Distributions {dwp}

Probability Distributions for Carcasses Versus Distance from Turbine

Description

PDFs and CDFs that are required by ddd, pdd and qdd but are not included among the standard R distributions. Relying on custom code and included here are the Maxwell-Boltzmann (pmb and dmb), xep0 (Pareto), xep1, xepi0 (inverse gamma), xep2 (Rayleigh), xep02, xep12, xep012, xep123, and xep0123. Not included here are the distributions that can be calculated using standard probability functions from base R, namely the exponential, truncated normal, lognormal, gamma (xep01), and chisquared distributions and the inverse gaussian, which is calculated using statmod::dinvgauss and statmod::dinvgauss. The functions are designed for vector x or q and scalar parameters.

Usage

dmb(x, a)

pmb(q, a)

dxep1(x, b1)

pxep1(q, b1)

pxep02(q, b0, b2)

dxep02(x, b0, b2)

dxep12(x, b1, b2)

pxep12(x, b1, b2)

dxep123(x, b1, b2, b3, const = NULL)

pxep123(x, b1, b2, b3, const = NULL)

dxepi0(x, shape, scale)

pxepi0(x, shape, scale)

dxep0123(x, b0, b1, b2, b3, const = NULL)

pxep0123(x, b0, b1, b2, b3, const = NULL)

dxep012(x, b0, b1, b2, const = NULL)

pxep012(x, b0, b1, b2, const = NULL)

dxep2(x, s2)

pxep2(x, s2)

dxep0(x, a)

pxep0(x, a)

Arguments

x, q

vector of distances

a, b0, b1, b2, b3, shape, scale, s2

parameters used in the respective distributions.

const

(optional) scalar normalizing constant for distributions that are numerically integrated using integrate, namely. Providing a const is not necessary but will improve the speed of calculation under certain conditions.

Details

An xep distribution is calculated by dividing its kernel (for the densities) or the integral of its kernel (for the cumulative distributions) by the normalizing constant, which is the integral of the kernel from 0 to Inf. The kernel of an xep distribution is defined as x e^{P(x)}, where P(x) is a polynomial with terms defined by the suffix on xep. For example, the kernel of xep12 would be x e^{b_1*x + b_2*x^2}. A 0 in the suffix indicates a log(X) term and an i indicates a 1/x term. The parameters of the xep distributions are some combination of b_0, b_1, b_2, b_3. The parameterizations of the inverse gamma (xepi0), Rayleigh (xep2), and Pareto (xep0) follow the standard conventions of shape and scale for the inverse gamma, s2 = s^2 for the Rayleigh, and a = a for the Pareto (with a scale or location parameter of 1 and PDF = a/x^(x + 1) with support (1, Inf).

The Maxwell-Boltzmann is a one-parameter family with parameter a and PDF f(a) = \sqrt{2/\pi}\frac{x^2 e^{-x^2/(2a^2)}}{a^3}. The kernel is f(a) = x^2 e^{-x^2}, which has a simple closed-form integral that involves the error function (pracma::erf).

Value

vector of probability densities or cumulative probabilities