approxbvncdf {weightedScores} | R Documentation |
Approximation of bivariate standard normal cumulative distribution function (Johnson and Kotz, 1972).
approxbvncdf(r,x1,x2,x1s,x2s,x1c,x2c,x1f,x2f,t1,t2)
r |
The correlation parameter of bivariate standard normal distribution. |
x1 |
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x2 |
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x1s |
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x2s |
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x1c |
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x2c |
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x1f |
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x2f |
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t1 |
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t2 |
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The approximation for the bivariate normal cdf is from Johnson and Kotz (1972),
page 118.
Let \Phi_2(x_1,x_2;\rho)=Pr(Z_1\le x_1,\,Z_2\le x_2)
,
where (Z_1,Z_2)
is bivariate normal with means 0, variances 1 and
correlation \rho
.
An expansion, due to Pearson (1901), is
\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
\psi_j(z) = (-1)^{j-1} d^{j-1} \phi(z)/dz^{j-1}.
Since
\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) ,
\phi^{(4)}(z) = [3-3z^2-z(3z-z^3)]\phi(z) = (3-6z^2+z^4)\phi(z)
we have
\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
+\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 \rho^3
term for |\rho| \le 0.4
, and at \rho^5
term for 0.4 < |\rho|\le 0.7
.
Higher order terms may be required for |\rho| > 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.