Description Usage Arguments Details Value References See Also Examples

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).

1 2 3 4 |

`x` |
Vector of quantiles. |

`stableParamObj` |
An object of class |

`p` |
Vector of tail probabilities. |

`log` |
Logical; if |

`lower.tail` |
Logical; if |

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

`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`

.

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.

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`

.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 | ```
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)
}
``` |

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