gkwreg (version 1.0.7)

dgkw: Density of the Generalized Kumaraswamy Distribution

Description

Computes the probability density function (PDF) for the five-parameter Generalized Kumaraswamy (GKw) distribution, defined on the interval (0, 1).

Usage

dgkw(x, alpha, beta, gamma, delta, lambda, log_prob = FALSE)

Value

A vector of density values (\(f(x)\)) or log-density values (\(\log(f(x))\)). The length of the result is determined by the recycling rule applied to the arguments (x, alpha, beta, gamma, delta, lambda). Returns 0 (or -Inf

if log_prob = TRUE) for x outside the interval (0, 1), or NaN if parameters are invalid.

Arguments

x

Vector of quantiles (values between 0 and 1).

alpha

Shape parameter alpha > 0. Can be a scalar or a vector. Default: 1.0.

beta

Shape parameter beta > 0. Can be a scalar or a vector. Default: 1.0.

gamma

Shape parameter gamma > 0. Can be a scalar or a vector. Default: 1.0.

delta

Shape parameter delta >= 0. Can be a scalar or a vector. Default: 0.0.

lambda

Shape parameter lambda > 0. Can be a scalar or a vector. Default: 1.0.

log_prob

Logical; if TRUE, the logarithm of the density is returned. Default: FALSE.

Author

Lopes, J. E.

Details

The probability density function of the Generalized Kumaraswamy (GKw) distribution with parameters alpha (\(\alpha\)), beta (\(\beta\)), gamma (\(\gamma\)), delta (\(\delta\)), and lambda (\(\lambda\)) is given by: $$ f(x; \alpha, \beta, \gamma, \delta, \lambda) = \frac{\lambda \alpha \beta x^{\alpha-1}(1-x^{\alpha})^{\beta-1}} {B(\gamma, \delta+1)} [1-(1-x^{\alpha})^{\beta}]^{\gamma\lambda-1} [1-[1-(1-x^{\alpha})^{\beta}]^{\lambda}]^{\delta} $$ for \(x \in (0,1)\), where \(B(a, b)\) is the Beta function beta.

This distribution was proposed by Cordeiro & de Castro (2011) and includes several other distributions as special cases:

• Kumaraswamy (Kw): gamma = 1, delta = 0, lambda = 1

• Exponentiated Kumaraswamy (EKw): gamma = 1, delta = 0

• Beta-Kumaraswamy (BKw): lambda = 1

• Generalized Beta type 1 (GB1 - implies McDonald): alpha = 1, beta = 1

• Beta distribution: alpha = 1, beta = 1, lambda = 1

The function includes checks for valid parameters and input values x. It uses numerical stabilization for x close to 0 or 1.

References

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

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

See Also

pgkw, qgkw, rgkw (if these exist), dbeta, integrate

Examples

# \donttest{
# Simple density evaluation at a point
dgkw(0.5, alpha = 2, beta = 3, gamma = 1, delta = 0, lambda = 1) # Kw case

# Plot the PDF for various parameter sets
x_vals <- seq(0.01, 0.99, by = 0.01)

# Standard Kumaraswamy (gamma=1, delta=0, lambda=1)
pdf_kw <- dgkw(x_vals, alpha = 2, beta = 3, gamma = 1, delta = 0, lambda = 1)

# Beta equivalent (alpha=1, beta=1, lambda=1) - Beta(gamma, delta+1)
pdf_beta <- dgkw(x_vals, alpha = 1, beta = 1, gamma = 2, delta = 3, lambda = 1)
# Compare with stats::dbeta
pdf_beta_check <- stats::dbeta(x_vals, shape1 = 2, shape2 = 3 + 1)
# max(abs(pdf_beta - pdf_beta_check)) # Should be close to zero

# Exponentiated Kumaraswamy (gamma=1, delta=0)
pdf_ekw <- dgkw(x_vals, alpha = 2, beta = 3, gamma = 1, delta = 0, lambda = 2)

plot(x_vals, pdf_kw, type = "l", ylim = range(c(pdf_kw, pdf_beta, pdf_ekw)),
     main = "GKw Densities Examples", ylab = "f(x)", xlab="x", col = "blue")
lines(x_vals, pdf_beta, col = "red")
lines(x_vals, pdf_ekw, col = "green")
legend("topright", legend = c("Kw(2,3)", "Beta(2,4) equivalent", "EKw(2,3, lambda=2)"),
       col = c("blue", "red", "green"), lty = 1, bty = "n")

# Log-density
log_pdf_val <- dgkw(0.5, 2, 3, 1, 0, 1, log_prob = TRUE)
print(log_pdf_val)
print(log(dgkw(0.5, 2, 3, 1, 0, 1))) # Should match

# }