Range of a Hyperbolic Distribution

Description

Given the parameter vector param of a hyperbolic distribution, this function calculates the range outside of which the distribution has negligible probability, or the density function is negligible, to a specified tolerance. The parameterization used is the (alpha, beta) one (see dhyperb). To use another parameterization, use hyperbChangePars.

Usage

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hyperbCalcRange(mu = 0, delta = 1, alpha = 1, beta = 0,
                param = c(mu, delta, alpha, beta),
                tol = 10^(-5), density = TRUE, ...)

Arguments

mu

mu is the location parameter. By default this is set to 0.

delta

delta is the scale parameter of the distribution. A default value of 1 has been set.

alpha

alpha is the tail parameter, with a default value of 1.

beta

beta is the skewness parameter, by default this is 0.

param

Value of parameter vector specifying the hyperbolic distribution. This takes the form c(mu, delta, alpha, beta).

tol

Tolerance.

density

Logical. If FALSE, the bounds are for the probability distribution. If TRUE, they are for the density function.

...

Extra arguments for calls to uniroot.

Details

The particular hyperbolic distribution being considered is specified by the value of the parameter value param.

If density = FALSE, the function calculates the effective range of the distribution, which is used in calculating the distribution function and quantiles, and may be used in determining the range when plotting the distribution. By effective range is meant that the probability of an observation being greater than the upper end is less than the specified tolerance tol. Likewise for being smaller than the lower end of the range. Note that this has not been implemented yet.

If density = TRUE, the function gives a range, outside of which the density is less than the given tolerance. Useful for plotting the density.

Value

A two-component vector giving the lower and upper ends of the range.

Author(s)

David Scott d.scott@auckland.ac.nz, Jennifer Tso, Richard Trendall

References

Barndorff-Nielsen, O. and Bl<e6>sild, P (1983). Hyperbolic distributions. In Encyclopedia of Statistical Sciences, eds., Johnson, N. L., Kotz, S. and Read, C. B., Vol. 3, pp. 700–707. New York: Wiley.

See Also

dhyperb, hyperbChangePars

Examples

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par(mfrow = c(1, 2))
param <- c(0, 1, 3, 1)
hyperbRange <- hyperbCalcRange(param = param, tol = 10^(-3))
hyperbRange
curve(phyperb(x, param = param), hyperbRange[1], hyperbRange[2])
maxDens <- dhyperb(hyperbMode(param = param), param = param)
hyperbRange <- hyperbCalcRange(param = param, tol = 10^(-3) * maxDens, density = TRUE)
hyperbRange
curve(dhyperb(x, param = param), hyperbRange[1], hyperbRange[2])

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