This S4 class is the major object class for elliptic lambda distribution. It stores the ecd parameters, numerical constants that facilitates quadpack integration, statistical attributes, and optionally, an internal structure for the quantile function.
callThe match.call slot
alpha,gamma,sigma,beta,mua length-one numeric. These are core ecd parameters.
cuspa length-one numeric as cusp indicator.
0: not a cusp;
1: cusp specified by alpha;
2: cusp specified by gamma.
lambdaa length-one numeric, the leading exponent for the special model, default is 3.
R,thetaa length-one numeric. These are derived ecd parameters in polar coordinate.
use.mpfrlogical, internal flag indicating whether to use mpfr.
constA length-one numeric as the integral of \(exp(y(x))\) that normalizes the PDF.
const_left_xA length-one numeric marking the left point of PDF integration.
const_right_xA length-one numeric marking the right point of PDF integration.
statsA list of statistics, see ecd.stats for more information.
quantileAn object of ecdq class, for quantile calculation.
modelA vector of four strings representing internal classification:
long_name.skew, codelong_name,
short_name.skew, short_name.
This slot doesn't have formal use yet.
The elliptic lambda distribution is the early research target of what becomes the stable lambda distribution. It is inspired from elliptic curve, and is defined by a depressed cubic polynomial, $$ -y(z)^3 - \left(\gamma+\beta z\right) y(z) + \alpha = z^2, $$ where \(y(z)\) must approach minus infinity as \(z\) approaches plus or minus infinity. The density function is defined as $$ P\left(x; \alpha, \gamma, \sigma, \beta, \mu\right) \equiv\, \frac{1}{C\,\sigma} e^{y\left(\left|\frac{x-\mu}{\sigma}\right|\right)}, $$ and \(C\) is the normalization constant,$$ C = \int_{-\infty}^{\infty}e^{y(z)}\,dz, $$ where \(\alpha\) and \(\gamma\) are the shape parameters, \(\sigma\) is the scale parameter, \(\beta\) is the asymmetric parameter, \(\mu\) is the location parameter.
For elliptic lambda distribution, see Stephen Lihn (2015). On the Elliptic Distribution and Its Application to Leptokurtic Financial Data. SSRN: https://www.ssrn.com/abstract=2697911