Estable {FMStable}

Extremal or Maximally Skew Stable Distributions

Description

Density function, distribution function, quantile function and random generation for stable distributions which are maximally skewed to the right. These distributions are called Extremal by Zolotarev (1986).

Usage

  dEstable(x, stableParamObj, log=FALSE) 
  pEstable(x, stableParamObj, log=FALSE, lower.tail=TRUE)
  qEstable(p, stableParamObj, log=FALSE, lower.tail=TRUE)
  tailsEstable(x, stableParamObj)

Arguments

x

Vector of quantiles.

stableParamObj

An object of class stableParameters which describes a maximally skew stable distribution. It may, for instance, have been created by setParam or setMomentsFMstable.

p

Vector of tail probabilities.

log

Logical; if TRUE, the log density or log tail probability is returned by functions dEstable and pEstable; and logarithms of probabilities are input to function qEstable.

lower.tail

Logical; if TRUE, the lower tail probability is returned. Otherwise, the upper tail probability.

Details

The values are worked out by interpolation, with several different interpolation formulae in various regions.

Value

dEstable gives the density function; pEstable gives the distribution function or its complement; qEstable gives quantiles; tailsEstable returns a list with the following components which are all the same length as x:

density

The probability density function.

F

The probability distribution function. i.e. the probability of being less than or equal to x.

righttail

The probability of being larger than x.

logdensity

The probability density function.

logF

The logarithm of the probability of being less than or equal to x.

logrighttail

The logarithm of the probability of being larger than x.

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, 71, 340–344.

See Also

If x has an extremal stable distribution then \exp(-x) has a finite moment log stable distribution. The left hand tail probability computed using pEstable should be the same as the coresponding right hand tail probability computed using pFMstable.

Aspects of extremal stable distributions may also be computed (though more slowly) using tailsGstable with beta=1.

Functions for generation of random variables having stable distributions are available in package stabledist.

Examples

tailsEstable(-2:3, setMomentsFMstable(mean=1, sd=1.5, alpha=1.7))

# Compare Estable and FMstable
obj <- setMomentsFMstable(1.7, mean=.5, sd=.2)
x <- c(.001, 1, 10)
pFMstable(x,  obj, lower.tail=TRUE, log=TRUE)
pEstable(-log(x), obj, lower.tail=FALSE, log=TRUE)

x <- seq(from=-5, to=10, length=30)
plot(x, dEstable(x, setMomentsFMstable(alpha=1.5)), type="l", log="y",
  ylab="Log(density) for stable distribution",
  main="log stable distribution with alpha=1.5, mean=1, sd=1" )

x <- seq(from=-2, to=5, length=30)
plot(x, x, ylim=c(0,1), type="n", ylab="Distribution function")
for (i in 0:2)lines(x, pEstable(x,
  setParam(location=0, logscale=-.5, alpha=1.5, pm=i)), col=i+1)
legend("bottomright", legend=paste("S", 0:2, sep=""), lty=rep(1,3), col=1:3)

p <- c(1.e-10, .01, .1, .2, .5, .99, 1-1.e-10)
obj <- setMomentsFMstable(alpha=1.95)
result <- qEstable(p, obj)
pEstable(result, obj) - p

# Plot to illustrate continuity near alpha=1
y <- seq(from=-36, to=0, length=30)
logprob <- -exp(-y)
plot(0, 0, type="n", xlim=c(-25,0), ylim=c(-35, -1),
  xlab="x (M parametrization)", ylab="-log(-log(distribution function))")
for (oneminusalpha in seq(from=-.2, to=0.2, by=.02)){
  obj <- setParam(oneminusalpha=oneminusalpha, location=0, logscale=0, pm=0)
  type <- if(oneminusalpha==0) 2 else 1
  lines(qEstable(logprob, obj, log=TRUE), y, lty=type, lwd=type)
}