Description

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

Usage

Arguments

Details

• The normal approximation for the CDF of the r.v. X is defined as

  F_X(x) \approx Φ((x-μ)/σ)

  where μ and σ^2 are the mean and variance of X, respectively.

• This approximation can be improved when the skewness parameter

  ν=E((X-μ)^3)/σ^3

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

  F_X(x) \approx Φ(√{9/ν^2 + 6z/ν+1}-3/ν)

  for z=(x-μ)/σ≥ 1 and 9/ν^2 + 6z/ν+1≥ 0.

• The shifted Gamma approximation uses the approximation

  X \approx Γ(4/ν^2, 2/(ν\timesσ)) + μ -2σ/ν.

  Here, we need that ν>0.

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

  F_X(x) \approx Φ(√{16/ν^2 + 8z/ν}-√{16/ν^2-1})

  for z=(x-μ)/σ≥ 1. We need again that ν>0.

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

Value

Vector of estimates for the probabilities F(x)=P(X≤ x).

Author(s)

Tom Reynkens

References

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

See Also

pEdge, pGC

Examples