Quantile Function of the Kumaraswamy-Kumaraswamy (kkw) Distribution

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 = 1,
  beta = 1,
  delta = 0,
  lambda = 1,
  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.

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.

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.

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
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)
#> [1] 0.2425575 0.4631919 0.6851540

# 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)
#> [1] 0.6851540 0.4631919 0.2425575
# Check: qkkw(p, ..., lt=F) == qkkw(1-p, ..., lt=T)
print(qkkw(1 - p_vals, alpha_par, beta_par, delta_par, lambda_par))
#> [1] 0.6851540 0.4631919 0.2425575

# 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)
#> [1] 0.2425575 0.4631919 0.6851540
# Check: should match original quantiles
print(quantiles)
#> [1] 0.2425575 0.4631919 0.6851540

# 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
#> [1] "Max difference: 0"

# 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))
#> [1] "Original p: 0.75  Recalculated p: 0.75"
# 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
#> [1] 0 1
print(qkkw(c(-Inf, 0), alpha_par, beta_par, delta_par, lambda_par, log.p = TRUE)) # Should be 0, 1
#> [1] 0 1
# }