approxbvncdf {weightedScores}R Documentation

APPROXIMATION OF BIVARIATE STANDARD NORMAL DISTRIBUTION

Description

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

Usage

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

Arguments

r

The correlation parameter of bivariate standard normal distribution.

x1

x1,x_1, see details.

x2

x2,x_2, see details.

x1s

x12.x_1^2.

x2s

x22.x_2^2.

x1c

x13.x_1^3.

x2c

x23.x_2^3.

x1f

x14.x_1^4.

x2f

x24.x_2^4.

t1

Φ(x1)Φ(x2),\Phi(x_1)\Phi(x_2), where Φ()\Phi(\cdot) is the cdf of univariate standard normal distribution.

t2

ϕ(x1)ϕ(x2),\phi(x_1)\phi(x_2), where ϕ()\phi(\cdot) is the density of univariate stamdard normal distribution.

Details

The approximation for the bivariate normal cdf is from Johnson and Kotz (1972), page 118. Let Φ2(x1,x2;ρ)=Pr(Z1x1,Z2x2)\Phi_2(x_1,x_2;\rho)=Pr(Z_1\le x_1,\,Z_2\le x_2), where (Z1,Z2)(Z_1,Z_2) is bivariate normal with means 0, variances 1 and correlation ρ\rho. An expansion, due to Pearson (1901), is

Φ2(x1,x2;ρ)=Φ(x1)Φ(x2)+ϕ(x1)ϕ(x2)j=1ρjψj(x1)ψj(x2)/j! \Phi_2(x_1,x_2;\rho) =\Phi(x_1)\Phi(x_2) +\phi(x_1)\phi(x_2) \sum_{j=1}^\infty \rho^j \psi_j(x_1) \psi_j(x_2)/j!

where

ψj(z)=(1)j1dj1ϕ(z)/dzj1.\psi_j(z) = (-1)^{j-1} d^{j-1} \phi(z)/dz^{j-1}.

Since

ϕ(z)=zϕ(z),ϕ(z)=(z21)ϕ(z),ϕ(z)=[2zz(z21)]ϕ(z)=(3zz3)ϕ(z),\phi'(z) = -z\phi(z), \phi''(z) = (z^2-1)\phi(z) , \phi'''(z) = [2z-z(z^2-1)]\phi(z) = (3z-z^3)\phi(z) ,

ϕ(4)(z)=[33z2z(3zz3)]ϕ(z)=(36z2+z4)ϕ(z)\phi^{(4)}(z) = [3-3z^2-z(3z-z^3)]\phi(z) = (3-6z^2+z^4)\phi(z)

we have

Φ2(x1,x2;ρ)=Φ(x1)Φ(x2)+ϕ(x1)ϕ(x2)[ρ+ρ2x1x2/2+ρ3(x121)(x221)/6+ρ4(x133x1)(x233x2)/24 \Phi_2(x_1,x_2;\rho) = \Phi(x_1)\Phi(x_2)+\phi(x_1)\phi(x_2) [\rho+ \rho^2x_1x_2/2 + \rho^3 (x_1^2-1)(x_2^2-1)/6 +\rho^4 (x_1^3-3x_1)(x_2^3-3x_2)/24

+ρ5(x146x12+3)(x246x22+3)/120+]+\rho^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 ρ3\rho^3 term for ρ0.4|\rho| \le 0.4, and at ρ5\rho^5 term for 0.4<ρ0.70.4 < |\rho|\le 0.7. Higher order terms may be required for ρ>0.7|\rho| > 0.7.

Value

An approximation of bivariate normal cumulative distribution function.

References

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.

See Also

scoreCov


[Package weightedScores version 0.9.5.3 Index]