stableParameters {FMStable}

Setting up Parameters to Describe both Extremal Stable Distributions and Finite Moment Log Stable Distributions

Description

Functions which create stable distributions having specified properties. Each of these functions takes scalar arguments and produces a description of a single stable distribution.

Usage

setParam(alpha, oneminusalpha, twominusalpha, location, logscale, pm)
setMomentsFMstable(mean=1, sd=1, alpha, oneminusalpha, twominusalpha)
fitGivenQuantile(mean, sd, prob, value, tol=1.e-10)
matchQuartiles(quartiles, alpha, oneminusalpha, twominusalpha, tol=1.e-10)

Arguments

alpha

Stable distribution parameter which must be a single value satisfying 0 < \alpha <= 2.

oneminusalpha

Alternative specification of stable distribution parameter: Specify 1-alpha.

twominusalpha

Alternative specification of stable distribution parameter: Specify 2-alpha.

location

Location parameter of stable distribution.

logscale

Logarithm of scale parameter of stable distribution.

pm

Parametrization used in specifying stable distribution which is maximally skewed to the right. Allowable values are 0, "S0", "M", 1, "S1", "A", 2, "CMS" or "C" for some common parametrizations.

mean

Mean of logstable distribution.

sd

Standard deviation of logstable distribution.

value, prob

Required probability distribution function (> 0) for a logstable distribution at a value (> 0).

quartiles

Vector of two quartiles to be matched by a logstable distribution.

tol

Tolerance for matching of quantile or quartiles.

Details

The parametrizations used internally by this package are Nolan's "S0" (or Zolotarev's "M") parametrization when alpha >= 0.5, and the Zolotarev's "C" parametrization (which was used by Chambers, Mallows and Struck (1976) when alpha < 0.5.

By using objects of class stableParameters to store descriptions of stable distributions, it will generally be possible to write code in a way which is not affected by the internal representation. Such usage is encouraged.

Value

Each of the functions described here produces an object of class stableParameters which describes a maximally skew stable distribution. Its components include at least the shape parameter alpha, a location parameter referred to as location and the logarithm of a scale parameter referred to as logscale.

Currently objects of this class also store information about how they were created, as well as storing the numbers 1-alpha and 2-alpha in order to improve computational precision.

References

Chambers, J.M., Mallows, C.L. and Stuck, B.W. (1976). A method for simulating stable random variables. Journal of the American Statistical Association, Vol. 71, 340–344.

Nolan, J.P. (2012). Stable Distributions. ISBN 9780817641597

Zolotarev, V.M. (1986). One-Dimensional Stable Distributions. Amer. Math. Soc. Transl. of Math. Monographs, Vol. 65. Amer Math. Soc., Providence, RI. (Original Russian version was published in 1983.)

See Also

Extremal stable distributions with parameters set up using these procedures can be used by functions such as dEstable. The corresponding finite moment log stable distributions can be dealt with using functions such as dFMstable.

Examples

setParam(alpha=1.5, location=1, logscale=-.6, pm="M")
setParam(alpha=.4, location=1, logscale=-.6, pm="M")
setMomentsFMstable(alpha=1.7, mean=.5, sd=.2)
fitGivenQuantile(mean=5, sd=1, prob=.001, value=.01, tol=1.e-10)
fitGivenQuantile(mean=20, sd=1, prob=1.e-20, value=1, tol=1.e-24)
matchQuartiles(quartiles=c(9,11), alpha=1.8)