distribution is a natural choice. However, count data arising in various fields of scientific research are often under-reported. In such situations, inference carried out on the basis of Poisson model will result in biased parameter estimates and suboptimal tests. A modified Poisson model is developed to accommodate the possible undercount. For model-identifiability a double sampling scheme of data collection has been adopted. The focus of this paper is to develop asymptotically optimal tests for the Poisson mean in presence of undercount. Simulation study is conducted to compare the performance of the tests with respect to level and power and also to investigate the impact of ignoring undercount on each of the tests. The findings are validated using real life data.
{"title":"Testing of Poisson mean with under-reported counts","authors":"Debjit Sengupta, Surupa Roy, Tathagata Banerjee","doi":"10.1214/20-bjps493","DOIUrl":"https://doi.org/10.1214/20-bjps493","url":null,"abstract":"For modelling unbounded count data, Poisson distribution is a natural choice. However, count data arising in various fields of scientific research are often under-reported. In such situations, inference carried out on the basis of Poisson model will result in biased parameter estimates and suboptimal tests. A modified Poisson model is developed to accommodate the possible undercount. For model-identifiability a double sampling scheme of data collection has been adopted. The focus of this paper is to develop asymptotically optimal tests for the Poisson mean in presence of undercount. Simulation study is conducted to compare the performance of the tests with respect to level and power and also to investigate the impact of ignoring undercount on each of the tests. The findings are validated using real life data.","PeriodicalId":51242,"journal":{"name":"Brazilian Journal of Probability and Statistics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66081375","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}
A partition of a finite set is a collection of disjoint and exhaustive subsets of that set. This paper concerns the distributions occasioned by partitioning a multivariate distribution, in particular to find the resultant distribution within each partition subset, and the distribution between them. If the multivariate distribution has independent univariate components, then partitioning leads to independent univariate components within each partition subset. Consequently, our attention is drawn to distributions without independent univariate components. The multivariate normal distribution is often the first such distribution that one thinks of. When a partition consists of only two subsets, the normal distribution can be partitioned into a conditional of one subset conditioned on the other, and a marginal distribution on the second subset. And each of these can be recursively partitioned. The result, however, is a product of normal densities that apparently depends on the order in which the partitioning is done, although it cannot in fact so depend, as each such product re-expresses the original normal multivariate distribution. Among partitions, there are two extreme cases. The first is the coarsest possible partition, consisting of the whole space. Then trivially the distribution over this partition is the original multivariate distribution, whatever it was. The other extreme case is a finest possible partition, each subset of which has only a single component. Then the distribution on such a subset takes the value of the random variable with probability one. So the partitions at issue in this paper are the partitions in-between, neither the coarsest nor the finest.
{"title":"Partitioning some multivariate distributions","authors":"J. Kadane","doi":"10.1214/21-bjps501","DOIUrl":"https://doi.org/10.1214/21-bjps501","url":null,"abstract":"A partition of a finite set is a collection of disjoint and exhaustive subsets of that set. This paper concerns the distributions occasioned by partitioning a multivariate distribution, in particular to find the resultant distribution within each partition subset, and the distribution between them. If the multivariate distribution has independent univariate components, then partitioning leads to independent univariate components within each partition subset. Consequently, our attention is drawn to distributions without independent univariate components. The multivariate normal distribution is often the first such distribution that one thinks of. When a partition consists of only two subsets, the normal distribution can be partitioned into a conditional of one subset conditioned on the other, and a marginal distribution on the second subset. And each of these can be recursively partitioned. The result, however, is a product of normal densities that apparently depends on the order in which the partitioning is done, although it cannot in fact so depend, as each such product re-expresses the original normal multivariate distribution. Among partitions, there are two extreme cases. The first is the coarsest possible partition, consisting of the whole space. Then trivially the distribution over this partition is the original multivariate distribution, whatever it was. The other extreme case is a finest possible partition, each subset of which has only a single component. Then the distribution on such a subset takes the value of the random variable with probability one. So the partitions at issue in this paper are the partitions in-between, neither the coarsest nor the finest.","PeriodicalId":51242,"journal":{"name":"Brazilian Journal of Probability and Statistics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66085825","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}
In this paper, we consider a model for data analysis with measurement errors. The main objective of this work is to develop statistical inference tools, such as parameter estimation and hypothesis tests in a linear functional relationship with replicated observations. For this purpose, we use the maximum likelihood method in the presence of incidental parameters, and the unbiased estimating equations approach. Both approaches lead to explicit expressions for the asymptotic covariance matrices of the estimators of the model parameters. A simulation study is performed to assess the empirical behavior of estimators and of a Wald statistic. The methodology is illustrated with a real data set.
{"title":"Inference in a linear functional relationship with replications","authors":"Julio Hokama, P. Morettin, H. Bolfarine, M. Galea","doi":"10.1214/21-bjps498","DOIUrl":"https://doi.org/10.1214/21-bjps498","url":null,"abstract":"In this paper, we consider a model for data analysis with measurement errors. The main objective of this work is to develop statistical inference tools, such as parameter estimation and hypothesis tests in a linear functional relationship with replicated observations. For this purpose, we use the maximum likelihood method in the presence of incidental parameters, and the unbiased estimating equations approach. Both approaches lead to explicit expressions for the asymptotic covariance matrices of the estimators of the model parameters. A simulation study is performed to assess the empirical behavior of estimators and of a Wald statistic. The methodology is illustrated with a real data set.","PeriodicalId":51242,"journal":{"name":"Brazilian Journal of Probability and Statistics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66085761","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}
We compute higher derivatives of the Fréchet function on spheres with an absolutely continuous and rotationally symmetric probability distribution. Consequences include (i) a practical condition to test if the mode of the symmetric distribution is a local Fréchet mean; (ii) a Central Limit Theorem on spheres with practical assumptions and an explicit limiting distribution; and (iii) an answer to the question of whether the smeary effect can occur on spheres with absolutely continuous and rotationally symmetric distributions: with the method presented here, it can in dimension at least 4.
{"title":"Behavior of the Fréchet mean and Central Limit Theorems on spheres","authors":"Do Tran","doi":"10.1214/21-bjps499","DOIUrl":"https://doi.org/10.1214/21-bjps499","url":null,"abstract":"We compute higher derivatives of the Fréchet function on spheres with an absolutely continuous and rotationally symmetric probability distribution. Consequences include (i) a practical condition to test if the mode of the symmetric distribution is a local Fréchet mean; (ii) a Central Limit Theorem on spheres with practical assumptions and an explicit limiting distribution; and (iii) an answer to the question of whether the smeary effect can occur on spheres with absolutely continuous and rotationally symmetric distributions: with the method presented here, it can in dimension at least 4.","PeriodicalId":51242,"journal":{"name":"Brazilian Journal of Probability and Statistics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66085811","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}
The present article represents a review of the geometric generated family of distributions. Based on this family of distribution, several distributions are proposed. The family can be proposed by using the compounding concept of zero truncated geometric distribution with any other model or family of distributions. Here, we provide a complete survey on this family of distributions and also listed the contributory related research work, their sub-models, hazard rates, and utilized real datasets. We also address 10 power series distributions, 60 distributions based on the geometric family of distribution. These numbers show the importance of the geometric family of distribution.
{"title":"Geometric generated family of distributions: A review","authors":"Teena Goyal, S. Maurya, S. Nadarajah","doi":"10.1214/20-bjps485","DOIUrl":"https://doi.org/10.1214/20-bjps485","url":null,"abstract":"The present article represents a review of the geometric generated family of distributions. Based on this family of distribution, several distributions are proposed. The family can be proposed by using the compounding concept of zero truncated geometric distribution with any other model or family of distributions. Here, we provide a complete survey on this family of distributions and also listed the contributory related research work, their sub-models, hazard rates, and utilized real datasets. We also address 10 power series distributions, 60 distributions based on the geometric family of distribution. These numbers show the importance of the geometric family of distribution.","PeriodicalId":51242,"journal":{"name":"Brazilian Journal of Probability and Statistics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66081249","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}
In this work we prove that for any dimension $dgeq 1$ and any $gamma in (0,1)$ super-Brownian motion corresponding to the log-Laplace equation begin{equation*} begin{split} frac{partial v(t,x)}{partial t } & = frac{1}{2}bigtriangleup v(t,x) + v^gamma (t,x) ,: (t,x) in mathbb{R}_+times mathbb{R}^d, v(0,x)&= f(x) end{split} end{equation*} is absolutely continuous with respect to the Lebesgue measure at any fixed time $t>0$. Our proof is based on properties of solutions of the LL equation. We also prove that when initial datum $v(0,cdot)$ is a finite, non-zero measure, then the LL equation has a unique, continuous solution. Moreover this solution continuously depends on initial data.
{"title":"Absolute continuity of the super-Brownian motion with infinite mean","authors":"R. Mamin, L. Mytnik","doi":"10.1214/21-bjps508","DOIUrl":"https://doi.org/10.1214/21-bjps508","url":null,"abstract":"In this work we prove that for any dimension $dgeq 1$ and any $gamma in (0,1)$ super-Brownian motion corresponding to the log-Laplace equation begin{equation*} begin{split} frac{partial v(t,x)}{partial t } & = frac{1}{2}bigtriangleup v(t,x) + v^gamma (t,x) ,: (t,x) in mathbb{R}_+times mathbb{R}^d, v(0,x)&= f(x) end{split} end{equation*} is absolutely continuous with respect to the Lebesgue measure at any fixed time $t>0$. Our proof is based on properties of solutions of the LL equation. \u0000We also prove that when initial datum $v(0,cdot)$ is a finite, non-zero measure, then the LL equation has a unique, continuous solution. Moreover this solution continuously depends on initial data.","PeriodicalId":51242,"journal":{"name":"Brazilian Journal of Probability and Statistics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2020-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44355568","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}
In this paper, we consider the problem of simultaneously estimating parameters of independent Poisson distributions in the presence of possibly unbalanced sample sizes under weighted standardized squared error loss. A class of heterogeneous Bayesian shrinkage estimators that utilize the unbalanced nature of sample sizes is proposed. To provide a theoretical justification, we first derive a necessary and sufficient condition for an estimator in the class to be proper Bayes and hence admissible and then obtain sufficient conditions for minimaxity that are compatible with the admissibility condition. Heterogeneous and homogeneous shrinkage estimators are compared by simulation. Several estimation methods are applied to data relating to the standardized mortality ratio.
{"title":"Proper Bayes minimax estimation of parameters of Poisson distributions in the presence of unbalanced sample sizes","authors":"Y. Hamura, T. Kubokawa","doi":"10.1214/19-BJPS459","DOIUrl":"https://doi.org/10.1214/19-BJPS459","url":null,"abstract":"In this paper, we consider the problem of simultaneously estimating parameters of independent Poisson distributions in the presence of possibly unbalanced sample sizes under weighted standardized squared error loss. A class of heterogeneous Bayesian shrinkage estimators that utilize the unbalanced nature of sample sizes is proposed. To provide a theoretical justification, we first derive a necessary and sufficient condition for an estimator in the class to be proper Bayes and hence admissible and then obtain sufficient conditions for minimaxity that are compatible with the admissibility condition. Heterogeneous and homogeneous shrinkage estimators are compared by simulation. Several estimation methods are applied to data relating to the standardized mortality ratio.","PeriodicalId":51242,"journal":{"name":"Brazilian Journal of Probability and Statistics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48926993","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}
Abstract. A test of fit for the exponential distribution is presented, which is based on transformed observations and a new estimator for the scale parameter. The asymptotic null distribution of the test statistic is obtained and the consistency of the test is discussed. Monte Carlo simulation results on a power comparison study show that the proposed test is competitive under the considered families of alternatives and sample sizes.
{"title":"On testing exponentiality based on a new estimator for the scale parameter","authors":"J. Villaseñor, E. González-Estrada","doi":"10.1214/19-BJPS461","DOIUrl":"https://doi.org/10.1214/19-BJPS461","url":null,"abstract":"Abstract. A test of fit for the exponential distribution is presented, which is based on transformed observations and a new estimator for the scale parameter. The asymptotic null distribution of the test statistic is obtained and the consistency of the test is discussed. Monte Carlo simulation results on a power comparison study show that the proposed test is competitive under the considered families of alternatives and sample sizes.","PeriodicalId":51242,"journal":{"name":"Brazilian Journal of Probability and Statistics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41442785","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}
Following the methodology developed by Hyman and Shashkov (1997), we define a discrete version of gradient vector and associated line integral along arbitrary path connecting two nodes of uniform grid. An exponential representation of joint survival function of bivariate discrete non-negative integer-valued random variables in terms of discrete line integral is established. We apply it to generate a discrete analogue of the Sibuya-type aging property, incorporating many classical and new bivariate discrete models. Several characterizations and closure properties of this class of bivariate discrete distributions are presented.
{"title":"Discrete line integral on uniform grids: Probabilistic interpretation and applications","authors":"N. Kolev","doi":"10.1214/19-BJPS454","DOIUrl":"https://doi.org/10.1214/19-BJPS454","url":null,"abstract":"Following the methodology developed by Hyman and Shashkov (1997), we define a discrete version of gradient vector and associated line integral along arbitrary path connecting two nodes of uniform grid. An exponential representation of joint survival function of bivariate discrete non-negative integer-valued random variables in terms of discrete line integral is established. We apply it to generate a discrete analogue of the Sibuya-type aging property, incorporating many classical and new bivariate discrete models. Several characterizations and closure properties of this class of bivariate discrete distributions are presented.","PeriodicalId":51242,"journal":{"name":"Brazilian Journal of Probability and Statistics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46643372","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：