Function to return delta as used to estimate confidence intervals for flood frequency analysis

Delta.r

# Function to return delta as used to estimate
# confidence intervals for flood frequency analysis
 
# Usage
#   Delta(g, aep)
#
# g = skewness
# aep = Annual exceedance probability
 
#
# Implementing equation 9-54 from Kite (2004) 
# Frequency and risk analysis in hydrology.  Water Resources Publications
# and Equation 2.17 Book IV Australian Rainfall and Runoff
 
 
Delta <- function(g, aep){
  
  require(lmomco)
  require(numDeriv)
  
  Ky_Qp3 <- function(g, aep){ 
    # Return the quantile of the Pearson III distribution given 
    # skewness of g, and annual exceedance probability, aep
    
    param <- vec2par(c(0,1,g), type='pe3')
    quape3(1 - aep, param)  
  }
  
  
  K <- Ky_Qp3(g, aep)
  dK <- grad(Ky_Qp3, g, method="Richardson", aep = aep)
  sqrt(1 + K*g + 0.5*(K^2)*(0.75*(g^2) + 1) + 3*K*dK*(g + 0.25*g^3) + 3*(dK^2)*(2 + 3*g^2 + (5/8)*g^4))
}