djsbb: Computing the probability density function of bivariate...

In ForestFit: Statistical Modelling for Plant Size Distributions

Computing the probability density function of bivariate Johnson's SB (JSBB) distribution

Description

Computes the probability density function of the 9-parameter JSBB distibution given by

f_{Y_{1},Y_{2}}\bigl(y_1,y_2\big \vert\Theta\bigr) = f_{Y_1, Y_2}(y_1, y_2) =\frac{\delta_1\delta_2\lambda_1\lambda_2\exp\Bigl\{\frac{-z^{2}_{1}-z^{2}_{2} +2\rho z_{1}z_{2}}{2(1-\rho^2)}\Bigr\}}{2\pi \sqrt{1-\rho^2}\bigl(y_1-\xi_1\bigr)\bigl(y_2-\xi_2\bigr)\bigl(\lambda_1+\xi_1-y_1\bigr)\bigl(\lambda_2+\xi_2-y_2\bigr)},

where

z_{i}=\delta_i \log \Bigl(\frac{y_{i}-{\xi}_i}{{\xi}_i+{\lambda}_i-y_{i}}\Bigr)+\gamma_{i},

for i=1,2. The parameter space of SBB distribution is \Theta=({\bf{\delta}},{\bf{\gamma}},{\bf{\lambda}},{\bf{\xi}}, \rho)^{\top} in which {\bf{\delta}}=(\delta_1,\delta_2)^{\top}, {\bf{\gamma}}=(\gamma_1,\gamma_2, \rho)^{\top}, {\bf{\lambda}}=(\lambda_1,\lambda_2)^{\top}, and {\bf{\xi}}=(\xi_1,\xi_2)^{\top}. The supports of marginals are \xi_1<y_1<\lambda_1+\xi_1 and \xi_2<y_2<\lambda_2+\xi_2. The support of the parameter space is \delta_1>0,\delta_2>0,-\infty<\gamma_1<+\infty,-\infty<\gamma_2<+\infty, \lambda_1>0,\lambda_2>0, -\infty<\xi_1<+\infty, -\infty<\xi_2<+\infty and -1<\rho<+1.

Usage

djsbb(data, param, log = FALSE)

Arguments

data	Vector of observations.
param	Vector of the parameters {\bf{\delta}}, {\bf{\gamma}}, {\bf{\lambda}}, {\bf{\xi}}, \rho.
log	If TRUE, then log of density function is returned.

Value

A vector of length n, giving the density function of JSBB distribution.

Author(s)

Mahdi Teimouri

Examples

Delta <- c(2.5, 3)
Gamma <- c(2, 1)
Lambda <- c(1, 3)
Xi <- c(0, 2)
rho <- -0.5
param <- c(Delta[1], Gamma[1], Lambda[1], Xi[1], Delta[2], Gamma[2], Lambda[2], Xi[2], rho)
data <- rjsbb(20, param)
djsbb(data, param, log = FALSE)

ForestFit documentation built on April 3, 2025, 5:27 p.m.