qkkw: Quantile Function of the Kumaraswamy-Kumaraswamy (kkw)...

In gkwreg: Generalized Kumaraswamy Regression Models for Bounded Data

Quantile Function of the Kumaraswamy-Kumaraswamy (kkw) Distribution

Description

Computes the quantile function (inverse CDF) for the Kumaraswamy-Kumaraswamy (kkw) distribution with parameters alpha (\alpha), beta (\beta), delta (\delta), and lambda (\lambda). It finds the value q such that P(X \le q) = p. This distribution is a special case of the Generalized Kumaraswamy (GKw) distribution where the parameter \gamma = 1.

Usage

qkkw(p, alpha, beta, delta, lambda, lower_tail = TRUE, log_p = FALSE)

Arguments

p	Vector of probabilities (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.
lower_tail	Logical; if TRUE (default), probabilities are p = P(X \le q), otherwise, probabilities are p = P(X > q).
log_p	Logical; if TRUE, probabilities p are given as \log(p). Default: FALSE.

Details

The quantile function Q(p) is the inverse of the CDF F(q). The CDF for the kkw (\gamma=1) distribution is (see pkkw):

F(q) = 1 - \bigl\{1 - \bigl[1 - (1 - q^\alpha)^\beta\bigr]^\lambda\bigr\}^{\delta + 1}

Inverting this equation for q yields the quantile function:

Q(p) = \left[ 1 - \left\{ 1 - \left[ 1 - (1 - p)^{1/(\delta+1)} \right]^{1/\lambda} \right\}^{1/\beta} \right]^{1/\alpha}

The function uses this closed-form expression and correctly handles the lower_tail and log_p arguments by transforming p appropriately before applying the formula.

Value

A vector of quantiles corresponding to the given probabilities p. The length of the result is determined by the recycling rule applied to the arguments (p, alpha, beta, delta, lambda). Returns:

• 0 for p = 0 (or p = -Inf if log_p = TRUE, when lower_tail = TRUE).

• 1 for p = 1 (or p = 0 if log_p = TRUE, when lower_tail = TRUE).

• NaN for p < 0 or p > 1 (or corresponding log scale).

• NaN for invalid parameters (e.g., alpha <= 0, beta <= 0, delta < 0, lambda <= 0).

Boundary return values are adjusted accordingly for lower_tail = FALSE.

Author(s)

Lopes, J. E.

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.

See Also

qgkw (parent distribution quantile function), dkkw, pkkw, rkkw, qbeta

Examples

# Example values
p_vals <- c(0.1, 0.5, 0.9)
alpha_par <- 2.0
beta_par <- 3.0
delta_par <- 0.5
lambda_par <- 1.5

# Calculate quantiles
quantiles <- qkkw(p_vals, alpha_par, beta_par, delta_par, lambda_par)
print(quantiles)

# Calculate quantiles for upper tail probabilities P(X > q) = p
# e.g., for p=0.1, find q such that P(X > q) = 0.1 (90th percentile)
quantiles_upper <- qkkw(p_vals, alpha_par, beta_par, delta_par, lambda_par,
                         lower_tail = FALSE)
print(quantiles_upper)
# Check: qkkw(p, ..., lt=F) == qkkw(1-p, ..., lt=T)
print(qkkw(1 - p_vals, alpha_par, beta_par, delta_par, lambda_par))

# Calculate quantiles from log probabilities
log_p_vals <- log(p_vals)
quantiles_logp <- qkkw(log_p_vals, alpha_par, beta_par, delta_par, lambda_par,
                        log_p = TRUE)
print(quantiles_logp)
# Check: should match original quantiles
print(quantiles)

# Compare with qgkw setting gamma = 1
quantiles_gkw <- qgkw(p_vals, alpha_par, beta_par, gamma = 1.0,
                      delta_par, lambda_par)
print(paste("Max difference:", max(abs(quantiles - quantiles_gkw)))) # Should be near zero

# Verify inverse relationship with pkkw
p_check <- 0.75
q_calc <- qkkw(p_check, alpha_par, beta_par, delta_par, lambda_par)
p_recalc <- pkkw(q_calc, alpha_par, beta_par, delta_par, lambda_par)
print(paste("Original p:", p_check, " Recalculated p:", p_recalc))
# abs(p_check - p_recalc) < 1e-9 # Should be TRUE

# Boundary conditions
print(qkkw(c(0, 1), alpha_par, beta_par, delta_par, lambda_par)) # Should be 0, 1
print(qkkw(c(-Inf, 0), alpha_par, beta_par, delta_par, lambda_par, log_p = TRUE)) # Should be 0, 1

gkwreg documentation built on April 16, 2025, 1:10 a.m.