dBinormalPR {PosRatioDist}

BinormalPR

Description

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

Usage

dBinormalPR(x, a, b, rho)

Arguments

x

vector of positive quantiles.

a

parameter

b

parameter

rho

correlation coefficient,-1<\rho<1

Details

Probability density function

f_R (r \mid X > 0, Y > 0) =\frac {1}{2 \pi \sqrt{1 - \rho^2} \Pr (X > 0, Y > 0)}\exp \left[ -\frac {a^2 + b^2 - 2 \rho a b}{2 \left( 1 - \rho^2 \right)} \right]I_1 \left( \frac {1 + C r + r^2}{2 \left( 1 - \rho^2 \right)},\frac {A r + B}{2 \left( 1 - \rho^2 \right)} \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 = -2 \rho

Value

dBinormalPR gives the probability density function for quotient of Bivariate normal 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.

Balakrishna, N. and Shiji, K. (2014). On a class of bivariate exponential distributions.Statistics and Probability Letters, 85, pp153-160.

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 <- dBinormalPR(x, 2, 1, 0.5)
plot(x,y,type = 'l')