Probability density and distribution functions for Polynomial-Tail Laplace distribution
dPTL(x,alpha,beta,gamma)
pPTL(q,alpha,beta,gamma)Numeric vector of quantiles
Linear tail adjustment coefficient for PTL distribution
Exponential decay term for PTL distribution, similar to beta parameter in Laplace distribution
Polynomial tail adjustment coefficient for PTL distribution
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.
The PTL distribution has density $$ f(x) = \left\{\begin{array}{cc} 0 & x < -2\\ \displaystyle \frac{\alpha(\frac{x^2}{2}+2x+2) + \beta(e^{\frac{x}{\beta}}-e^{\frac{-2}{\beta}}) + \gamma(\frac{x^3}{3}+4x+\frac{16}{3})}{4\alpha + 2\beta(1-e^{\frac{-2}{\beta}}) + \frac{32\gamma}{3}} & -2 \leq x \leq 0\\ \displaystyle \frac{\alpha(2x-\frac{x^2}{2}-2) + \beta(e^{\frac{-2}{\beta}}-e^{\frac{x}{\beta}}) + \gamma(4x-\frac{x^3}{3}-\frac{16}{3})}{4\alpha + 2\beta(1-e^{\frac{-2}{\beta}}) + \frac{32\gamma}{3}} & 0 < x \leq 2\\ 1 & x > 2 \end{array}\right. $$