# Quantile Function of the Kumaraswamy (Kw) Distribution 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 $P(X \le q) = p$. ```r 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`. ## Returns 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`. ## 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 $\gamma=1, \delta=0, \lambda=1$. ## Examples ```r # 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 ``` ## 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` ## Author(s) Lopes, J. E.

qkw function