approxbvncdf: APPROXIMATION OF BIVARIATE STANDARD NORMAL DISTRIBUTION
In weightedScores: Weighted Scores Method for Regression Models with Dependent Data

Description Usage Arguments Details Value References See Also

Approximation of bivariate standard normal cumulative distribution function (Johnson and Kotz, 1972).

1	approxbvncdf(r,x1,x2,x1s,x2s,x1c,x2c,x1f,x2f,t1,t2)

`r`	The correlation parameter of bivariate standard normal distribution.
`x1`	x_1, see details.
`x2`	x_2, see details.
`x1s`	x_1^2.
`x2s`	x_2^2.
`x1c`	x_1^3.
`x2c`	x_2^3.
`x1f`	x_1^4.
`x2f`	x_2^4.
`t1`	Φ(x_1)Φ(x_2), where Φ(\cdot) is the cdf of univariate standard normal distribution.
`t2`	φ(x_1)φ(x_2), where φ(\cdot) is the density of univariate stamdard normal distribution.

The approximation for the bivariate normal cdf is from Johnson and Kotz (1972), page 118. Let Φ_2(x_1,x_2;ρ)=Pr(Z_1≤ x_1,\,Z_2≤ x_2), where (Z_1,Z_2) is bivariate normal with means 0, variances 1 and correlation ρ. An expansion, due to Pearson (1901), is

Φ_2(x_1,x_2;ρ) =Φ(x_1)Φ(x_2) +φ(x_1)φ(x_2) ∑_{j=1}^∞ ρ^j ψ_j(x_1) ψ_j(x_2)/j!

where

ψ_j(z) = (-1)^{j-1} d^{j-1} φ(z)/dz^{j-1}.

Since

φ'(z) = -zφ(z), φ''(z) = (z^2-1)φ(z) , φ'''(z) = [2z-z(z^2-1)]φ(z) = (3z-z^3)φ(z) ,

φ^{(4)}(z) = [3-3z^2-z(3z-z^3)]φ(z) = (3-6z^2+z^4)φ(z)

we have

Φ_2(x_1,x_2;ρ) = Φ(x_1)Φ(x_2)+φ(x_1)φ(x_2) [ρ+ ρ^2x_1x_2/2 + ρ^3 (x_1^2-1)(x_2^2-1)/6 +ρ^4 (x_1^3-3x_1)(x_2^3-3x_2)/24

+ρ^5 (x_1^4-6x_1^2+3)(x_2^4-6x_2^2+3)/120+\cdots ]

A good approximation is obtained truncating the series at ρ^3 term for |ρ| ≤ 0.4, and at ρ^5 term for 0.4 < |ρ|≤ 0.7. Higher order terms may be required for |ρ| > 0.7.

An approximation of bivariate normal cumulative distribution function.

Johnson, N. L. and Kotz, S. (1972) Continuous Multivariate Distributions. Wiley, New York.

Pearson, K. (1901) Mathematical contributions to the theory of evolution-VII. On the correlation of characters not quantitatively measureable. Philosophical Transactions of the Royal Society of London, Series A, 195, 1–47.

scoreCov

weightedScores documentation built on March 24, 2020, 1:07 a.m.