dBilomaxPR {PosRatioDist}

BilomaxPR

Description

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

Usage

dBilomaxPR(x, a, b, c, alpha, beta, theta)

Arguments

x

single positive scalar for quotient

a

parameter for Bivariate lomax distribution

b

parameter for Bivariate lomax distribution

c

parameter for Bivariate lomax distribution

alpha

parameter for Bivariate lomax distribution

beta

parameter for Bivariate lomax distribution

theta

parameter for Bivariate lomax distribution

Details

Probability density function

f_R (r \mid X > 0, Y > 0) = \frac {c^2 \theta^2 r}{\Pr (X > 0, Y > 0)} J_3 \left( \theta r, \beta - \theta a + \left( \alpha - \theta b \right) r, 1 - \alpha a - \beta b + \theta a b, c + 2 \right) +\frac {c^2 \theta \left[ (\alpha - \theta b) r + \beta - \theta a \right]} {\Pr (X > 0, Y > 0)} J_2 \left( \theta r, \beta - \theta a + \left( \alpha - \theta b \right) r, 1 - \alpha a - \beta b + \theta a b, c + 2 \right) +\frac {c \left[ c (\alpha - \theta b) (\beta - \theta a) + \alpha \beta - \theta \right]}{\Pr (X > 0, Y > 0)}J_1 \left( \theta r, \beta - \theta a + \left( \alpha - \theta b \right) r,1 - \alpha a - \beta b + \theta a b, c + 2 \right)

For r > 0,\alpha > 0, \beta > 0, \theta > 0, 0 \leq \theta \leq (c + 1) \alpha \beta where J_1,J_2,J_3 are given by first reference paper section (2.5)

Value

dBilomaxPR gives the probability density function for bivariate lomax 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.