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The Unit-Gompertz Quantile Regression Model for the Bounded Responses

  • Josmar Mazucheli , Bruna Alves and Mustafa Ç. Korkmaz EMAIL logo
Published/Copyright: August 4, 2023
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 73 Issue 4

ABSTRACT

This paper proposes a regression model for the continuous responses bounded to the unit interval which is based on the unit-Gompertz distribution as an alternative to quantile regression models based on the unit-Birnbaum-Saunders, unit-Weibull, L-Logistic, Kumaraswamy and Johnson SB distributions. Re-parameterizing the unit-Gompertz distribution as a function of its quantile allows us to model the effect of covariates across the entire response distribution, rather than only at the mean. Our proposal sometimes outperforms the other distributions available in the literature. These discoveries are provided by Monte Carlo simulations and one application using a real data set. An R package, including parameter estimation, model checking as well as density, cumulative distribution, quantile and random number generating functions of the unit-Gompertz distribution are developed and can be readily used in applications.

2020 Mathematics Subject Classification: Primary 60E05; 62E10; 62N05
Keywords: Bounded data; Gompertz distribution; likelihood function; Monte Carlo simulation; parametric quantile regression

(Communicated by Gejza Wimmer)

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Appendix

  1. The UBSA distribution [27] is obtained from the transformation Y = exp(–X), where X ~ Birnbaum-Saunders(α, θ) denotes a Birnbaum-Saunders [7] distributed random variable. The corresponding p.d.f., c.d.f. and q.f. are written as

    f(y∣α,θ)=12yαθ2π[(−αlog(y))12+(−αlog(y))32]exp[12θ2(2+log(y)α+αlog(y))],F(y∣α,θ)=1−Φ{1θ[(−log(y)α)12−(−αlog(y))12]}

    and

    Q(τ∣α,θ)=exp{−2α2+[θΦ−1(1−τ)]2−θΦ−1(1−τ)4+[θΦ−1(1−τ)]2},

    respectively, where 0 < y < 1, θ > 0, α > 0 and Φ(·) is the c.d.f. of the standard normal distribution.

  2. The ULOG distribution [35] is obtained from the transformation Y=exp(X−αθ)1+exp(X−αθ)) , where X ~ Log (0,1) denotes a standard Logistic distributed random variable [6]. The corresponding p.d.f., c.d.f. and q.f. are written as

    f(y∣α,θ)=θexp(α)(y1−y)θ−1[1+exp(α)(y1−y)θ]2,F(y∣α,θ)=exp(α)(y1−y)θ1+exp(α)(y1−y)θ,

    and

    Q(τ∣α,θ)=exp(−αθ)(τ1−τ)1θ1+exp(−αθ)(τ1−τ)1θ,

    respectively, where 0 < y, τ < 1, α > 0 and θ > 0. From above q.f., the parameter α can be re-parameterized as α=g−1(μ)=logτ1−τ−θlogμ1−μ .

  3. The JOSB distribution [16] can be obtained from the transformation Y=exp(X−αθ)1+exp (X−αθ) ) , where X ~ N (0, 1) denotes a standard Normal random variable. The corresponding p.d.f., c.d.f. and q.f. are written as

    f(y∣α,θ)=θ2π1y(1−y)exp{−12[α+θlog(y1−y)]2},F(y∣α,θ)=Φ[α+θlog(y1−y)],Q(τ∣α,θ)=exp[Φ−1(τ)−αθ]1+exp[Φ−1(τ)−αθ],

    respectively, where 0 < y, τ < 1, α ∈ ℝ and θ > 0 and Φ–1(·) is the qf of the standard Normal distribution. From above q.f., the parameter α can be re-parameterized as α=g−1(μ)=Φ−1(τ)−θlog(μ1−μ) .

  4. The KUMA distribution [24] can be obtained from the transformation Y = exp(–X), where X ~ EE(α, θ) denotes a Exponentiated-Exponential distributed random variable [14]. The corresponding p.d.f., c.d.f. and q.f. are written as

    f(y|α, θ)=αθyθ−1(1−yθ)α−1,F(y|α, θ)=1−(1−yθ)α,Q(τ|α, θ)=[1−(1−τ)1α]1θ,

    respectively, where 0 < y, τ < 1, α > 0 and θ > 0 are shapes parameters. Prom above q.f., the parameter α can be re-parameterized as α=g−1(μ)=log⁡(1−τ)log⁡(1−μθ) .

  5. The UWEI distribution [28] is obtained from the transformation Y = exp(–X), where X ~ WEI(α, θ) denotes a Weibull distributed random variable [36]. The corresponding p.d.f., c.d.f. and q.f. are written as

    f(y∣α,θ)=αθy−log⁡(y)θ−1exp−α−log⁡(y)θ,F(y∣α,θ)=exp−α−log⁡(y)θ,Qτ∣α,θ=exp−−log⁡(τ)α1θ,

    respectively, where 0 < y, τ < 1, α > 0 and θ > 0. Prom above q.f., the parameter α can be re-parameterized as α=g−1(μ)=−log⁡(τ)[−log⁡(μ)]θ .
Received: 2021-09-14
Accepted: 2022-01-26
Published Online: 2023-08-04

© 2023 Mathematical Institute Slovak Academy of Sciences

Articles in the same Issue

  1. A Note on Special Subsets of the Rudin-Frolík Order for Regulars
  2. The 2-Class Group of Certain Families of Imaginary Triquadratic Fields
  3. The Deranged Bell Numbers
  4. On Index and Monogenity of Certain Number Fields Defined by Trinomials
  5. The k-Generalized Lucas Numbers Close to a Power of 2
  6. Shifted Power of a Polynomial with Integral Roots
  7. Further Insights into the Mysteries of the Values of Zeta Functions at Integers
  8. Memoryless Properties on Time Scales
  9. A Study of the Higher-Order Schwarzian Derivatives of Hirotaka Tamanoi
  10. Besov and Triebel-Lizorkin Capacity in Metric Spaces
  11. Oscillation of Odd Order Linear Differential Equations with Deviating Arguments with Dominating Delay Part
  12. An Elliptic Type Inclusion Problem on the Heisenberg Lie Group
  13. Existence Result for a Double Phase Problem Involving the (p(x), q(x))-Laplacian Operator
  14. A New Series Space Derived by Absolute Generalized Nörlund Means
  15. Examples of Weinstein Domains in the Complement of Smoothed Total Toric Divisors
  16. The Uniform Effros Property and Local Homogeneity
  17. Limit Theorems for Weighted Sums of Asymptotically Negatively Associated Random Variables Under Some General Conditions
  18. The Unit-Gompertz Quantile Regression Model for the Bounded Responses
  19. An Extended Gamma-Lindley Model and Inference for the Prediction of Covid-19 in Tunisia
  20. Modeling Bivariate Data Using Linear Exponential and Weibull Distributions as Marginals
https://doi.org/10.1515/ms-2023-0077
Keywords for this article
Bounded data; Gompertz distribution; likelihood function; Monte Carlo simulation; parametric quantile regression

Articles in the same Issue

  1. A Note on Special Subsets of the Rudin-Frolík Order for Regulars
  2. The 2-Class Group of Certain Families of Imaginary Triquadratic Fields
  3. The Deranged Bell Numbers
  4. On Index and Monogenity of Certain Number Fields Defined by Trinomials
  5. The k-Generalized Lucas Numbers Close to a Power of 2
  6. Shifted Power of a Polynomial with Integral Roots
  7. Further Insights into the Mysteries of the Values of Zeta Functions at Integers
  8. Memoryless Properties on Time Scales
  9. A Study of the Higher-Order Schwarzian Derivatives of Hirotaka Tamanoi
  10. Besov and Triebel-Lizorkin Capacity in Metric Spaces
  11. Oscillation of Odd Order Linear Differential Equations with Deviating Arguments with Dominating Delay Part
  12. An Elliptic Type Inclusion Problem on the Heisenberg Lie Group
  13. Existence Result for a Double Phase Problem Involving the (p(x), q(x))-Laplacian Operator
  14. A New Series Space Derived by Absolute Generalized Nörlund Means
  15. Examples of Weinstein Domains in the Complement of Smoothed Total Toric Divisors
  16. The Uniform Effros Property and Local Homogeneity
  17. Limit Theorems for Weighted Sums of Asymptotically Negatively Associated Random Variables Under Some General Conditions
  18. The Unit-Gompertz Quantile Regression Model for the Bounded Responses
  19. An Extended Gamma-Lindley Model and Inference for the Prediction of Covid-19 in Tunisia
  20. Modeling Bivariate Data Using Linear Exponential and Weibull Distributions as Marginals
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