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, see details.
|
x2 |
x2, see details.
|
x1s |
x12.
|
x2s |
x22.
|
x1c |
x13.
|
x2c |
x23.
|
x1f |
x14.
|
x2f |
x24.
|
t1 |
Φ(x1)Φ(x2), where Φ(⋅) is the cdf of univariate
standard normal distribution.
|
t2 |
ϕ(x1)ϕ(x2), where ϕ(⋅) 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(Z1≤x1,Z2≤x2)
,
where (Z1,Z2)
is bivariate normal with means 0, variances 1 and
correlation ρ
.
An expansion, due to Pearson (1901), is
Φ2(x1,x2;ρ)=Φ(x1)Φ(x2)+ϕ(x1)ϕ(x2)∑j=1∞ρjψj(x1)ψj(x2)/j!
where
ψj(z)=(−1)j−1dj−1ϕ(z)/dzj−1.
Since
ϕ′(z)=−zϕ(z),ϕ′′(z)=(z2−1)ϕ(z),ϕ′′′(z)=[2z−z(z2−1)]ϕ(z)=(3z−z3)ϕ(z),
ϕ(4)(z)=[3−3z2−z(3z−z3)]ϕ(z)=(3−6z2+z4)ϕ(z)
we have
Φ2(x1,x2;ρ)=Φ(x1)Φ(x2)+ϕ(x1)ϕ(x2)[ρ+ρ2x1x2/2+ρ3(x12−1)(x22−1)/6+ρ4(x13−3x1)(x23−3x2)/24
+ρ5(x14−6x12+3)(x24−6x22+3)/120+⋯]
A good approximation is obtained truncating the series
at ρ3
term for ∣ρ∣≤0.4
, and at ρ5
term for 0.4<∣ρ∣≤0.7
.
Higher order terms may be required for ∣ρ∣>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]