dTLCAR: Probability Density Function (PDF) for the TLCAR Distribution
Description
Calculate the probability density at a given value using the TLCAR distribution.
Usage
dTLCAR(x, alpha, a, b, theta, m)
Value
Probability density at the given value.
Arguments
x
Value at which to calculate the PDF.
alpha
Parameter representing the distribution of the Topp-Leone component.
a
Parameter representing the scale (a) of the Cauchy component.
b
Parameter representing the position (b) of the Cauchy component.
theta
Parameter representing the scale of the Rayleigh component.
m
Additional parameter.
Details
The probability density function (PDF) for the TLCAR distribution is defined as follows:
$$f(x)=\frac{2\alpha}{\pi a}\left[\frac{1+\left(\frac{x^2}{\theta^2}-1\right)e^{-\frac{x^2}{2\theta^2}}+m}{1+\left(\frac{x\left(1-e^{-\frac{x^2}{2\theta^2}}+m\right) -b}{a}\right)^2}\right]\left[\frac{1}{2}-\frac{1}{\pi}\arctan\frac{x\left(1-e^{-\frac{x^2}{2\theta^2}}+m\right) -b}{a}\right]\left[ 1-\left(\frac{1}{2}-\frac{1}{\pi}\arctan\frac{x\left(1-e^{-\frac{x^2}{2\theta^2}}+m\right)-b}{a}\right)^2\right]^{\alpha-1}$$