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Modelling Bimodal Data Using a Multivariate Triangular-Linked Distribution. MATHEMATICS 2022. [DOI: 10.3390/math10142370] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/05/2023]
Abstract
Bimodal distributions have rarely been studied although they appear frequently in datasets. We develop a novel bimodal distribution based on the triangular distribution and then expand it to the multivariate case using a Gaussian copula. To determine the goodness of fit of the univariate model, we use the Kolmogorov–Smirnov (KS) and Cramér–von Mises (CVM) tests. The contributions of this work are that a simplistic yet robust distribution was developed to deal with bimodality in data, a multivariate distribution was developed as a generalisation of this univariate distribution using a Gaussian copula, a comparison between parametric and semi-parametric approaches to modelling bimodality is given, and an R package called btld is developed from the workings of this paper.
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A Symmetric/Asymmetric Bimodal Extension Based on the Logistic Distribution: Properties, Simulation and Applications. MATHEMATICS 2022. [DOI: 10.3390/math10121968] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/04/2022]
Abstract
In this paper, we introduce bimodal extensions, one symmetric and one asymmetric, of the logistic distribution. We define this new density and study some basic properties. We draw inferences from the moment estimator and maximum likelihood approaches. We present a simulation study to assess the behaviour of the moment and maximum likelihood estimators. We also study the singularity of the Fisher information matrix for particular cases. We offer applications in real data and compare them with a mixture of logistics distributions.
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Azzalini A. An overview on the progeny of the skew-normal family— A personal perspective. J MULTIVARIATE ANAL 2022. [DOI: 10.1016/j.jmva.2021.104851] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
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Martínez-Flórez G, Tovar-Falón R, Elal-Olivero D. Some new flexible classes of normal distribution for fitting multimodal data. STATISTICS-ABINGDON 2022. [DOI: 10.1080/02331888.2022.2041642] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
Affiliation(s)
- Guillermo Martínez-Flórez
- Departamento de Matemáticas y Estadística, Facultad de Ciencias Básicas, Universidad de Córdoba, Montería, Colombia
| | - Roger Tovar-Falón
- Departamento de Matemáticas y Estadística, Facultad de Ciencias Básicas, Universidad de Córdoba, Montería, Colombia
| | - David Elal-Olivero
- Departamento de Matemática, Facultad de Ingeniería, Universidad de Atacama, Copiapó, Chile
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TOVAR-FALÓN ROGER, MARTÍNEZ-FLÓREZ GUILLERMO. A New Class of Exponentiated Beta-Skew-Laplace Distribution. AN ACAD BRAS CIENC 2022; 94:e20191597. [DOI: 10.1590/0001-3765202220191597] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/22/2019] [Accepted: 06/02/2020] [Indexed: 11/21/2022] Open
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Meraou MA, Al-Kandari NM, Raqab MZ, Kundu D. Analysis of skewed data by using compound Poisson exponential distribution with applications to insurance claims. J STAT COMPUT SIM 2021. [DOI: 10.1080/00949655.2021.1981324] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
Affiliation(s)
| | | | | | - Debasis Kundu
- Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, India
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A Multivariate Flexible Skew-Symmetric-Normal Distribution: Scale-Shape Mixtures and Parameter Estimation via Selection Representation. Symmetry (Basel) 2021. [DOI: 10.3390/sym13081343] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/16/2022] Open
Abstract
Multivariate skew-symmetric-normal (MSSN) distributions have been recognized as an appealing tool for modeling data with non-normal features such as asymmetry and heavy tails, rendering them suitable for applications in diverse areas. We introduce a richer class of MSSN distributions based on a scale-shape mixture of (multivariate) flexible skew-symmetric normal distributions, called the SSMFSSN distributions. This very general class of SSMFSSN distributions can capture various shapes of multimodality, skewness, and leptokurtic behavior in the data. We investigate some of its probabilistic characterizations and distributional properties which are useful for further methodological developments. An efficient EM-type algorithm designed under the selection mechanism is advocated to compute the maximum likelihood (ML) estimates of parameters. Simulation studies as well as applications to a real dataset are employed to illustrate the usefulness of the presented methods. Numerical results show the superiority of our proposed model in comparison to several existing competitors.
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Abstract
This paper introduces a new family of distributions for modelling censored multimodal data. The model extends the widely known tobit model by introducing two parameters that control the shape and the asymmetry of the distribution. Basic properties of this new family of distributions are studied in detail and a model for censored positive data is also studied. The problem of estimating parameters is addressed by considering the maximum likelihood method. The score functions and the elements of the observed information matrix are given. Finally, three applications to real data sets are reported to illustrate the developed methodology.
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Mahdavi A, Amirzadeh V, Jamalizadeh A, Lin TI. Maximum likelihood estimation for scale-shape mixtures of flexible generalized skew normal distributions via selection representation. Comput Stat 2021. [DOI: 10.1007/s00180-021-01079-2] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/22/2022]
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Affiliation(s)
- Piotr Sulewski
- Institute of Exact and Technical Sciences, Pomeranian University, Słupsk, Poland
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Vila R, Leão J, Saulo H, Shahzad MN, Santos-Neto M. On a bimodal Birnbaum–Saunders distribution with applications to lifetime data. BRAZ J PROBAB STAT 2020. [DOI: 10.1214/19-bjps448] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Bakouch HS, Cadena M, Chesneau C. A new class of skew distributions with climate data analysis. J Appl Stat 2020; 48:3002-3024. [DOI: 10.1080/02664763.2020.1791804] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/23/2022]
Affiliation(s)
- Hassan S. Bakouch
- Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt
| | - Meitner Cadena
- DECE, Universidad de las Fuerzas Armadas, Sangolqui, Ecuador
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Abstract
Data from some research fields tend to exhibit a positive skew. For example, in experimental psychology, reaction times (RTs) are characterised as being positively skewed. However, it is not unlikely that RTs can take a normal or, even, a negative shape. While the Ex-Gaussian distribution is suitable to model positively skewed data, it cannot cope with negatively skewed data. This manuscript proposes a distribution that can deal with both negative and positive skews: the exponential-centred skew-normal (ECSN) distribution. The mathematical properties of the proposed distribution are reported, and it is featured in two non-synthetic datasets.
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Venegas O, Salinas HS, Gómez HW. A note on the Fisher information matrix for the flexible generalized-skew-normal model. J Korean Stat Soc 2020. [DOI: 10.1007/s42952-019-00025-9] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
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Abstract
The main object of this paper is to develop an alternative construction for the bimodal skew-normal distribution. The construction is based upon a study of the mixture of skew-normal distributions. We study some basic properties of this family, its stochastic representations and expressions for its moments. Parameters are estimated using the maximum likelihood estimation method. A simulation study is carried out to observe the performance of the maximum likelihood estimators. Finally, we compare the efficiency of the new distribution with other distributions in the literature using a real data set. The study shows that the proposed approach presents satisfactory results.
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Likelihood-Based Inference for the Asymmetric Beta-Skew Alpha-Power Distribution. Symmetry (Basel) 2020. [DOI: 10.3390/sym12040613] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/16/2022] Open
Abstract
This paper introduces a new family of asymmetric distributions that allows to fit unimodal as well as bimodal and trimodal data sets. The model extends the normal model by introducing two parameters that control the shape and the asymmetry of the distribution. Basic properties of this new distribution are studied in detail. The problem of estimating parameters is addressed by considering the maximum likelihood method and Fisher information matrix is derived. A small Monte Carlo simulation study is conducted to examine the performance of the obtained estimators. Finally, two data set are considered to illustrate the developed methodology.
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Ownuk J, Baghishani H, Nezakati A. Heavy or semi-heavy tail, that is the question. J Appl Stat 2020; 48:646-668. [PMID: 35706985 PMCID: PMC9041666 DOI: 10.1080/02664763.2020.1738360] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/13/2019] [Accepted: 03/01/2020] [Indexed: 10/24/2022]
Abstract
While there has been considerable research on the analysis of extreme values and outliers by using heavy-tailed distributions, little is known about the semi-heavy-tailed behaviors of data when there are a few suspicious outliers. To address the situation where data are skewed possessing semi-heavy tails, we introduce two new skewed distribution families of the hyperbolic secant with exciting properties. We extend the semi-heavy-tailedness property of data to a linear regression model. In particular, we investigate the asymptotic properties of the ML estimators of the regression parameters when the error term has a semi-heavy-tailed distribution. We conduct simulation studies comparing the ML estimators of the regression parameters under various assumptions for the distribution of the error term. We also provide three real examples to show the priority of the semi-heavy-tailedness of the error term comparing to heavy-tailedness. Online supplementary materials for this article are available. All the new proposed models in this work are implemented by the shs R package, which can be found on the GitHub webpage.
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Affiliation(s)
- Jamil Ownuk
- Department of Statistics, Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran
| | - Hossein Baghishani
- Department of Statistics, Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran
| | - Ahmad Nezakati
- Department of Statistics, Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran
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Affiliation(s)
- Wen-Jang Huang
- Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung, Taiwan
| | - Nan-Cheng Su
- Department of Statistics, National Taipei University, Taipei, Taiwan
| | - Hui-Yi Teng
- Institute of Statistics, National University of Kaohsiung, Kaohsiung, Taiwan
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Abstract
In this article, we study an extension of the sinh Cauchy model in order to obtain asymmetric bimodality. The behavior of the distribution may be either unimodal or bimodal. We calculate its cumulative distribution function and use it to carry out quantile regression. We calculate the maximum likelihood estimators and carry out a simulation study. Two applications are analyzed based on real data to illustrate the flexibility of the distribution for modeling unimodal and bimodal data.
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Multivariate skew distributions with mode-invariance through the transformation of scale. JAPANESE JOURNAL OF STATISTICS AND DATA SCIENCE 2019. [DOI: 10.1007/s42081-019-00047-x] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
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Lee D, Sinha S. Identifiability and bias reduction in the skew-probit model for a binary response. J STAT COMPUT SIM 2019. [DOI: 10.1080/00949655.2019.1590579] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
Affiliation(s)
- DongHyuk Lee
- Department of Statistics, Texas A&M University, College Station, TX, USA
| | - Samiran Sinha
- Department of Statistics, Texas A&M University, College Station, TX, USA
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Dey R, Cadigan N, Zheng N. Estimation of the Von Bertalanffy growth model when ages are measured with error. J R Stat Soc Ser C Appl Stat 2019. [DOI: 10.1111/rssc.12340] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Rajib Dey
- Memorial University of Newfoundland; St John's Canada
| | - Noel Cadigan
- Memorial University of Newfoundland; St John's Canada
| | - Nan Zheng
- Memorial University of Newfoundland; St John's Canada
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Amiri M, Gómez HW, Jamalizadeh A, Towhidi M. Bimodal extension based on the skew-$t$-normal distribution. BRAZ J PROBAB STAT 2019. [DOI: 10.1214/17-bjps372] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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26
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Sakhanenko LA. Testing a Multivariate Distribution for Generalized Skew Ellipticity. THEORY OF PROBABILITY AND ITS APPLICATIONS 2019. [DOI: 10.1137/s0040585x97t989490] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/20/2022]
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Acitas S. A new weighted distribution as an extension of the generalized half-normal distribution with applications. J STAT COMPUT SIM 2018. [DOI: 10.1080/00949655.2018.1462812] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/17/2022]
Affiliation(s)
- S. Acitas
- Department of Statistics, Anadolu University, Eskisehir, Turkey
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Altun E, Tatlıdil H, Özel G. Conditional ASGT-GARCH Approach to Value-at-Risk. IRANIAN JOURNAL OF SCIENCE AND TECHNOLOGY, TRANSACTIONS A: SCIENCE 2018. [DOI: 10.1007/s40995-018-0484-1] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
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Altun E, Tatlidil H, Ozel G, Nadarajah S. A new generalization of skew- T distribution with volatility models. J STAT COMPUT SIM 2018. [DOI: 10.1080/00949655.2018.1427240] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
Affiliation(s)
- Emrah Altun
- Department of Statistics, Hacettepe University, Ankara, Turkey
| | | | - Gamze Ozel
- Department of Statistics, Hacettepe University, Ankara, Turkey
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Rozliman NA, Ibrahim AIN, Yunus RM. Bayesian approach to errors-in-variables in count data regression models with departures from normality and overdispersion. J STAT COMPUT SIM 2017. [DOI: 10.1080/00949655.2017.1381845] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
Affiliation(s)
- Nur Aainaa Rozliman
- Institute of Mathematical Sciences, University of Malaya, Kuala Lumpur, Malaysia
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Yilmaz A. The flexible skew Laplace distribution. COMMUN STAT-THEOR M 2016. [DOI: 10.1080/03610926.2014.974821] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
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Sharafi M, Sajjadnia Z, Behboodian J. A new generalization of alpha-skew-normal distribution. COMMUN STAT-THEOR M 2016. [DOI: 10.1080/03610926.2015.1117639] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/21/2022]
Affiliation(s)
- Maryam Sharafi
- Department of Statistics, Shiraz University, Shiraz, Iran
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Sastry DVS, Bhati D. A new skew logistic distribution: Properties and applications. BRAZ J PROBAB STAT 2016. [DOI: 10.1214/14-bjps278] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Bouaddi M, Belhachemi R. The Continuous Hidden threshold Mixed Skew-Symmetric Distribution. COMMUN STAT-THEOR M 2016. [DOI: 10.1080/03610926.2015.1040500] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/22/2022]
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Shafiei S, Doostparast M, Jamalizadeh A. The alpha–beta skew normal distribution: properties and applications. STATISTICS-ABINGDON 2015. [DOI: 10.1080/02331888.2015.1096938] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/22/2022]
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37
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A flexible generalization of the skew normal distribution based on a weighted normal distribution. STAT METHOD APPL-GER 2015. [DOI: 10.1007/s10260-015-0337-4] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/27/2022]
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Acitas S, Senoglu B, Arslan O. Alpha-Skew Generalized t Distribution. REVISTA COLOMBIANA DE ESTADÍSTICA 2015. [DOI: 10.15446/rce.v38n2.51665] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022] Open
Abstract
<p>The alpha-skew normal (ASN) distribution has been proposed recently in the literature by using standard normal distribution and a skewing approach. Although ASN distribution is able to model both skew and bimodal data, it is shortcoming when data has thinner or thicker tails than normal. Therefore, we propose an alpha-skew generalized t (ASGT) by using the generalized t (GT) distribution and a new skewing procedure. From this point of view, ASGT can be seen as an alternative skew version of GT distribution. However, ASGT differs from the previous skew versions of GT distribution since it is able to model bimodal data sest as well as it nests most commonly used density functions. In this paper, moments and maximum likelihood estimation of the parameters of ASGT distribution are given. Skewness and kurtosis measures are derived based on the first four noncentral moments. The cumulative distribution function (cdf) of ASGT distribution is also obtained. In the application part of the study, two real life problems taken from the literature are modeled by using ASGT distribution.</p>
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Jones MC. Rejoinder. Int Stat Rev 2014. [DOI: 10.1111/insr.12084] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- M. C. Jones
- Department of Mathematics and Statistics; The Open University; Walton Hall, Milton Keynes MK7 6AA UK
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Hallin M, Ley C. Skew-symmetric distributions and Fisher information: The double sin of the skew-normal. BERNOULLI 2014. [DOI: 10.3150/13-bej528] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Using skew-logistic probability density function as a model for age-specific fertility rate pattern. BIOMED RESEARCH INTERNATIONAL 2014; 2014:790294. [PMID: 24967404 PMCID: PMC4055447 DOI: 10.1155/2014/790294] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 02/04/2014] [Revised: 04/13/2014] [Accepted: 04/27/2014] [Indexed: 11/17/2022]
Abstract
Fertility rate is one of the most important global indexes. Past researchers found models which fit to age-specific fertility rates. For example, mixture probability density functions have been proposed for situations with bi-modal fertility patterns. This model is less useful for unimodal age-specific fertility rate patterns, so a model based on skew-symmetric (skew-normal) pdf was proposed by Mazzuco and Scarpa (2011) which was flexible for unimodal and bimodal fertility patterns. In this paper, we introduce skew-logistic probability density function as a better model: its residuals are less than those of the skew-normal model and it can more precisely estimate the parameters of the model.
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Shafiei S, Doostparast M. Balakrishnan Skew- tDistribution and Associated Statistical Characteristics. COMMUN STAT-THEOR M 2014. [DOI: 10.1080/03610926.2012.701697] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
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Ghosh S, Dey D. Heuristically Deciding Between Normal and Skew Normal Distributions for Describing the Data on a Response Variable and an Explanatory Variable. JOURNAL OF STATISTICAL THEORY AND PRACTICE 2014. [DOI: 10.1080/15598608.2013.823581] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
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Rocha GH, Loschi RH, Arellano-Valle RB. Inference in flexible families of distributions with normal kernel. STATISTICS-ABINGDON 2013. [DOI: 10.1080/02331888.2012.688207] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/28/2022]
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Ma Y, Kim M, Genton MG. Semiparametric Efficient and Robust Estimation of an Unknown Symmetric Population Under Arbitrary Sample Selection Bias. J Am Stat Assoc 2013. [DOI: 10.1080/01621459.2013.816184] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
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Nadarajah S, Nassiri V, Mohammadpour A. Truncated-exponential skew-symmetric distributions. STATISTICS-ABINGDON 2013. [DOI: 10.1080/02331888.2013.821474] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
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Nekoukhou V, Alamatsaz MH, Aghajani AH. A Flexible Skew-Generalized Normal Distribution. COMMUN STAT-THEOR M 2013. [DOI: 10.1080/03610926.2011.599003] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
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