TY - JOUR

T1 - Vasicek Quantile and Mean Regression Models for Bounded Data

T2 - New Formulation, Mathematical Derivations, and Numerical Applications

AU - Mazucheli, Josmar

AU - Alves, Bruna

AU - Korkmaz, Mustafa

AU - Leiva, Víctor

N1 - Publisher Copyright:
© 2022 by the authors. Licensee MDPI, Basel, Switzerland.

PY - 2022/5/1

Y1 - 2022/5/1

N2 - 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.

AB - 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.

KW - Monte Carlo simulation

KW - R software

KW - maximum likelihood method

KW - mean regression

KW - parametric quantile regression

UR - http://www.scopus.com/inward/record.url?scp=85129334467&partnerID=8YFLogxK

U2 - 10.3390/math10091389

DO - 10.3390/math10091389

M3 - Article

AN - SCOPUS:85129334467

SN - 2227-7390

VL - 10

JO - Mathematics

JF - Mathematics

IS - 9

M1 - 1389

ER -