cfN_Quinkert: Characteristic function of the Quinkert distribution

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

Characteristic function of the Quinkert distribution

Description

cfN_Quinkert(t,a,b) evaluates the characteristic function cf(t) of the Quinkert distribution, with the parameters a (a > 0) and b (b > 0), i.e.

cf(t) = cfN_Quinkert(t,a,b) = 1F1(a,a+b,e^(1i*t)-1), where 1F1 denotes the confluent hypergeometric (Kummer's) function. For more details see [4], p. 564.

Usage

cfN_Quinkert(t, a = 1, b = 1, cfX)

Arguments

t	vector or array of real values, where the CF is evaluated.
a	vector of the 'shape' parameters a > 0. If empty, default value is a = 1.
b	vector of the 'shape' parameters b > 0. If empty, default value is b = 1.
cfX	function.

Value

Characteristic function cf(t) of the Quinkert distribution.

Note

ver.: 31-Jul-2021 12:36:48 (consistent with Matlab CharFunTool v1.5.1, 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_GeneralizedPoisson(), cfN_Geometric(), cfN_Logarithmic(), cfN_NegativeBinomial(), cfN_Poisson(), cfN_PolyaEggenberger(), cfN_Waring()

Examples

## EXAMPLE 1
# CF of the Quinkert distribution with the parameter a=3, b=5
  a <- 3;
  b <- 5;
  t <- seq(from = -15,
           to = 15,
           length.out =501)
  plotReIm(function(t)
          cfN_Quinkert(t, a, b),
          t,
          title = "CF of the Quinkert distribution with a = 3, b = 5")

  ## EXAMPLE 2
  # CF of the compound Quinkert-Exponential distribution
  a <- 3
  b <- 5
  lambda <- 5
  cfX <- function(t)
          cfX_Exponential(t, lambda)
  t <- seq(from = -15,
           to = 15,
           length.out = 501)
  plotReIm(function(t)
          cfN_Quinkert(t, a, b, cfX),
          t,
          title = "CF of the compound Quinkert-Exponential distribution")

 ## EXAMPLE3
 # PDF/CDF of the compound Quinkert-Exponential distribution
    a <- 3;
    b <- 5;
    lambda <- 5;
    cfX <- function(t)
            cfX_Exponential(t, lambda)
    cf <- function(t)
            cfN_Quinkert(t,a,b,cfX)
    x <- seq(0,1.5, 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.