Vasicek Quantile and Mean Regression Models for Bounded Data: New Formulation, Mathematical Derivations, and Numerical Applications

Abstract

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.

Modeling Income Data via New Parametric Quantile Regressions: Formulation, Computational Statistics, and Application

“…As mentioned, several parametric quantile regression models have been proposed in the literature considering diverse distributions [3,[23][24][25][26][27][28][29][30][31][32][33][34][35]. The reader is referred to [36] for a full overview of parametric quantile regressions, their applications, and computational implementations, which are helpful to model indexes, proportions, and rates.…”

The unit generalized half-normal quantile regression model: formulation, estimation, diagnostics, and numerical applications

“…2019a , b ), unit second degree improved Lindley (Altun and Cordeiro 2020 ), and Vasicek (Mazucheli et al. 2022a ) distributions. For other exponentiated distributions, see, for example, Akgul ( 2021 ).…”

“… 2022a ) and Vasicek (Mazucheli et al. 2022a ) distributions. For a complete overview on parametric quantile regression models and their computational implementation with diverse applications, see Mazucheli et al.…”

Another unit Burr XII quantile regression model based on the different reparameterization applied to dropout in Brazilian undergraduate courses

“…[8,9] proposed the unit Burr XII and reflected unit Burr XII, respectively. Other quantile regressions were introduced by [10,11], and [12]. One may also see [13][14][15][16] for unit regressions applied to educational measurements.…”