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.
dBitPR(x, a, b, rho, v)single positive scalar,for quotient of Bivariate t random variables conditioned to the positive quadrant
parameter for Bivariate t distribution
parameter for Bivariate t distribution
correlation coefficient,\(-1<\rho<1\)
parameter, degree of freedom of Bivariate t distribution
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.
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).
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.
# NOT RUN {
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')
# }
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