Vasicek Quantile and Mean Regression Models for Bounded Data: New Formulation, Mathematical Derivations, and Numerical Applications
The Vasicek distribution is a two-parameter probability model with bounded support on the open unit interval. This distribution allows for different and flexible shapes and plays an important role in many statistical applications, especially for modeling default rates in the field of finance. Althou...
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MDPI AG
2022-04-01
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| author | Josmar Mazucheli Bruna Alves Mustafa Ç. Korkmaz Víctor Leiva |
| author_facet | Josmar Mazucheli Bruna Alves Mustafa Ç. Korkmaz Víctor Leiva |
| author_sort | Josmar Mazucheli |
| collection | DOAJ |
| description | The Vasicek distribution is a two-parameter probability model with bounded support on the open unit interval. This distribution allows for different and flexible shapes and plays an important role in many statistical applications, especially for modeling default rates in the field of finance. Although its probability density function resembles some well-known distributions, such as the beta and Kumaraswamy models, the Vasicek distribution has not been considered to analyze data on the unit interval, especially when we have, in addition to a response variable, one or more covariates. In this paper, we propose to estimate quantiles or means, conditional on covariates, assuming that the response variable is Vasicek distributed. Through appropriate link functions, two Vasicek regression models for data on the unit interval are formulated: one considers a quantile parameterization and another one its original parameterization. Monte Carlo simulations are provided to assess the statistical properties of the maximum likelihood estimators, as well as the coverage probability. An R package developed by the authors, named vasicekreg, makes available the results of the present investigation. Applications with two real data sets are conducted for illustrative purposes: in one of them, the unit Vasicek quantile regression outperforms the models based on the Johnson-SB, Kumaraswamy, unit-logistic, and unit-Weibull distributions, whereas in the second one, the unit Vasicek mean regression outperforms the fits obtained by the beta and simplex distributions. Our investigation suggests that unit Vasicek quantile and mean regressions can be of practical usage as alternatives to some well-known models for analyzing data on the unit interval. |
| first_indexed | 2024-03-10T03:56:51Z |
| format | Article |
| id | doaj.art-b139f9453bfa4d1ca2d6e4bcc36a2699 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T03:56:51Z |
| publishDate | 2022-04-01 |
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| series | Mathematics |
| spelling | doaj.art-b139f9453bfa4d1ca2d6e4bcc36a26992023-11-23T08:43:34ZengMDPI AGMathematics2227-73902022-04-01109138910.3390/math10091389Vasicek Quantile and Mean Regression Models for Bounded Data: New Formulation, Mathematical Derivations, and Numerical ApplicationsJosmar Mazucheli0Bruna Alves1Mustafa Ç. Korkmaz2Víctor Leiva3Department of Statistics, Universidade Estadual de Maringá, Maringá 87020-900, BrazilDepartment of Statistics, Universidade Estadual de Maringá, Maringá 87020-900, BrazilDepartment of Measurement and Evaluation, Artvin Coruh University, Artvin 08100, TurkeySchool of Industrial Engineering, Pontificia Universidad Católica de Valparaíso, Valparaíso 2362807, ChileThe Vasicek distribution is a two-parameter probability model with bounded support on the open unit interval. This distribution allows for different and flexible shapes and plays an important role in many statistical applications, especially for modeling default rates in the field of finance. Although its probability density function resembles some well-known distributions, such as the beta and Kumaraswamy models, the Vasicek distribution has not been considered to analyze data on the unit interval, especially when we have, in addition to a response variable, one or more covariates. In this paper, we propose to estimate quantiles or means, conditional on covariates, assuming that the response variable is Vasicek distributed. Through appropriate link functions, two Vasicek regression models for data on the unit interval are formulated: one considers a quantile parameterization and another one its original parameterization. Monte Carlo simulations are provided to assess the statistical properties of the maximum likelihood estimators, as well as the coverage probability. An R package developed by the authors, named vasicekreg, makes available the results of the present investigation. Applications with two real data sets are conducted for illustrative purposes: in one of them, the unit Vasicek quantile regression outperforms the models based on the Johnson-SB, Kumaraswamy, unit-logistic, and unit-Weibull distributions, whereas in the second one, the unit Vasicek mean regression outperforms the fits obtained by the beta and simplex distributions. Our investigation suggests that unit Vasicek quantile and mean regressions can be of practical usage as alternatives to some well-known models for analyzing data on the unit interval.https://www.mdpi.com/2227-7390/10/9/1389maximum likelihood methodMonte Carlo simulationparametric quantile regressionmean regressionR software |
| spellingShingle | Josmar Mazucheli Bruna Alves Mustafa Ç. Korkmaz Víctor Leiva Vasicek Quantile and Mean Regression Models for Bounded Data: New Formulation, Mathematical Derivations, and Numerical Applications Mathematics maximum likelihood method Monte Carlo simulation parametric quantile regression mean regression R software |
| title | Vasicek Quantile and Mean Regression Models for Bounded Data: New Formulation, Mathematical Derivations, and Numerical Applications |
| title_full | Vasicek Quantile and Mean Regression Models for Bounded Data: New Formulation, Mathematical Derivations, and Numerical Applications |
| title_fullStr | Vasicek Quantile and Mean Regression Models for Bounded Data: New Formulation, Mathematical Derivations, and Numerical Applications |
| title_full_unstemmed | Vasicek Quantile and Mean Regression Models for Bounded Data: New Formulation, Mathematical Derivations, and Numerical Applications |
| title_short | Vasicek Quantile and Mean Regression Models for Bounded Data: New Formulation, Mathematical Derivations, and Numerical Applications |
| title_sort | vasicek quantile and mean regression models for bounded data new formulation mathematical derivations and numerical applications |
| topic | maximum likelihood method Monte Carlo simulation parametric quantile regression mean regression R software |
| url | https://www.mdpi.com/2227-7390/10/9/1389 |
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