moments {FMStable}

Convolutions of Finite Moment Log Stable Distributions and the Moments of such Distributions

Description

If X_1, \dots , X_n are independent random variables with the same stable distribution then X_1+ \dots +X_n has a stable distribution with the same alpha. The function iidcombine allows the parameters of the resulting stable distribution to be computed. Because stable distributions are infinitely divisible, it is also easy to find the parameters describing the distribution of X_1 from the parameters describing the distribution of X_1+ \dots +X_n.

Convolutions of maximally skew stable distributions correspond to products of logstable distributions. The raw moments of these distributions (i.e. moments about zero, not moments about the mean) can be readily computed using the function moments. Note that the raw moments of the convolution of two independent distributions are the products of the corresponding moments of the component distributions, so the accuracy of iidcombine can be checked by using moments.

Usage

iidcombine(n, stableParamObj)
moments(powers, stableParamObj, log=FALSE)

Arguments

n

Number of random variables to be convoluted. May be any positive number.

powers

Raw moments of logstable distributions to be computed.

stableParamObj

An object of class stableParameters which describes a maximally skew stable distribution.

log

Logical; if TRUE, the logarithms of moments are returned.

Value

The value returned by iidcombine is another object of class stableParameters. The value returned by moments is a numeric vector giving the values of the specified raw moments.

See Also

Objects of class stableParameters can be created using functions such as setParam. The taking of convolutions is sometimes associated with the computing of values of options using functions such as callFMstable.

Examples

yearDsn <- fitGivenQuantile(mean=1, sd=2, prob=.7, value=.1)
upper <- exp(-yearDsn$location)	# Only sensible for alpha<.5
x <- exp(seq(from=log(.0001), to=log(upper), length=50))
plot(x, pFMstable(x, yearDsn), type="l", ylim=c(.2,1), lwd=2, xlab="Price",
  ylab="Distribution function of future price")
half <- iidcombine(.5, yearDsn)
lines(x, pFMstable(x, half), lty=2, lwd=2)
quarter <- iidcombine(.25, yearDsn)
lines(x, pFMstable(x, quarter), lty=3, lwd=2)
legend("bottomright", legend=paste(c("1","1/2","1/4"),"year"), lty=c(1,2,3),
  lwd=c(2,2,2))
moments(1:2, yearDsn)
moments(1:2, half)
moments(1:2, quarter)

# Check logstable against lognormal
iidcombine(2, setMomentsFMstable(.5, .2, alpha=2))
p <- lnorm.param(.5, .2)
2*p$meanlog		# Gives the mean
log(p$sdlog)	# Gives the logscale