Sibuya Distribution - Sampling and Probabilities
The Sibuya distribution Sib(alpha) can be defined by its Laplace transform
1-(1-exp(-t))^alpha, t in [0,Inf),
its distribution function
F(k)=1-(-1)^k * choose(alpha-1,k)=1-1/(k * beta(k, 1-alpha)), k in N,
(where beta denotes the beta function) or its probability mass function
p_k = choose(alpha,k)*(-1)^(k-1), k in N,
where alpha in (0,1].
pSibuya evaluates the distribution function.
dSibuya evaluates the probability mass function.
rSibuya generates random variates from
Sib(alpha) with
the algorithm described in Hofert (2011), Proposition 3.2.
dsumSibuya gives the probability mass function of the
n-fold convolution of Sibuya variables, that is, the sum of n
independent Sibuya random variables,
S = sum(i=1..n) X[i], where
X[i] ~ Sib(alpha).
This probability mass function can be shown (see Hofert (2010, pp. 99)) to be
sum(j=1..n) choose(n,j) * choose(j*alpha,k) * (-1)^(k-j), k >= n.
rSibuya(n, alpha)
dSibuya(x, alpha, log=FALSE)
pSibuya(x, alpha, lower.tail=TRUE, log.p=FALSE)
dsumSibuya(x, n, alpha,
method=c("log", "direct", "diff", "exp.log",
"Rmpfr", "Rmpfr0", "RmpfrM", "Rmpfr0M"),
mpfr.ctrl = list(minPrec = 21, fac = 1.25, verbose=TRUE),
log=FALSE)n |
for |
alpha |
parameter in (0,1]. |
x |
vector of |
log, log.p |
|
lower.tail |
|
method |
character string specifying which computational method is to be applied. Implemented are:
|
mpfr.ctrl |
for |
The Sibuya distribution has no finite moments, that is, specifically infinite mean and variance.
For documentation and didactical purposes, rSibuyaR is a pure-R
implementation of rSibuya, of course slower than rSibuya
as the latter is implemented in C.
Note that the sum to evaluate for dsumSibuya is numerically
highly challenging, even already for small
alpha values (for example, n >= 10),
and therefore should be used with care. It may require high-precision
arithmetic which can be accessed with method="Rmpfr" (and the
Rmpfr package).
A vector of positive integers of
length n containing the generated random variates.
a vector of
probabilities of the same length as x.
a vector of probabilities, positive if and only if
x >= n and of the same length as x (or n if
that is longer).
Hofert, M. (2010). Sampling Nested Archimedean Copulas with Applications to CDO Pricing. Südwestdeutscher Verlag fuer Hochschulschriften AG & Co. KG.
Hofert, M. (2011). Efficiently sampling nested Archimedean copulas. Computational Statistics & Data Analysis 55, 57–70.
## Sample n random variates from a Sibuya(alpha) distribution and plot a ## histogram n <- 1000 alpha <- .4 X <- rSibuya(n, alpha) hist(log(X), prob=TRUE); lines(density(log(X)), col=2, lwd=2)
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