dBitPR {PosRatioDist}

BitPR

Description

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

Usage

dBitPR(x, a, b, rho, v)

Arguments

x

single positive scalar,for quotient of Bivariate t random variables conditioned to the positive quadrant

a

parameter for Bivariate t distribution

b

parameter for Bivariate t distribution

rho

correlation coefficient,-1<\rho<1

v

parameter, degree of freedom of Bivariate t distribution

Details

Probability density function

f_R (r \mid X > 0, Y > 0) =\frac {\Gamma \left( \frac {\nu + 2}{2} \right) \nu^{\frac {\nu}{2}}\left( 1 - \rho^2 \right)^{\frac {\nu + 1}{2}}}{\Gamma \left( \frac {\nu}{2} \right) \pi \Pr (X > 0, Y > 0)}J_1 \left( r^2 - 2 \rho r + 1, A r + B, C + \nu \left( 1 - \rho^2 \right),\frac {\nu}{2} + 1 \right)

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

Value

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

Invalid arguments will return an error message.

Author(s)

Saralees Nadarajah & Yuancheng Si [e-mail]

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.

Arnold, B. C. and Strauss, D. (1988).Pseudolikelihood estimation.Sankhya B , 53, pp233-243.

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

x <- seq(0.1,5,0.1)
y <- c()
for (i in x){y=c(y,dBitPR(i,1,2,0.5,2))}
plot(x,y,type = 'l')