pGC function

Gram-Charlier approximation

Gram-Charlier approximation of the CDF using the first four moments.

pGC(x, moments = c(0, 1, 0, 3), raw = TRUE, lower.tail = TRUE, log.p = FALSE)

Arguments

• x: Vector of points to approximate the CDF in.
• moments: The first four raw moments if raw=TRUE. By default the first four raw moments of the standard normal distribution are used. When raw=FALSE, the mean $\mu=E(X)$, variance $\sigma^2=E((X-\mu)^2)$, skewness (third standardised moment, $\nu=E((X-\mu)^3)/\sigma^3$) and kurtosis (fourth standardised moment, $k=E((X-\mu)^4)/\sigma^4$).
• raw: When TRUE (default), the first four raw moments are provided in moments. Otherwise, the mean, variance, skewness and kurtosis are provided in moments.
• lower.tail: Logical indicating if the probabilities are of the form $P(X\le x)$ (TRUE) or $P(X>x)$ (FALSE). Default is TRUE.
• log.p: Logical indicating if the probabilities are given as $\log(p)$, default is FALSE.

Details

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 for the CDF of $X$ ($F(x)$) is given by

with $h_2(z)=z^2-1$, $h_3(z)=z^3-3z$ and $z=(x-\mu)/\sigma$.

See Section 6.2 of Albrecher et al. (2017) for more details.

Returns

Vector of estimates for the probabilities $F(x)=P(X\le x)$.

References

Albrecher, H., Beirlant, J. and Teugels, J. (2017). Reinsurance: Actuarial and Statistical Aspects, Wiley, Chichester.

Cheah, P.K., Fraser, D.A.S. and Reid, N. (1993). "Some Alternatives to Edgeworth." The Canadian Journal of Statistics, 21(2), 131--138.

Author(s)

Tom Reynkens

See Also

pEdge, pClas

Examples

# Chi-squared sample
X <- rchisq(1000, 2)

x <- seq(0, 10, 0.01)

# Empirical moments
moments = c(mean(X), mean(X^2), mean(X^3), mean(X^4))


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)