It is often very useful to approximate a distribution using the first
moments. See Section 6.2 in Albrecher et al. (2017) for more details on
the approximations discussed in this section.
Classical approximations
Several classical approximations are implemented in the function
pClas:
The normal approximation
(method="normal") for the CDF of the r.v. \(X\) is defined as \[F_X(x) \approx \Phi((x-\mu)/\sigma)\]
where \(\mu\) and \(\sigma^2\) are the mean and variance of
\(X\), respectively, and \(\Phi\) is the CDF of the standard normal
distribution.
This approximation can be improved when the skewness
parameter
\[\nu=E((X-\mu)^3/\sigma^3)\] is
available. The normal-power approximation
(method="normal-power") of the CDF is then given by \[F_X(x) \approx \Phi\left(\sqrt{9/\nu^2 +
6z/\nu+1}-3/\nu\right)\] for \(z=(x-\mu)/\sigma\geq 1\) and \(9/\nu^2 + 6z/\nu+1\geq 0\).
The shifted Gamma approximation
(method="shifted Gamma") uses the approximation \[X \approx \Gamma\left(4/\nu^2,
2/(\nu\times\sigma)\right) + \mu -2\sigma/\nu.\] Here, we need
again that \(\nu>0\).
The normal approximation to the shifted Gamma
distribution (method="shifted Gamma normal")
approximates the CDF of \(X\) as \[F_X(x) \approx \Phi\left(\sqrt{16/\nu^2 +
8z/\nu}-\sqrt{16/\nu^2-1}\right)\] for \(z=(x-\mu)/\sigma\geq 1\). We need again
that \(\nu>0\).
These methods need the mean, variance and possibly the skewness
coefficient (all four methods except the normal approximation) as input.
Below, the distribution function of the chi-squared distribution is
approximated using the four methods.
# Chi-squared sample
X <- rchisq(1000, 2)
x <- seq(0,10,0.01)
# Classical approximations
p1 <- pClas(x, mean(X), var(X))
p2 <- pClas(x, mean(X), var(X), mean((X-mean(X))^3)/sd(X)^3, method="normal-power")
p3 <- pClas(x, mean(X), var(X), mean((X-mean(X))^3)/sd(X)^3, method="shifted Gamma")
p4 <- pClas(x, mean(X), var(X), mean((X-mean(X))^3)/sd(X)^3, method="shifted Gamma normal")
# True probabilities
p <- pchisq(x, 2)
# Plot true and estimated probabilities
plot(x, p, type="l", ylab="F(x)", ylim=c(0,1), col="red")
lines(x, p1, lty=2)
lines(x, p2, lty=3, col="green")
lines(x, p3, lty=4)
lines(x, p4, lty=5, col="blue")
legend("bottomright", c("True CDF", "normal approximation", "normal-power approximation",
"shifted Gamma approximation", "shifted Gamma normal approximation"),
lty=1:5, col=c("red", "black", "green", "black", "blue"), lwd=2)
Approximations using orthogonal polynomials
Based on the theory of orthogonal polynomial expansions, the normal
approximation can be improved. The Gram-Charlier
approximation (pGC) is an improvement of the
normal approximation using Hermite polynomials. Another commonly used
approximation is the Edgeworth expansion
(pEdge).
Denote the standard normal PDF and CDF respectively by \(\phi\) and \(\Phi\). Let \(\mu\) be the first moment, \(\sigma^2=E((X-\mu)^2)\) the variance, \(\mu_3=E((X-\mu)^3)\) the third central
moment and \(\mu_4=E((X-\mu)^4)\) the
fourth central moment of the random variable \(X\). The corresponding cumulants are given
by \(\kappa_1=\mu\), \(\kappa_2=\sigma^2\), \(\kappa_3=\mu_3\) and \(\kappa_4=\mu_4-3\sigma^4\). Now consider
the random variable \(Z=(X-\mu)/\sigma\), which has cumulants 0,
1, \(\nu=\kappa_3/\sigma^3\) and \(k=\kappa_4/\sigma^4=\mu_4/\sigma^4-3\).
The Gram-Charlier approximation (pGC)
for the CDF of \(X\) is given by \[\hat{F}_{GC}(x) = \Phi(z) + \phi(z) (-\nu/6
h_2(z)- k/24h_3(z))\] with \(h_2(z)=z^2-1\), \(h_3(z)=z^3-3z\) and \(z=(x-\mu)/\sigma\).
The Edgeworth expansion (pEdge) for the
CDF of \(X\) is given by \[\hat{F}_{E}(x) = \Phi(z) + \phi(z) (-\nu/6
h_2(z)- (3k \times h_3(z)+\nu^2h_5(z))/72)\] with \(h_2(z)=z^2-1\), \(h_3(z)=z^3-3z\), \(h_5(z)=z^5-10z^3+15z\) and \(z=(x-\mu)/\sigma\).
Both approximations need the first four raw moments as input when
raw=TRUE or otherwise the mean, variance, skewness
coefficient and kurtosis. Applying the two approximations to the same
chi-squared sample as before gives following plot.
x <- seq(0, 10, 0.01)
# Empirical moments
moments = c(mean(X), mean(X^2), mean(X^3), mean(X^4))
# Gram-Charlier approximation
p1 <- pGC(x, moments)
# Edgeworth approximation
p2 <- pEdge(x, moments)
# Normal approximation
p3 <- pClas(x, mean(X), var(X))
# True probabilities
p <- pchisq(x, 2)
# Plot true and estimated probabilities
plot(x, p, type="l", ylab="F(x)", ylim=c(0,1), col="red")
lines(x, p1, lty=2)
lines(x, p2, lty=3)
lines(x, p3, lty=4, col="blue")
legend("bottomright", c("True CDF", "GC approximation",
"Edgeworth approximation", "Normal approximation"),
col=c("red", "black", "black", "blue"), lty=1:4, lwd=2)
