gkwreg: Generalized Kumaraswamy Regression Models

Overview

The gkwreg package provides a robust and efficient framework for modeling data restricted to the standard unit interval $(0, 1)$, such as proportions, rates, fractions, or indices. While the Beta distribution is commonly used for such data, gkwreg focuses on the Generalized Kumaraswamy (GKw) distribution family, offering enhanced flexibility by encompassing several important bounded distributions (including Beta and Kumaraswamy) as special cases.

The package facilitates both distribution fitting and regression modeling with potentially all distribution parameters modeled as functions of covariates using various link functions. Estimation is performed efficiently via Maximum Likelihood leveraging the Template Model Builder (TMB) framework, which utilizes automatic differentiation for superior speed, accuracy, and stability.

Key Features

• Flexible Distribution Family: Model data using the 5-parameter Generalized Kumaraswamy (GKw) distribution and its seven key nested sub-families:

  Distribution	Code	Parameters Modeled	Fixed Parameters	# Par.
  Generalized Kumaraswamy	gkw	alpha, beta, gamma, delta, lambda	None	5
  Beta-Kumaraswamy	bkw	alpha, beta, gamma, delta	lambda = 1	4
  Kumaraswamy-Kumaraswamy	kkw	alpha, beta, delta, lambda	gamma = 1	4
  Exponentiated Kumaraswamy	ekw	alpha, beta, lambda	gamma = 1, delta = 0	3
  McDonald / Beta Power	mc	gamma, delta, lambda	alpha = 1, beta = 1	3
  Kumaraswamy	kw	alpha, beta	gamma = 1, delta = 0, lambda = 1	2
  Beta	beta	gamma, delta	alpha = 1, beta = 1, lambda = 1	2

• Advanced Regression Modeling (gkwreg): Independently model each relevant distribution parameter as a function of covariates using a flexible formula interface:

  y ~ alpha_terms | beta_terms | gamma_terms | delta_terms | lambda_terms

• Multiple Link Functions: Choose appropriate link functions for each parameter, including:

  • log (default for all parameters)
  • logit, probit, cloglog (with optional scaling)
  • identity, inverse, sqrt

• Efficient Estimation: Utilizes the TMB package for fast and stable Maximum Likelihood Estimation, leveraging automatic differentiation for precise gradient and Hessian calculations.

• Standard R Interface: Provides familiar methods like summary(), predict(), plot(), coef(), vcov(), logLik(), AIC(), BIC(), residuals() for model inspection, inference, and diagnostics.

• Distribution Utilities: Implements standard d*, p*, q*, r* also as analytical log-likelihood ll*, gradient gr* and hessian hs* functions for all supported distributions in C++/RcppArmadillo.

Installation

# Install the stable version from CRAN:
install.packages("gkwreg")

# Or install the development version from GitHub:
# install.packages("devtools")
devtools::install_github("evandeilton/gkwreg")

Mathematical Background

The Generalized Kumaraswamy (GKw) Distribution

The GKw distribution is a flexible five-parameter distribution for variables on $(0, 1)$. Its cumulative distribution function (CDF) is given by:

$$F(x; \alpha, \beta, \gamma, \delta, \lambda) = I_{[1-(1-x^{\alpha})^{\beta}]^{\lambda}}(\gamma, \delta)$$

where $I_z(a,b)$ is the regularized incomplete beta function, and $\alpha, \beta, \gamma, \delta, \lambda > 0$ are the distribution parameters. The corresponding probability density function (PDF) is:

$$f(x; \alpha, \beta, \gamma, \delta, \lambda) = \frac{\lambda \alpha \beta x^{\alpha-1}}{B(\gamma, \delta)} (1-x^{\alpha})^{\beta-1} [1-(1-x^{\alpha})^{\beta}]^{\gamma\lambda-1} {1-[1-(1-x^{\alpha})^{\beta}]^{\lambda}}^{\delta-1}$$

where $B(\gamma, \delta)$ is the beta function.

The five parameters collectively provide exceptional flexibility in modeling distributions on $(0, 1)$: - Parameters alpha and beta primarily govern the basic shape inherited from the Kumaraswamy distribution - Parameters gamma and delta affect tail behavior and concentration around modes - Parameter lambda introduces additional flexibility, influencing skewness and peak characteristics

This parameterization allows the GKw distribution to capture a wide spectrum of shapes, including symmetric, skewed, unimodal, bimodal, J-shaped, U-shaped, and bathtub-shaped forms.

Regression Framework

In the regression setting, we assume that the response variable $y_i \in (0,1)$ follows a distribution from the GKw family with parameters $\theta_i = (\alpha_i, \beta_i, \gamma_i, \delta_i, \lambda_i)^{\top}$. Each parameter $\theta_{ip}$ (where $p \in {$alpha, beta, gamma, delta, lambda$}$) can depend on covariates through a link function $g_p(\cdot)$:

$$g_p(\theta_{ip}) = \eta_{ip} = \mathbf{x}_{ip}^{\top}\boldsymbol{\beta}_p$$

where $\eta_{ip}$ is the linear predictor, and $\boldsymbol{\beta}p$ is the vector of regression coefficients. Equivalently, $\theta{ip} = g_p^{-1}(\eta_{ip})$. The default link function is log for all parameters, ensuring the positivity constraint.

Parameters are estimated using maximum likelihood, with the log-likelihood function:

$$\ell(\Theta; \mathbf{y}, \mathbf{X}) = \sum_{i=1}^{n} \log f(y_i; \theta_i)$$

where each parameter $\theta_{ip}$ depends on $\Theta$ (the complete set of regression coefficients) via the link functions and linear predictors.

Computational Engine: TMB

The package uses Template Model Builder (TMB) (Kristensen et al. 2016) as its computational backend. TMB translates the statistical model into C++ templates and uses Automatic Differentiation (AD) to compute exact gradients and Hessians, providing several advantages:

• Speed: AD combined with compiled C++ is significantly faster than numerical differentiation or pure R implementations
• Accuracy: AD provides derivatives accurate to machine precision
• Stability: Precise derivatives improve optimization stability and convergence reliability
• Scalability: Efficiently handles models with many parameters

Examples

Regression Modeling

Model parameters of a GKw family distribution as functions of covariates:

library(gkwreg)

# Simulate data for a Kumaraswamy regression model
set.seed(123)
n <- 100
x1 <- runif(n, -2, 2)
x2 <- rnorm(n)

# Simulate true parameters (using log link)
alpha_true <- exp(0.8 + 0.3 * x1 - 0.2 * x2) 
beta_true  <- exp(1.2 - 0.4 * x1 + 0.1 * x2)

# Generate response
y <- rkw(n, alpha = alpha_true, beta = beta_true)
y <- pmax(pmin(y, 1 - 1e-7), 1e-7)  # Ensure y in (0, 1)
df1 <- data.frame(y = y, x1 = x1, x2 = x2)

# Fit Kumaraswamy regression: alpha ~ x1 + x2, beta ~ x1 + x2
kw_model <- gkwreg(y ~ x1 + x2 | x1 + x2, data = df1, family = "kw")
summary(kw_model)

Real Data Analysis

# Food Expenditure Data
library(gkwreg)
data("FoodExpenditure", package = "betareg")
FoodExpenditure$y <- FoodExpenditure$food/FoodExpenditure$income

# Fit models from different GKw families
kkw_model <- gkwreg(y ~ income, data = FoodExpenditure, family = "kkw")
ekw_model <- gkwreg(y ~ income, data = FoodExpenditure, family = "ekw")
kw_model <- gkwreg(y ~ income, data = FoodExpenditure, family = "kw")

# Compare models
data.frame(
  logLik = rbind(logLik(kkw_model), logLik(ekw_model), logLik(kw_model)),
  AIC = rbind(AIC(kkw_model), AIC(ekw_model), AIC(kw_model)),
  BIC = rbind(BIC(kkw_model), BIC(ekw_model), BIC(kw_model))
)

# Summary
summary(kw_model)

res <- residuals(kw_model, type = "quantile")

# Visual diagnostics
plot(kw_model)

# Predicted
pred <- predict(kw_model)

Distribution Fitting

Fit a GKw family distribution to univariate data (no covariates):

# Simulate data from Beta(2, 3)
set.seed(2203)
y_beta <- rbeta_(1000, gamma = 2, delta = 3)

# Fit Beta and Kumaraswamy distributions
fit_beta <- gkwfit(data = y_beta, family = "beta")
fit_kw <- gkwfit(data = y_beta, family = "kw")

# Compare models
summary(fit_beta)
summary(fit_kw)
AIC(fit_beta)
AIC(fit_kw)

Diagnostic Methods

The package provides several diagnostic tools for model assessment:

# Residual analysis
model <- gkwreg(y ~ x1 | x2, data = mydata, family = "kw")
res <- residuals(model, type = "quantile")  # Randomized quantile residuals

# Visual diagnostics
plot(model)  # QQ-plot, residuals vs. fitted, etc.

pred <- predict(model, type = "response")

References

• Cordeiro, G. M., & de Castro, M. (2011). A new family of generalized distributions. Journal of Statistical Computation and Simulation, 81(7), 883-898.

• Carrasco, J. M. F., Ferrari, S. L. P., & Cordeiro, G. M. (2010). A new generalized Kumaraswamy distribution. arXiv preprint arXiv:1004.0911.

• Jones, M. C. (2009). Kumaraswamy’s distribution: A beta-type distribution with some tractability advantages. Statistical Methodology, 6(1), 70-81.

• Kristensen, K., Nielsen, A., Berg, C. W., Skaug, H., & Bell, B. M. (2016). TMB: Automatic Differentiation and Laplace Approximation. Journal of Statistical Software, 70(5), 1-21.

• Kumaraswamy, P. (1980). A generalized probability density function for double-bounded random processes. Journal of Hydrology, 46(1-2), 79-88.

• Ferrari, S. L. P., & Cribari-Neto, F. (2004). Beta regression for modelling rates and proportions. Journal of Applied Statistics, 31(7), 799-815.

• Cribari-Neto, F., & Zeileis, A. (2010). Beta Regression in R. Journal of Statistical Software, 34(2), 1-24.

• Lopes, J. E. (2025). Generalized Kumaraswamy Regression Models with gkwreg. Journal of Statistical Software, forthcoming.

Comparing with Other Packages

The gkwreg package complements and extends existing approaches for modeling bounded data:

Contributing

Contributions to gkwreg are welcome! Please feel free to submit issues or pull requests on the GitHub repository.

License

This package is licensed under the MIT License. See the LICENSE file for details.

Author and Maintainer

Lopes, J. E. (evandeilton@gmail.com)
LEG - Laboratório de Estatística e Geoinformação
UFPR - Universidade Federal do Paraná, Brazil