Polynomial-Tail Laplace

Description

Probability density and distribution functions for Polynomial-Tail Laplace distribution

Usage

1
2
dPTL(x,alpha,beta,gamma)
pPTL(q,alpha,beta,gamma)

Arguments

x,q

Numeric vector of quantiles

alpha

Linear tail adjustment coefficient for PTL distribution

beta

Exponential decay term for PTL distribution, similar to beta parameter in Laplace distribution

gamma

Polynomial tail adjustment coefficient for PTL distribution

Details

The PTL distribution has density

f(x) = ≤ft\{\begin{array}{cc} 0 & x < -2\\ \displaystyle \frac{α(\frac{x^2}{2}+2x+2) + β(e^{\frac{x}{β}}-e^{\frac{-2}{β}}) + γ(\frac{x^3}{3}+4x+\frac{16}{3})}{4α + 2β(1-e^{\frac{-2}{β}}) + \frac{32γ}{3}} & -2 ≤q x ≤q 0\\ \displaystyle \frac{α(2x-\frac{x^2}{2}-2) + β(e^{\frac{-2}{β}}-e^{\frac{x}{β}}) + γ(4x-\frac{x^3}{3}-\frac{16}{3})}{4α + 2β(1-e^{\frac{-2}{β}}) + \frac{32γ}{3}} & 0 < x ≤q 2\\ 1 & x > 2 \end{array}\right.

Value

dnorm gives the density, pnorm gives the distribution function.

The length of the result is the maximum of the lengths of the numerical parameters for the other functions. The numerical parameters are recycled to the length of the result.

Author(s)

Ed Curry e.curry@imperial.ac.uk