qkw: Quantile Function of the Kumaraswamy (Kw) Distribution

In gkwreg: Generalized Kumaraswamy Regression Models for Bounded Data

Quantile Function of the Kumaraswamy (Kw) Distribution

Description

Computes the quantile function (inverse CDF) for the two-parameter Kumaraswamy (Kw) distribution with shape parameters alpha (\alpha) and beta (\beta). It finds the value q such that P(X \le q) = p.

Usage

qkw(p, alpha, beta, 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.
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 Kumaraswamy distribution is F(q) = 1 - (1 - q^\alpha)^\beta (see pkw). Inverting this equation for q yields the quantile function:

Q(p) = \left\{ 1 - (1 - p)^{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. This is equivalent to the general GKw quantile function (qgkw) evaluated with \gamma=1, \delta=0, \lambda=1.

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). 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).

Boundary return values are adjusted accordingly for lower_tail = FALSE.

Author(s)

Lopes, J. E.

References

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

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

See Also

qgkw (parent distribution quantile function), dkw, pkw, rkw (other Kw functions), qbeta, qunif

Examples

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

# Calculate quantiles using qkw
quantiles <- qkw(p_vals, alpha_par, beta_par)
print(quantiles)

# Calculate quantiles for upper tail probabilities P(X > q) = p
quantiles_upper <- qkw(p_vals, alpha_par, beta_par, lower_tail = FALSE)
print(quantiles_upper)
# Check: qkw(p, ..., lt=F) == qkw(1-p, ..., lt=T)
print(qkw(1 - p_vals, alpha_par, beta_par))

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

# Compare with qgkw setting gamma = 1, delta = 0, lambda = 1
quantiles_gkw <- qgkw(p_vals, alpha = alpha_par, beta = beta_par,
                     gamma = 1.0, delta = 0.0, lambda = 1.0)
print(paste("Max difference:", max(abs(quantiles - quantiles_gkw)))) # Should be near zero

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

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

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