ecd (version 0.9.2.4)

dstablecnt: Stable Count distribution

Description

Implements the stable count distribution (based on stabledist package) for stable random walk simulation. Quartic stable distribution is implemented through gamma distribution.

Usage

dstablecnt(x, alpha = NULL, nu0 = 0, theta = 1, lambda = NULL)
pstablecnt(x, alpha = NULL, nu0 = 0, theta = 1, lambda = NULL)
rstablecnt(n, alpha = NULL, nu0 = 0, theta = 1, lambda = NULL)
qstablecnt(q, alpha = NULL, nu0 = 0, theta = 1, lambda = NULL)
cfstablecnt(s, alpha = NULL, nu0 = 0, theta = 1, lambda = NULL)
kstablecnt(alpha = NULL, nu0 = 0, theta = 1, lambda = NULL)

Arguments

x

numeric, vector of responses.

alpha

numeric, the shape parameter, default is NULL. User must provide either alpha or lambda.

nu0

numeric, the location parameter, default is 0.

theta

numeric, the scale parameter, default is 1.

lambda

numeric, alternative shape parameter, default is NULL.

n

numeric, number of observations.

q

numeric, vector of quantiles.

s

numeric, vector of responses for characteristic function.

Value

numeric, standard convention is followed: d* returns the density, p* returns the distribution function, q* returns the quantile function, and r* generates random deviates. The following are our extensions: k* returns the first 4 cumulants, skewness, and kurtosis, cf* returns the characteristic function.

Details

The stable count distribution is the conjugate prior of the stable distribution. The density function is defined as $$ \mathit{N}_{\alpha}\left(\nu;\nu_{0},\theta\right) \equiv\frac{\alpha}{\mathit{\Gamma}\left(\frac{1}{\alpha}\right)}\, \frac{1}{\nu-\nu_{0}}\,L_{\alpha}\left(\frac{\theta}{\nu-\nu_{0}}\right), \:\mathrm{where}\,\nu>\nu_{0}. $$ where \(\nu>\nu_0\). \(\alpha\) is the stability index, \(\nu_0\) is the location parameter, and \(\theta\) is the scale parameter.

At \(\alpha=0.5\) aka \(\lambda=4\), it is called "quartic stable count distribution", which is a gamma distribution with shape of 3/2. It has the closed form of $$ \mathit{N}_{\frac{1}{2}}\left(\nu;\nu_{0},\theta\right) \equiv\frac{1}{4\sqrt{\pi}\,\theta^{3/2}} \left(\nu-\nu_{0}\right)^{\frac{1}{2}} e^{-\frac{\nu-\nu_{0}}{4\theta}} $$

References

For more detail, see Section 2.4 and Section 3.3 of Stephen Lihn (2017). A Theory of Asset Return and Volatility under Stable Law and Stable Lambda Distribution. SSRN: 3046732, https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3046732. This distribution is also documented formally in Wikipedia: https://en.wikipedia.org/wiki/Stable_count_distribution.

Examples

# NOT RUN {
  # generate the pdf of the VIX distribution
  x <- c(0, 100, by=0.1)
  pdf <- dstablecnt(x, nu0=10.4, theta=1.6, lambda=4)
  
# }