cfN_GeneralizedPoisson: Characteristic function of Generalized Poisson distribution

In gajdosandrej/CharFunToolR: Numerical Computation Cumulative Distribution Function and Probability Density Function from Characteristic Function

Characteristic function of Generalized Poisson distribution

Description

cfN_GeneralizedPoisson(t, a, p, cfX) evaluates the characteristic function cf(t) of the Generalized-Poisson distribution, with the parameters a (variable mean, a > 0), and p (success probability, 0 \le p \le 1) i.e.

cfN_GeneralizedPoisson(t, a, p) = exp(a*(sum_{j=1}^Inf ((p*j)^(j-1)*e^(-p*j)/j!)*e^(1i*t*j)-1)).

For more details see [4], p. 93.

The Generalized-Poisson distribution is equivalent with the Borel-Tanner distribution with parameters (p,m), see WIMMER & ALTMANN (1999), where m ~ Poisson(a).

cfN_GeneralizedPoisson(t, a, p, cfX) evaluates the compound characteristic function

cf(t) = cfN_GeneralizedPoisson(-1i*log(cfX(t)), a, p),

where cfX is function handle of the characteristic function cfX(t) of a continuous distribution and/or random variable X.

Note that such CF is characteristic function of the compound distribution, i.e. distribution of the random variable Y = X_1 + ... + X_N, where X_i ~ F_X are i.i.d. random variables with common CF cfX(t), and N ~ F_N is independent RV with its CF given by cfN(t).

Usage

cfN_GeneralizedPoisson(t, a, p, cfX)

Arguments

t	vector or array of real values, where the CF is evaluated.
a	variable mean, a > 0.
p	success probability, 0 \le p \le 1, default value p = 1/2.
cfX	function.

Value

Characteristic function cf(t) of the Poisson distribution.

Note

Ver.: 16-Sep-2018 18:59:23 (consistent with Matlab CharFunTool v1.3.0, 15-Nov-2016 13:36:26).

References

[1] WITKOVSKY V., WIMMER G., DUBY T. (2016). Computing the aggregate loss distribution based on numerical inversion of the compound empirical characteristic function of frequency and severity. Preprint submitted to Insurance: Mathematics and Economics.

[2] DUBY T., WIMMER G., WITKOVSKY V.(2016). MATLAB toolbox CRM for computing distributions of collective risk models. Preprint submitted to Journal of Statistical Software.

[3] WITKOVSKY V. (2016). Numerical inversion of a characteristic function: An alternative tool to form the probability distribution of output quantity in linear measurement models. Acta IMEKO, 5(3), 32-44.

[4] WIMMER G., ALTMANN G. (1999). Thesaurus of univariate discrete probability distributions. STAMM Verlag GmbH, Essen, Germany. ISBN 3-87773-025-6.

See Also

Other Discrete Probability Distribution: cfN_Binomial(), cfN_Delaporte(), cfN_Geometric(), cfN_Logarithmic(), cfN_NegativeBinomial(), cfN_Poisson(), cfN_PolyaEggenberger(), cfN_Quinkert(), cfN_Waring()

Examples

## EXAMPLE1
# CF of the Generalized-Poisson distribution with a = 10, p = 0.5
a <- 10
p <- 0.5
t <- seq(-10, 10, length.out = 501)
plotReIm(function(t)
        cfN_GeneralizedPoisson(t, a, p), t,
        title = "CF of the Generalized-Poisson distribution with a = 10, p = 0.5")

## EXAMPLE2
# CF of the compound Generalized-Poisson-Exponential distribution
a <- 10
p <- 0.5
lambda <- 5
cfX <- function(t)
        cfX_Exponential(t, lambda)
t <- seq(-10, 10, length.out = 501)
plotReIm(function(t)
        cfN_GeneralizedPoisson(t, a, p, cfX), t,
        title = "CF of the compound Generalized-Poisson-Exponential distribution")

## EXAMPLE3
# PDF/CDF of the compound Generalized-Poisson-Exponential distribution
a <- 10
p <- 0.5
lambda <- 5
cfX <- function(t)
        cfX_Exponential(t, lambda)
cf <- function(t)
        cfN_GeneralizedPoisson(t, a, p, cfX)
x <- seq(0, 15, length.out = 101)
prob <- c(0.9, 0.95, 0.99)
options <- list()
options$isCompound <- TRUE
result <- cf2DistGP(cf, x, prob, options)

gajdosandrej/CharFunToolR documentation built on June 3, 2024, 7:46 p.m.