R/RLambda.R

In HyperbolicDist: The Hyperbolic Distribution

### RLambda & SLambda - functions used in calculation of the
### theoretical mean and variance of a hyperbolic distribution 
### (as given in Barndorff-Nielsen & Blaesild, 1983).
RLambda <- function(zeta, lambda = 1){
  besselK(zeta, lambda + 1)/besselK(zeta, lambda)
}

SLambda <- function(zeta,lambda = 1){
  (besselK(zeta, lambda + 2)*besselK(zeta, lambda) -
   besselK(zeta, lambda + 1)^2)/besselK(zeta, lambda)^2
}

MLambda <- function(zeta, lambda = 1){
  besselK(zeta, lambda + 2)/besselK(zeta, lambda)
}


### WLambda1, WLambda2, WLambda3 and WLambda4 from
### Barndorff-Nielsen and Blaesild (1981), p. 41
WLambda1 <- function(zeta, lambda = 1){
  RLambda(zeta, lambda)
}

WLambda2 <- function(zeta, lambda = 1){
  -RLambda(zeta, lambda)^2 + 2*(lambda + 1)*RLambda(zeta, lambda)/zeta + 1 
}

WLambda3 <- function(zeta, lambda = 1){
  2*RLambda(zeta, lambda)^3 - 6*(lambda + 1)*RLambda(zeta, lambda)^2/zeta + 
    (4*(lambda + 1)*(lambda + 2)/zeta^2 - 2)*RLambda(zeta, lambda) + 
    2*(lambda + 2)/zeta 
}

WLambda4 <- function(zeta, lambda = 1){
  -6*RLambda(zeta, lambda)^4 + 24*(lambda + 1)*RLambda(zeta, lambda)^3/zeta + 
    (-4*(lambda + 1)*(7*lambda + 11)/zeta^2 + 8)*RLambda(zeta, lambda)^2 +
    (8*(lambda + 1)*(lambda + 2)*(lambda + 3)/zeta^3 - 4*(4*lambda +5)/zeta)*
    RLambda(zeta, lambda) + 4*(lambda + 2)*(lambda + 3)/zeta^2 - 2 
}

### Theoretical skewness and kurtosis
### Barndorff-Nielsen and Blaesild (1981), p.42
gammaLambda1 <- function(hyperbPi, zeta, lambda = 1){
  (hyperbPi^3*WLambda3(zeta, lambda) + 3*hyperbPi*WLambda2(zeta, lambda)/zeta)/
    (hyperbPi^2*WLambda2(zeta, lambda) + WLambda1(zeta)/zeta)^(3/2)
}

gammaLambda2 <- function(hyperbPi, zeta, lambda = 1){
  (hyperbPi^4*WLambda4(zeta, lambda) + 
  6*hyperbPi^2*WLambda3(zeta, lambda)/zeta + 
  3*WLambda2(zeta, lambda)/zeta^2)/
    (hyperbPi^2*WLambda2(zeta, lambda) + WLambda1(zeta)/zeta)^2
}