besselK_inc_err: Calculation of the Incomplete BesselK Functions in Terms of...

In frmqa: The Generalized Hyperbolic Distribution, Related Distributions and Their Applications in Finance

Description

Calculates incomplete BesselK functions by evaluating explicit expressions for the lower and upper incomplete BesselK in terms of the complementary error function by calling CalIncLapInt.

Usage

   besselK_inc_err(x, z, lambda, bit, lower = FALSE)

Arguments

x	Argument, x > 0.
z	Argument, z > 0.
lambda	Argument, λ = \pm(j + \frac{1}{2}) with j = 0, 1, 2, ….
lower	Logical. Lower incomplete Bessel function is calculated if TRUE.
bit	Precision bit. A positive integer greater or equal 100.

Details

One of the integral representations of the lower incomplete BesselK is given by

\widehat K_{λ}(z, x) = \frac{1}{≤ft(2z\right)^{λ}}\,\int_0^x\, e^{-≤ft\{z^2\,ξ^2 \,+\, 1/(4\, ξ^2)\right\}}\, ξ^{-2λ -1}\, dξ,

which appears in the distribution function of the generalized inverse Gaussian distribution, see Barndorff-Nielsen(1977).

Note

Currently, analytical formulae for the incomplete BesselK functions are not available for any value of lambda.

Author(s)

Thanh T. Tran [e-mail]

References

Barndorff-Nielsen, O. E (1977) Exponentially decreasing distributions for the logarithm of particle size. Proceedings of the Royal Society of London. Series A, 353, 401–419.

Olver, F.W.J., Lozier, D.W., Boisver, R.F. and Clark, C.W (2010) Handbook of Mathematical Functions. New York: National Institute of Standards and Technology, and Cambridge University Press.

Tran, T. T., Yee, W.T. and Tee, J.G (2012) Formulae for the Extended Laplace Integral and Their Statistical Applications. Working Paper.

Watson, G.N (1931) A Treatise on the Theory of Bessel Functions and Their Applications to Physics. London: MacMillan and Co.

See Also

besselK_inc_clo, gamma_inc_clo, pgig

Examples

  ## Accuracy tests
  x <- 2
  z <- 5
  lambda <- -c(1/2, 3/2)
  lower <- sapply(lambda, function(w.)
    besselK_inc_err(x, z, lambda = w., 200, lower = TRUE))
  upper <- sapply(lambda, function(w.)
    besselK_inc_err(x, z, lambda = w., 200, lower = FALSE)) 
  ## sum of two parts   
  (lower + upper)
  ## equals the whole function
  (besselK(z, nu = lambda))
  

frmqa documentation built on May 2, 2019, 12:22 p.m.