# Extremal or Maximally Skew Stable Distributions 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). ```r 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. ## Returns `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 ```r 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) } ```

Estable function