We develop here a multivariate generalization of Birnbaum-Saunders (BS) distribution based on the multivariate skew-normal distribution. Some distributional characteristics and properties are presented, as well as a simple and efficient EM algorithm for the iterative computation of the maximum likelihood (ML) estimates of model parameters, through the hierarchical representation of the proposed model. The standard errors of the maximum likelihood estimates are calculated from the observed Fisher information matrix. Moreover, by using the tools, we present a log-linear regression model, where the the ML estimates are once again obtained using an EM algorithm. Finally, simulation studies and two applications to real data sets are presented for illustrating the model and the inferential results developed here. List of Abbreviations AIC Akaike information criterion BS Birnbaum-Saunders BVBS Bivariate Birnbaum-Saunders cdf cumulative distribution function CI Confidence Interval CM Conditional Maximization ECM Expectation-Conditional Maximization EM Expectation-Maximization LR Likelihood Ratio ML Maximum Likelihood MSE root Mean Squared Error NBS Non-Central Birnbaum-Saunders pdf probability density function RB Relative Bias SD Standard Deviation SE Standard Error SIC Schwarz information criterion SMN Scale Mixtures of Normal SN Skew-Normal SNBS Skew-Normal Birnbaum-Saunders TTT Total Time on Test
{"title":"Multivariate Birnbaum–Saunders distribution based on a skewed distribution and associated EM-estimation","authors":"F. Vilca, Camila Borelli Zeller, N. Balakrishnan","doi":"10.1214/22-bjps559","DOIUrl":"https://doi.org/10.1214/22-bjps559","url":null,"abstract":"We develop here a multivariate generalization of Birnbaum-Saunders (BS) distribution based on the multivariate skew-normal distribution. Some distributional characteristics and properties are presented, as well as a simple and efficient EM algorithm for the iterative computation of the maximum likelihood (ML) estimates of model parameters, through the hierarchical representation of the proposed model. The standard errors of the maximum likelihood estimates are calculated from the observed Fisher information matrix. Moreover, by using the tools, we present a log-linear regression model, where the the ML estimates are once again obtained using an EM algorithm. Finally, simulation studies and two applications to real data sets are presented for illustrating the model and the inferential results developed here. List of Abbreviations AIC Akaike information criterion BS Birnbaum-Saunders BVBS Bivariate Birnbaum-Saunders cdf cumulative distribution function CI Confidence Interval CM Conditional Maximization ECM Expectation-Conditional Maximization EM Expectation-Maximization LR Likelihood Ratio ML Maximum Likelihood MSE root Mean Squared Error NBS Non-Central Birnbaum-Saunders pdf probability density function RB Relative Bias SD Standard Deviation SE Standard Error SIC Schwarz information criterion SMN Scale Mixtures of Normal SN Skew-Normal SNBS Skew-Normal Birnbaum-Saunders TTT Total Time on Test","PeriodicalId":51242,"journal":{"name":"Brazilian Journal of Probability and Statistics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46622314","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}
Wei-Ming Wang, Chun-Che Wen, Tsai-Hsiang Hung, Junjiang Zhong, M. Wen
. The analysis of means (ANOM) is a method that can compare the mean of each treatment to the overall mean. According to the graphical result of a statistical data analysis, we can specify which one is different from another. One of the assumptions of the classical ANOM model is that the variances are equal. However, it is not always true for the practice. To solve unknown and unequal population variances, Nelson and Dudewicz (2002) proposed a two-stage sampling procedure. However, additional samples need to be added in the second stage of the two-stage sampling procedure, so it is not practical all the time due to limited time and insufficient budget. Thus, under heteroscedasticity, we applied Chen and Lam’s (1989) single-stage sampling procedure to solve the drawback of the two-stage sampling procedure. In addition, we also provided an illustrative example and critical values for practical uses. In order to make the procedure user-friendly, we built an interface by using R Shiny.
{"title":"Single-stage sampling procedure for heteroscedasticity analysis of means","authors":"Wei-Ming Wang, Chun-Che Wen, Tsai-Hsiang Hung, Junjiang Zhong, M. Wen","doi":"10.1214/22-bjps550","DOIUrl":"https://doi.org/10.1214/22-bjps550","url":null,"abstract":". The analysis of means (ANOM) is a method that can compare the mean of each treatment to the overall mean. According to the graphical result of a statistical data analysis, we can specify which one is different from another. One of the assumptions of the classical ANOM model is that the variances are equal. However, it is not always true for the practice. To solve unknown and unequal population variances, Nelson and Dudewicz (2002) proposed a two-stage sampling procedure. However, additional samples need to be added in the second stage of the two-stage sampling procedure, so it is not practical all the time due to limited time and insufficient budget. Thus, under heteroscedasticity, we applied Chen and Lam’s (1989) single-stage sampling procedure to solve the drawback of the two-stage sampling procedure. In addition, we also provided an illustrative example and critical values for practical uses. In order to make the procedure user-friendly, we built an interface by using R Shiny.","PeriodicalId":51242,"journal":{"name":"Brazilian Journal of Probability and Statistics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43952607","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 fully parametric estimation procedure is proposed for robust estimation of the structural parameter in a functional comparative calibration model, under normality. The proposed estimator is obtained, based on maximum L q -likelihood approach, first re-placing the incidental parameters by estimators depending on the structural parameter. The estimator, called ML q E, depends on a single distortion parameter q , which controls the bal-ance between robustness and efficiency. If q tends to 1, the maximum likelihood estimator (MLE) is obtained. The estimation procedure can be implemented easily by a simple and fast re-weighting algorithm. For applying the method to practical and real-data scenarios, a data-based choice of an appropriate value of q is proposed. Consistency and asymptotic normality is established and the covariance matrix is given. The influence function is derived, to show the local robustness properties. Theoretical properties, ease of imple-mentabiliy and empirical results on simulated and real data show the satisfactory behavior of the ML q E and its vantages over the MLE in presence of observations discordant with the assumed model.
{"title":"Robust estimation in functional comparative calibration models via maximum Lq-likelihood","authors":"Patricia Giménez, Lucas Guarracino, M. Galea","doi":"10.1214/22-bjps552","DOIUrl":"https://doi.org/10.1214/22-bjps552","url":null,"abstract":". A fully parametric estimation procedure is proposed for robust estimation of the structural parameter in a functional comparative calibration model, under normality. The proposed estimator is obtained, based on maximum L q -likelihood approach, first re-placing the incidental parameters by estimators depending on the structural parameter. The estimator, called ML q E, depends on a single distortion parameter q , which controls the bal-ance between robustness and efficiency. If q tends to 1, the maximum likelihood estimator (MLE) is obtained. The estimation procedure can be implemented easily by a simple and fast re-weighting algorithm. For applying the method to practical and real-data scenarios, a data-based choice of an appropriate value of q is proposed. Consistency and asymptotic normality is established and the covariance matrix is given. The influence function is derived, to show the local robustness properties. Theoretical properties, ease of imple-mentabiliy and empirical results on simulated and real data show the satisfactory behavior of the ML q E and its vantages over the MLE in presence of observations discordant with the assumed model.","PeriodicalId":51242,"journal":{"name":"Brazilian Journal of Probability and Statistics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46025245","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}
Clécio da Silva Ferreira, Michel H. Montoril, G. Paula
{"title":"Partially linear models with p-order autoregressive skew-normal errors","authors":"Clécio da Silva Ferreira, Michel H. Montoril, G. Paula","doi":"10.1214/22-bjps556","DOIUrl":"https://doi.org/10.1214/22-bjps556","url":null,"abstract":"","PeriodicalId":51242,"journal":{"name":"Brazilian Journal of Probability and Statistics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42218160","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 develop a new survival model induced by discrete frailty with Katz distribution. The new model encompasses as particular cases the mixture cure rate model and promotion cure rate model and has a proportional hazards structure when the covariates are modeled through mean frailty. Furthermore, we construct a regression model to evaluate the effects of covariates on both the cured fraction and risk of the event of interest. We discuss inference aspects of the proposed model in a classical approach, where an expectation maximization algorithm is developed to determine the maximum likelihood estimates of the models parameters. Finally, the model is fully illustrated with a dataset on cervical cancer.
{"title":"A survival model for lifetime with long-term survivors and unobserved heterogeneity","authors":"A. D. E. Santo, V. Cancho","doi":"10.1214/22-bjps549","DOIUrl":"https://doi.org/10.1214/22-bjps549","url":null,"abstract":". In this paper, we develop a new survival model induced by discrete frailty with Katz distribution. The new model encompasses as particular cases the mixture cure rate model and promotion cure rate model and has a proportional hazards structure when the covariates are modeled through mean frailty. Furthermore, we construct a regression model to evaluate the effects of covariates on both the cured fraction and risk of the event of interest. We discuss inference aspects of the proposed model in a classical approach, where an expectation maximization algorithm is developed to determine the maximum likelihood estimates of the models parameters. Finally, the model is fully illustrated with a dataset on cervical cancer.","PeriodicalId":51242,"journal":{"name":"Brazilian Journal of Probability and Statistics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44378197","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 third-order response surface designs are designs used to estimate the parameters of the third-order polynomial model. In this paper, the orthogonal uniform composite designs that combine two-level orthogonal array and four-level uniform design denoted by OUCD 4 for estimating the parameters of the third-order models are proposed. The OUCD 4 are good space-filling and efficient designs, that uses resolution IV as the initial design, provide more flexible run sizes, have the ability to execute in-depth analysis and perform multiple analysis with different parts of the data for cross-validation.
{"title":"Orthogonal uniform composite designs for the third-order models","authors":"Brenda Mbouamba Yankam, A. Oladugba","doi":"10.1214/22-bjps555","DOIUrl":"https://doi.org/10.1214/22-bjps555","url":null,"abstract":": The third-order response surface designs are designs used to estimate the parameters of the third-order polynomial model. In this paper, the orthogonal uniform composite designs that combine two-level orthogonal array and four-level uniform design denoted by OUCD 4 for estimating the parameters of the third-order models are proposed. The OUCD 4 are good space-filling and efficient designs, that uses resolution IV as the initial design, provide more flexible run sizes, have the ability to execute in-depth analysis and perform multiple analysis with different parts of the data for cross-validation.","PeriodicalId":51242,"journal":{"name":"Brazilian Journal of Probability and Statistics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46845995","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 note we establish when the bivariate discrete Schur-constant models possess the Sibuya-type aging property. It happens that the corresponding class is large, solv-ing the counterpart of classical Sincov’s functional equation on the set of non-negative integers.
{"title":"Probability solutions of the Sincov’s functional equation on the set of nonnegative integers","authors":"N. Kolev, S. Mulinacci","doi":"10.1214/22-bjps548","DOIUrl":"https://doi.org/10.1214/22-bjps548","url":null,"abstract":". In this note we establish when the bivariate discrete Schur-constant models possess the Sibuya-type aging property. It happens that the corresponding class is large, solv-ing the counterpart of classical Sincov’s functional equation on the set of non-negative integers.","PeriodicalId":51242,"journal":{"name":"Brazilian Journal of Probability and Statistics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47027452","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}
Carolina Grejo, Fábio Lopes, F. Machado, A. Roldán-Correa
. We propose a stochastic model for evolution through mutation and natural se-lection of a population that evolves on a T + d tree. We think of this model as a way of describing the evolution fitness landscape of a population. We obtain sharp and distinct conditions on the set of parameters for extinction and survival.
{"title":"Evolution with mass extinction on Td+","authors":"Carolina Grejo, Fábio Lopes, F. Machado, A. Roldán-Correa","doi":"10.1214/22-bjps554","DOIUrl":"https://doi.org/10.1214/22-bjps554","url":null,"abstract":". We propose a stochastic model for evolution through mutation and natural se-lection of a population that evolves on a T + d tree. We think of this model as a way of describing the evolution fitness landscape of a population. We obtain sharp and distinct conditions on the set of parameters for extinction and survival.","PeriodicalId":51242,"journal":{"name":"Brazilian Journal of Probability and Statistics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41528622","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}
. There is ever growing demand of modeling overdispersed count data generated by various disiplines. Excessive number of zeros and heterogeneity in the population are two main sources of the overdispersion problem. Development of new count models that are more flexible than conventional Poisson model is thus necessary in order to address such sources. This study fullfils this need by proposing a new heterogeneous Poisson model with a capture of excess zeros, namely zero-inflated Poisson-Ailamujia (ZIPA) model. In line with the aim of curing overdispersion, a censored variant of this newly suggested model is also here developed. An extensive simulation study is conducted to assess the performances of both forms of new models in terms of bias, precision and accuracy measures. Additionally, two real world applications are presented to illustrate practical implications of zero-inflated (censored) Poisson-Ailamujia models in comparison to some alternatives.
{"title":"New zero-inflated regression models with a variant of censoring","authors":"Y. Altinisik, E. Çankaya","doi":"10.1214/22-bjps544","DOIUrl":"https://doi.org/10.1214/22-bjps544","url":null,"abstract":". There is ever growing demand of modeling overdispersed count data generated by various disiplines. Excessive number of zeros and heterogeneity in the population are two main sources of the overdispersion problem. Development of new count models that are more flexible than conventional Poisson model is thus necessary in order to address such sources. This study fullfils this need by proposing a new heterogeneous Poisson model with a capture of excess zeros, namely zero-inflated Poisson-Ailamujia (ZIPA) model. In line with the aim of curing overdispersion, a censored variant of this newly suggested model is also here developed. An extensive simulation study is conducted to assess the performances of both forms of new models in terms of bias, precision and accuracy measures. Additionally, two real world applications are presented to illustrate practical implications of zero-inflated (censored) Poisson-Ailamujia models in comparison to some alternatives.","PeriodicalId":51242,"journal":{"name":"Brazilian Journal of Probability and Statistics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43682406","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}
G. Cordeiro, J. S. Vasconcelos, Denize P. dos Santos, E. M. Ortega, R. A. Sermarini
{"title":"Three mixed-effects regression models using an extended Weibull with applications on games in differential and integral calculus","authors":"G. Cordeiro, J. S. Vasconcelos, Denize P. dos Santos, E. M. Ortega, R. A. Sermarini","doi":"10.1214/22-bjps553","DOIUrl":"https://doi.org/10.1214/22-bjps553","url":null,"abstract":"","PeriodicalId":51242,"journal":{"name":"Brazilian Journal of Probability and Statistics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43697374","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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