PosRatioDist (version 1.2.1)

dBicauchyPR: BicauchyPR

Description

probability density function of quotient of Bivariate cauchy random variables conditioned to the positive quadrant.For more detailed information please read the first reference paper.

Usage

dBicauchyPR(x, a, b)

Arguments

x

single real positive scalar

a

parameter for bivaraite cauchy distribution

b

parameter for bivaraite cauchy distribution

Value

dBicauchyPR gives the probability density function for quotient of Bivariate cauchy random variables conditioned to the positive quadrant.

Invalid arguments will return an error message.

Details

Probability density function $$f_R (r \mid X > 0, Y > 0) =\frac {1}{2 \pi \Pr (X > 0, Y > 0)}J_1 \left( r^2 + 1, A r + B, C, \frac {3}{2} \right)$$

For \(-\infty < x < \infty\),\(-\infty < y < \infty,r > 0,-\infty < a < \infty,-\infty < b < \infty\),where \(A = -2 a, B = -2 b,C = 1 + a^2 + b^2\) and \(J_1\) is given by first reference paper section (2.5).

References

Yuancheng Si and Saralees Nadarajah and Xiaodong Song, (2020). On the distribution of quotient of random variables conditioned to the positive quadrant. Communications in Statistics - Theory and Methods, 49, pp2514-2528.

Balakrishnan, N. and Lai, C. -D. (2009).Continuous Bivariate Distributions.Springer Verlag, New York.

Caginalp, C. and Caginalp, G. (2018).The quotient of normal random variables and application to asset price fat tails.Physica A---Statistical Mechanics and Its Applications, 499, pp457-471.

Louzada, F., Ara, A. and Fernandes, G. (2017).The bivariate alpha-skew-normal distribution.Communications in Statistics - Theory and Methods, 46, pp7147-7156.

Nadarajah, S. (2009).A bivariate Pareto model for drought.Stochastic Environmental Research and Risk Assessment, 23, pp811-822.

Nadarajah, S. and Kotz, S. (2006).Reliability models based on bivariate exponential distributions.Probabilistic Engineering Mechanics, 21, pp338-351.

Nadarajah, S. and Kotz, S. (2007).Financial Pareto ratios.Quantitative Finance, 7, pp257-260.

Examples

# NOT RUN {
x <- seq(0.1,5,0.1)
y <- c()
for (i in x){y=c(y,dBicauchyPR(i,1,2))}
plot(x,y,type = 'l')






# }