Density of the Kumaraswamy-Kumaraswamy (kkw) Distribution

Computes the probability density function (PDF) for the Kumaraswamy-Kumaraswamy (kkw) distribution with parameters alpha ($\alpha$), beta ($\beta$), delta ($\delta$), and lambda ($\lambda$). This distribution is defined on the interval (0, 1).

Usage

dkkw(x, alpha, beta, delta, lambda, log_prob = FALSE)

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.
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 ($\log(f(x))$). Default: 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, delta, lambda). Returns 0 (or -Inf if log_prob = TRUE) for x outside the interval (0, 1), or NaN if parameters are invalid (e.g., alpha <= 0, beta <= 0, delta < 0, lambda <= 0).

Details

The Kumaraswamy-Kumaraswamy (kkw) distribution is a special case of the five-parameter Generalized Kumaraswamy distribution (dgkw) obtained by setting the parameter $\gamma = 1$.

The probability density function is given by: $$ f(x; \alpha, \beta, \delta, \lambda) = (\delta + 1) \lambda \alpha \beta x^{\alpha - 1} (1 - x^\alpha)^{\beta - 1} \bigl[1 - (1 - x^\alpha)^\beta\bigr]^{\lambda - 1} \bigl\{1 - \bigl[1 - (1 - x^\alpha)^\beta\bigr]^\lambda\bigr\}^{\delta} $$ for $0 < x < 1$. Note that $1/(\delta+1)$ corresponds to the Beta function term $B(1, \delta+1)$ when $\gamma=1$.

Numerical evaluation follows similar stability considerations as dgkw.

References

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

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

Author

Lopes, J. E.

Examples

# \donttest{
# Example values
x_vals <- c(0.2, 0.5, 0.8)
alpha_par <- 2.0
beta_par <- 3.0
delta_par <- 0.5
lambda_par <- 1.5

# Calculate density
densities <- dkkw(x_vals, alpha_par, beta_par, delta_par, lambda_par)
print(densities)
#> [1] 0.8281038 2.1612055 0.3594057

# Calculate log-density
log_densities <- dkkw(x_vals, alpha_par, beta_par, delta_par, lambda_par,
                       log_prob = TRUE)
print(log_densities)
#> [1] -0.1886168  0.7706662 -1.0233034
# Check: should match log(densities)
print(log(densities))
#> [1] -0.1886168  0.7706662 -1.0233034

# Compare with dgkw setting gamma = 1
densities_gkw <- dgkw(x_vals, alpha_par, beta_par, gamma = 1.0,
                      delta_par, lambda_par)
print(paste("Max difference:", max(abs(densities - densities_gkw)))) # Should be near zero
#> [1] "Max difference: 8.88178419700125e-16"

# Plot the density
curve_x <- seq(0.01, 0.99, length.out = 200)
curve_y <- dkkw(curve_x, alpha_par, beta_par, delta_par, lambda_par)
plot(curve_x, curve_y, type = "l", main = "kkw Density Example",
     xlab = "x", ylab = "f(x)", col = "blue")


# }