pClas function

Classical estimators for the CDF

Compute approximations of the CDF using the normal approximation, normal-power approximation, shifted Gamma approximation or normal approximation to the shifted Gamma distribution.

pClas(x, mean = 0, variance = 1, skewness = NULL, 
      method = c("normal", "normal-power", "shifted Gamma", "shifted Gamma normal"), 
      lower.tail = TRUE, log.p = FALSE)

Arguments

• x: Vector of points to approximate the CDF in.
• mean: Mean of the distribution, default is 0.
• variance: Variance of the distribution, default is 1.
• skewness: Skewness coefficient of the distribution, this argument is not used for the normal approximation. Default is NULL meaning no skewness coefficient is provided.
• method: Approximation method to use, one of "normal", "normal-power", "shifted Gamma" or "shifted Gamma normal". Default is "normal".
• 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

• The normal approximation 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.

• This approximation can be improved when the skewness parameter

$$\nu=E((X-\mu)^3)/\sigma^3$$

is available. The normal-power approximation of the CDF is then given by

$$F_X(x) \approx \Phi(\sqrt{9/\nu^2 + 6z/\nu+1}-3/\nu)$$

for $z=(x-\mu)/\sigma\ge 1$ and $9/\nu^2 + 6z/\nu+1\ge 0$.

• The shifted Gamma approximation uses the approximation

$$X \approx \Gamma(4/\nu^2, 2/(\nu\times\sigma)) + \mu -2\sigma/\nu.$$

Here, we need that $\nu\>0$.

• The normal approximation to the shifted Gamma distribution approximates the CDF of $X$ as

$$F_X(x) \approx \Phi(\sqrt{16/\nu^2 + 8z/\nu}-\sqrt{16/\nu^2-1})$$

for $z=(x-\mu)/\sigma\ge 1$. We need again that $\nu\>0$.

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.

Author(s)

Tom Reynkens

See Also

pEdge, pGC

Examples

# 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)