# PTL: Polynomial-Tail Laplace

Description Usage Arguments Details Value Author(s)

### 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

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