This work focusses on estimation the standard deviation (SD) and coefficient of variation (CV) in the normal distribution. These two measures are useful in applications and widely used to report the spread or variability of continuous data. We develop the confidence intervals for these parameters using the two pivotal quantity methods with the unbiased estimator of SD. The first confidence interval uses the pivot function based on a chi-square distribution and the second one is based on a generalized pivotal quantity. The performance of our approaches is conducted via simulations. We show that the confidence intervals for SD and CV based on the new pivots have coverage probabilities greater than existing confidence intervals. Furthermore, they have acceptable short expected lengths. We also provide two real-data sets on the SET50 index of Thailand to demonstrate the proposed methods.
{"title":"Alternative Confidence Interval for Variability Parameters in the Normal Distribution with Applications to Stock Exchange Index Data Set","authors":"Patarawan Sangnawakij, None Wanwarat Anlamlert","doi":"10.17713/ajs.v52i5.1534","DOIUrl":"https://doi.org/10.17713/ajs.v52i5.1534","url":null,"abstract":"This work focusses on estimation the standard deviation (SD) and coefficient of variation (CV) in the normal distribution. These two measures are useful in applications and widely used to report the spread or variability of continuous data. We develop the confidence intervals for these parameters using the two pivotal quantity methods with the unbiased estimator of SD. The first confidence interval uses the pivot function based on a chi-square distribution and the second one is based on a generalized pivotal quantity. The performance of our approaches is conducted via simulations. We show that the confidence intervals for SD and CV based on the new pivots have coverage probabilities greater than existing confidence intervals. Furthermore, they have acceptable short expected lengths. We also provide two real-data sets on the SET50 index of Thailand to demonstrate the proposed methods.","PeriodicalId":51761,"journal":{"name":"Austrian Journal of Statistics","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135982207","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}
The half-logistic (HL) distribution is a widely considered statistical model for studying lifetime phenomena arising in science, engineering, finance, and biomedical sciences. One of its weaknesses is that it has a decreasing probability density function and an increasing hazard rate function only. Due to that, researchers have been modifying the HL distribution to have more functional ability. This article provides an extensive overview of the HL distribution and its generalization (or extensions). The recent advancements regarding the HL distribution have led to numerous results in modern theory and statistical computing techniques across science and engineering. This work extended the body of literature in a summarized way to clarify some of the states of knowledge, potentials, and important roles played by the HL distribution and related models in probability theory and statistical studies in various areas and applications. In particular, at least sixty-seven flexible extensions of the HL distribution have been proposed in the past few years. We give a brief introduction to these distributions, emphasizing model parameters, properties derived, and the estimation method. Conclusively, there is no doubt that this summary could create a consensus between various related results in both theory and applications of the HL-related models to develop an interest in future studies.
{"title":"On the Type-I Half-logistic Distribution and Related Contributions: A Review","authors":"Mustapha Muhammad, M.H. Tahir, Lixia Liu, Farrukh Jamal, Christophe Chesneau, Badamasi Abba","doi":"10.17713/ajs.v52i5.1500","DOIUrl":"https://doi.org/10.17713/ajs.v52i5.1500","url":null,"abstract":"The half-logistic (HL) distribution is a widely considered statistical model for studying lifetime phenomena arising in science, engineering, finance, and biomedical sciences. One of its weaknesses is that it has a decreasing probability density function and an increasing hazard rate function only. Due to that, researchers have been modifying the HL distribution to have more functional ability. This article provides an extensive overview of the HL distribution and its generalization (or extensions). The recent advancements regarding the HL distribution have led to numerous results in modern theory and statistical computing techniques across science and engineering. This work extended the body of literature in a summarized way to clarify some of the states of knowledge, potentials, and important roles played by the HL distribution and related models in probability theory and statistical studies in various areas and applications. In particular, at least sixty-seven flexible extensions of the HL distribution have been proposed in the past few years. We give a brief introduction to these distributions, emphasizing model parameters, properties derived, and the estimation method. Conclusively, there is no doubt that this summary could create a consensus between various related results in both theory and applications of the HL-related models to develop an interest in future studies.","PeriodicalId":51761,"journal":{"name":"Austrian Journal of Statistics","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135939223","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}
Luiz Otávio de Oliveira Pala, Marcela de M. Carvalho, Thelma Sáfadi
Extensions of the Autoregressive Moving Average, ARMA(p, q), class for modeling non-Gaussian time series have been proposed in the literature in recent years, being applied in phenomena such as counts and rates. One of them is the Generalized Autoregressive Moving Average, GARMA(p, q), that is supported by the Generalized Linear Models theory and has been studied under the Bayesian perspective. This paper aimed to study models for time series of counts using the Poisson, Negative binomial and Poisson inverse Gaussian distributions, and adopting the Bayesian framework. To do so, we carried out a simulation study and, in addition, we showed a practical application and evaluation of these models by using a set of real data, corresponding to the number of vehicle thefts in Brazil.
{"title":"Analysis of Count Time Series: A Bayesian GARMA(p, q) Approach","authors":"Luiz Otávio de Oliveira Pala, Marcela de M. Carvalho, Thelma Sáfadi","doi":"10.17713/ajs.v52i5.1568","DOIUrl":"https://doi.org/10.17713/ajs.v52i5.1568","url":null,"abstract":"Extensions of the Autoregressive Moving Average, ARMA(p, q), class for modeling non-Gaussian time series have been proposed in the literature in recent years, being applied in phenomena such as counts and rates. One of them is the Generalized Autoregressive Moving Average, GARMA(p, q), that is supported by the Generalized Linear Models theory and has been studied under the Bayesian perspective. This paper aimed to study models for time series of counts using the Poisson, Negative binomial and Poisson inverse Gaussian distributions, and adopting the Bayesian framework. To do so, we carried out a simulation study and, in addition, we showed a practical application and evaluation of these models by using a set of real data, corresponding to the number of vehicle thefts in Brazil.","PeriodicalId":51761,"journal":{"name":"Austrian Journal of Statistics","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135939221","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 few years ago a paper appeared in this Journal that proposed a closed-form approximation of the p-value for Pearson's chi-squared statistic. Extensive empirical and simulation studies were performed and it was shown that the approximation provides very accurate p-values when compared with what we considered to be the ``true'' p-value (obtained using a base R function). It is important to note, however, that Pearson's chi-squared statistic is a special case of the Cressie-Read family of divergence statistics as is, for example, the log-likelihood ratio statistic, the Freeman-Tukey statistic and the Cressie-Read statistic. Therefore, this paper adapts the previously published closed form approximation of the p-value by demonstrating its applicability to any member of the Cressie-Read family of divergence statistics. We also give two further closed form approximations and assess their accuracy by analysing three contingency tables of varying sample size, degrees of freedom and statistical significance of the association.
{"title":"On the P-value for Members of the Cressie-Read Family of Divergence Statistics","authors":"Eric Beh, Wang Ting-Wu","doi":"10.17713/ajs.v52i5.1607","DOIUrl":"https://doi.org/10.17713/ajs.v52i5.1607","url":null,"abstract":"A few years ago a paper appeared in this Journal that proposed a closed-form approximation of the p-value for Pearson's chi-squared statistic. Extensive empirical and simulation studies were performed and it was shown that the approximation provides very accurate p-values when compared with what we considered to be the ``true'' p-value (obtained using a base R function). It is important to note, however, that Pearson's chi-squared statistic is a special case of the Cressie-Read family of divergence statistics as is, for example, the log-likelihood ratio statistic, the Freeman-Tukey statistic and the Cressie-Read statistic. Therefore, this paper adapts the previously published closed form approximation of the p-value by demonstrating its applicability to any member of the Cressie-Read family of divergence statistics. We also give two further closed form approximations and assess their accuracy by analysing three contingency tables of varying sample size, degrees of freedom and statistical significance of the association.","PeriodicalId":51761,"journal":{"name":"Austrian Journal of Statistics","volume":"57 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135981632","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}
Generalizations of the 1-factorial tests by Kruskal-Wallis and Friedman, as well as of the van der Waerden test are proposed for factorial split-plot designs, both allowing interactions. They are compared in regard to the type I error control and the power with the parametric F test, including the Huynh-Feldt adjustment, the inverse normal transform (INT), the ANOVA type statistic by Brunner et al. (ATS), the aligned rank transform (ART), the L statistic by Puri & Sen and a procedure by Koch. The two methods proposed show a perfect type I error control, except for two situations, and an attractive power, particularly in case of nonnormal distributions. The charm and advantage of these procedures are the possibility to apply them with statistical standard tools using only variable transformations and data management, and to receive results from well-known methods which are easy to understand.
Kruskal-Wallis和Friedman提出的1因子检验的推广，以及van der Waerden检验，用于因子分裂图设计，两者都允许相互作用。比较了它们的I型误差控制和参数F检验的功率，包括Huynh-Feldt平差、反正态变换(INT)、Brunner等人的ANOVA型统计量(ATS)、对齐秩变换(ART)、Puri &的L统计量;参议员和科赫的程序。除了两种情况外，所提出的两种方法都显示出完美的I型误差控制，并且具有吸引力，特别是在非正态分布的情况下。这些程序的魅力和优势在于可以将它们与仅使用变量转换和数据管理的统计标准工具一起应用，并从易于理解的众所周知的方法中获得结果。
{"title":"Generalizations of the Tests by Kruskal-Wallis, Friedman and van der Waerden for Split-plot Designs","authors":"Haiko Lüpsen","doi":"10.17713/ajs.v52i5.1545","DOIUrl":"https://doi.org/10.17713/ajs.v52i5.1545","url":null,"abstract":"Generalizations of the 1-factorial tests by Kruskal-Wallis and Friedman, as well as of the van der Waerden test are proposed for factorial split-plot designs, both allowing interactions. They are compared in regard to the type I error control and the power with the parametric F test, including the Huynh-Feldt adjustment, the inverse normal transform (INT), the ANOVA type statistic by Brunner et al. (ATS), the aligned rank transform (ART), the L statistic by Puri & Sen and a procedure by Koch. The two methods proposed show a perfect type I error control, except for two situations, and an attractive power, particularly in case of nonnormal distributions. The charm and advantage of these procedures are the possibility to apply them with statistical standard tools using only variable transformations and data management, and to receive results from well-known methods which are easy to understand.","PeriodicalId":51761,"journal":{"name":"Austrian Journal of Statistics","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135982080","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}
Composite indicators can be understood as a summary of well-chosen and relevant sub-indicators which are combined into a single number. Their aim is to represent a multi-dimensional construct and map the performance of entities such as countries or companies. These multidimensional constructs are for example sustainability, poverty or well-being. Composite indicators are widely applied in various disciplines such as social or economic research and benefit from their apparent ease of interpretation. In the context of the Sustainable Development Framework a composite indicator over all 17 sustainable development goals, as been proposed. As composite indicators are commonly applied in highly sensitive areas this, urges the need to discuss methodical advantages and disadvantages as well as their adequacy for performance comparisons. In this paper we discuss and illustrate quality issues with regard to aspects of the subjective choices made in the construction process of composite indicators, imputation of missing data and the survey design. As an example we construct a composite indicator on sustainable economic development using data of the Sustainable Development framework. Furthermore, we exemplify and discuss strategies and methods for the quality assessment of a composite indicator.
{"title":"Quality and Sensitivity of Composite Indicators for Sustainable Development","authors":"Laura Güdemann, Ralf Münnich","doi":"10.17713/ajs.v52i5.1539","DOIUrl":"https://doi.org/10.17713/ajs.v52i5.1539","url":null,"abstract":"Composite indicators can be understood as a summary of well-chosen and relevant sub-indicators which are combined into a single number. Their aim is to represent a multi-dimensional construct and map the performance of entities such as countries or companies. These multidimensional constructs are for example sustainability, poverty or well-being. Composite indicators are widely applied in various disciplines such as social or economic research and benefit from their apparent ease of interpretation. In the context of the Sustainable Development Framework a composite indicator over all 17 sustainable development goals, as been proposed. As composite indicators are commonly applied in highly sensitive areas this, urges the need to discuss methodical advantages and disadvantages as well as their adequacy for performance comparisons. In this paper we discuss and illustrate quality issues with regard to aspects of the subjective choices made in the construction process of composite indicators, imputation of missing data and the survey design. As an example we construct a composite indicator on sustainable economic development using data of the Sustainable Development framework. Furthermore, we exemplify and discuss strategies and methods for the quality assessment of a composite indicator.","PeriodicalId":51761,"journal":{"name":"Austrian Journal of Statistics","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135981629","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}
The notion of ageing plays an important role in reliability and survival analysis as it is an inherent property of all systems and products. Jiang, Ji, and Xiao (2003) proposed a new quantitative measure, known as ageing intensity (AI) function, an alternative measureto study the ageing pattern of probability models. In this paper, we propose a nonparametric estimator for ageing intensity function. Asymptotic properties of the estimator are established under suitable regularity conditions. A set of simulation studies are carriedout based on various probability models to examine the performance of estimator and to establish its efficiency over the classical estimator. The usefulness of the estimator is also examined through a real data set.
老化的概念在可靠性和生存分析中起着重要作用，因为它是所有系统和产品的固有属性。Jiang, Ji, and Xiao(2003)提出了一种新的定量度量，称为老化强度(AI)函数，这是研究概率模型老化模式的一种替代度量。本文提出了一种老化强度函数的非参数估计方法。在适当的正则性条件下，建立了估计量的渐近性质。在各种概率模型的基础上进行了一系列仿真研究，以检验该估计器的性能，并证明其优于经典估计器。通过一个实际数据集检验了该估计器的有效性。
{"title":"Kernel-based Estimation of Ageing Intensity Function: Properties and Applications","authors":"R S Rasin, None S M Sunoj, None Rakesh Poduval","doi":"10.17713/ajs.v52i5.1497","DOIUrl":"https://doi.org/10.17713/ajs.v52i5.1497","url":null,"abstract":"The notion of ageing plays an important role in reliability and survival analysis as it is an inherent property of all systems and products. Jiang, Ji, and Xiao (2003) proposed a new quantitative measure, known as ageing intensity (AI) function, an alternative measureto study the ageing pattern of probability models. In this paper, we propose a nonparametric estimator for ageing intensity function. Asymptotic properties of the estimator are established under suitable regularity conditions. A set of simulation studies are carriedout based on various probability models to examine the performance of estimator and to establish its efficiency over the classical estimator. The usefulness of the estimator is also examined through a real data set.","PeriodicalId":51761,"journal":{"name":"Austrian Journal of Statistics","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135937799","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}
Pub Date : 2023-08-15DOI: 10.17713/ajs.v52isi.1749
Oksana Braganets, A. Iksanov
We investigate an infinite balls-in-boxes scheme, in which boxes are arranged in nested hierarchy and random probabilities of boxes are defined in terms of iterated fragmentation of a unit mass. Gnedin and Iksanov (2020) obtained a multivariate functional central limit theorem with centering for the cumulative occupancy counts as the number of balls becomes large. We prove a counterpart of their result, in which centering is not needed and the limit processes are not Gaussian. An application is given to the scheme generated by a residual allocation model.
{"title":"A Limit Theorem for a Nested Infinite Occupancy Scheme in Random Environment","authors":"Oksana Braganets, A. Iksanov","doi":"10.17713/ajs.v52isi.1749","DOIUrl":"https://doi.org/10.17713/ajs.v52isi.1749","url":null,"abstract":"We investigate an infinite balls-in-boxes scheme, in which boxes are arranged in nested hierarchy and random probabilities of boxes are defined in terms of iterated fragmentation of a unit mass. Gnedin and Iksanov (2020) obtained a multivariate functional central limit theorem with centering for the cumulative occupancy counts as the number of balls becomes large. We prove a counterpart of their result, in which centering is not needed and the limit processes are not Gaussian. An application is given to the scheme generated by a residual allocation model.","PeriodicalId":51761,"journal":{"name":"Austrian Journal of Statistics","volume":"44 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86437165","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}
Pub Date : 2023-08-15DOI: 10.17713/ajs.v52isi.1753
O. Hopkalo, L. Sakhno, O. Vasylyk
The paper presents bounds for the distributions of suprema for particular classes of ϕ-sub-Gaussian random fields. Results stated depend on representations of bounds for increments of the fields in different metrics. Several examples of applications are provided to illustrate the results, in particular, to random fields related to stochastic partial differential equations and partial differential equations with random initial conditions.
{"title":"Bounds for the Tail Distributions of Suprema of Sub-Gaussian Type Random Fields","authors":"O. Hopkalo, L. Sakhno, O. Vasylyk","doi":"10.17713/ajs.v52isi.1753","DOIUrl":"https://doi.org/10.17713/ajs.v52isi.1753","url":null,"abstract":"The paper presents bounds for the distributions of suprema for particular classes of ϕ-sub-Gaussian random fields. Results stated depend on representations of bounds for increments of the fields in different metrics. Several examples of applications are provided to illustrate the results, in particular, to random fields related to stochastic partial differential equations and partial differential equations with random initial conditions.","PeriodicalId":51761,"journal":{"name":"Austrian Journal of Statistics","volume":"70 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77475310","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}
Pub Date : 2023-08-15DOI: 10.17713/ajs.v52isi.1752
V. Golomoziy, Olha Moskanova
This paper is devoted to establishing conditions that guarantee the existence of a p-th moment of the time it takes for a timeinhomogeneous Markov chain to hit some set C. We modified the well-known Drift Condition from the theory of homogeneous Markov chains. We demonstrated that the inhomogeneous Markov chain may be polynomially recurrent while exhibiting different dynamics from its homogeneous counterpart.
{"title":"Polynomial Recurrence of Time-inhomogeneous Markov Chains","authors":"V. Golomoziy, Olha Moskanova","doi":"10.17713/ajs.v52isi.1752","DOIUrl":"https://doi.org/10.17713/ajs.v52isi.1752","url":null,"abstract":"This paper is devoted to establishing conditions that guarantee the existence of a p-th moment of the time it takes for a timeinhomogeneous Markov chain to hit some set C. We modified the well-known Drift Condition from the theory of homogeneous Markov chains. We demonstrated that the inhomogeneous Markov chain may be polynomially recurrent while exhibiting different dynamics from its homogeneous counterpart.","PeriodicalId":51761,"journal":{"name":"Austrian Journal of Statistics","volume":"130 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73463487","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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