besselRatio function

Ratio of Bessel K Functions

Calculates the ratio of Bessel K functions of different orders, but the same value of the argument.

besselRatio(x, nu, orderDiff, useExpScaled = 700)

Arguments

• x: Numeric, $>= 0$. Value at which the numerator and denominator Bessel functions are evaluated.
• nu: Numeric. The order of the Bessel function in the denominator.
• orderDiff: Numeric. The order of the numerator Bessel function minus the order of the denominator Bessel function.
• useExpScaled: Numeric, $>= 0$. The smallest value of $x$ for which the ratio is calculated using the exponentially-scaled Bessel function values.

Returns

The ratio

of two modified Bessel functions of the third kind whose orders differ by $k$.

Details

Uses the function besselK to calculate the ratio of two modified Bessel function of the third kind whose orders are different. The calculation of Bessel functions will underflow if the value of $x$ is greater than around 740. To avoid underflow the exponentially-scaled Bessel functions can be returned by besselK. The ratio is actually unaffected by exponential scaling since the scaling cancels across numerator and denominator.

The Bessel function ratio is useful in calculating moments of the generalized inverse Gaussian distribution, and hence also for the moments of the hyperbolic and generalized hyperbolic distributions.

Author(s)

David Scott d.scott@auckland.ac.nz

See Also

besselK, gigMom

Examples

nus <- c(0:5, 10, 20)
x <- seq(1, 4, length.out = 11)
k <- 3

raw <- matrix(nrow = length(nus), ncol = length(x))
scaled <- matrix(nrow = length(nus), ncol = length(x))
compare <- matrix(nrow = length(nus), ncol = length(x))

for (i in 1:length(nus)){
    for (j in 1:length(x)) {
        raw[i,j] <- besselRatio(x[j], nus[i],
                                orderDiff = k)
        scaled[i,j] <- besselRatio(x[j], nus[i],
                                orderDiff = k, useExpScaled = 1)
        compare[i,j] <- raw[i,j]/scaled[i,j]
    }
}
raw
scaled
compare