I_1 {PosRatioDist}

Lemma

Description

Technical Lemmas for calculating quotient of random variables conditioned to the positive quadrant.For more detailed information please read the first reference paper section 2.2.

Usage

I_1(a, b)

I_2(a, b)

I_3(a, b)

J_1(a, b, c, alpha)

J_2(a, b, c, alpha)

J_3(a, b, c, alpha)

Arguments

a

parameter

b

parameter

c

parameter

alpha

parameter

Details

I_n Type I Integration

I_n (a, b) = \int_0^\infty y^n \exp \left( -a y^2 - b y \right) dy

For -\infty < a < \infty,-\infty < b < \infty,where n is positive integer.

In particular,for a > 0,we have expressions below

I_1 (a, b) = -\frac {\sqrt{\pi} b}{4 a^{3 / 2}}\exp \left( \frac {b^2}{4 a} \right)\ {\rm erfc} \left( \frac {b}{2 \sqrt{a}} \right) + \frac {1}{2 a}

I_2 (a, b) = \frac {\sqrt{\pi}}{4 a^{3 / 2}}\exp \left( \frac {b^2}{4 a} \right)\ {\rm erfc} \left( \frac {b}{2 \sqrt{a}} \right) +\frac {\sqrt{\pi} b^2}{8 a^{5 / 2}} \exp \left( \frac {b^2}{4 a} \right)\ {\rm erfc} \left( \frac {b}{2 \sqrt{a}} \right) - \frac {b}{4 a^2}

I_3 (a, b) = -\frac {3 \sqrt{\pi} b}{8 a^{5 / 2}}\exp \left( \frac {b^2}{4 a} \right)\ {\rm erfc} \left( \frac {b}{2 \sqrt{a}} \right) -\frac {\sqrt{\pi} b^3}{16 a^{7 / 2}}\exp \left( \frac {b^2}{4 a} \right)\ {\rm erfc} \left( \frac {b}{2 \sqrt{a}} \right) + \frac {1}{2 a^2} + \frac {b^2}{8 a^3}

J_n Type J Integration

J_n (a, b, c, \alpha) = \int_0^\infty y^n \left( a y^2 + b y + c \right)^{-\alpha} dy

In particular,for a > 0,b^2 < 4ac, -1 < n < 2\alpha - 1,we have expressions below

J_1 (a, b, c, \alpha) = a^{-1} c^{1 - \alpha} B \left( 2, 2 \alpha - 2 \right) \ {}_2F_1 \left( 1, \alpha - 1; \alpha + \frac {1}{2}; 1 - \frac {b^2}{4 a c} \right)

J_2 (a, b, c, \alpha) = a^{-\frac {3}{2}} c^{\frac {3}{2} - \alpha} B \left( 3, 2 \alpha - 3 \right) \ {}_2F_1 \left( \frac {3}{2}, \alpha - \frac {3}{2}; \alpha + \frac {1}{2}; 1 - \frac {b^2}{4 a c} \right)

J_3 (a, b, c, \alpha) = a^{-2} c^{2 - \alpha} B \left( 4, 2 \alpha - 4 \right) \ {}_2F_1 \left( 2, \alpha - 2; \alpha + \frac {1}{2}; 1 - \frac {b^2}{4 a c} \right)

Value

I_1 gives value of Type I integration with n = 1

I_2 gives value of Type I integration with n = 2

I_3 gives value of Type I integration with n = 3

J_1 gives value of Type J integration with n = 1

J_2 gives value of Type J integration with n = 2

J_3 gives value of Type J integration with n = 3

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

I_1(1,2)
I_2(1,2)
I_3(1,2)
J_1(1,2,3,3)
J_2(1,2,3,3)
J_3(1,2,3,3)