View source: R/cfN_NegativeBinomial.R
cfN_NegativeBinomial | R Documentation |
cfN_NegativeBinomial(t, r, p)
evaluates the characteristic function cf(t)
of the
Negative-Binomial distribution with the parameters r
(number of failures until
the experiment is stopped, r
is a natural number) and p
(success probability in each experiment,
p
in [0,1]
), i.e.
cfN_NegativeBinomial(t, r, p) = p^r * (1 - (1-p) * e^(1i*t))^(-r).
For more details see [4].
cfN_NegativeBinomial(t, r, p, cfX)
evaluates the compound characteristic function
cf(t) = cfN_NegativeBinomial(-1i*log(cfX(t)), r, 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)
.
cfN_NegativeBinomial(t, r = 10, p = 0.5, cfX)
t |
vector or array of real values, where the CF is evaluated. |
r |
number of trials. |
p |
success probability, |
cfX |
function. |
Characteristic function cf(t)
of the Negative-Binomial distribution.
Ver.: 16-Sep-2018 19:01:54 (consistent with Matlab CharFunTool v1.3.0, 15-Nov-2016 13:36:26).
[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.
For more details see WIKIPEDIA: https://en.wikipedia.org/wiki/Negative_binomial_distribution.
Other Discrete Probability Distribution:
cfN_Binomial()
,
cfN_Delaporte()
,
cfN_GeneralizedPoisson()
,
cfN_Geometric()
,
cfN_Logarithmic()
,
cfN_Poisson()
,
cfN_PolyaEggenberger()
,
cfN_Quinkert()
,
cfN_Waring()
## EXAMPLE1
# CF of the Negative Binomial distribution with r = 5, p = 0.3
r <- 5
p <- 0.3
t <- seq(-15, 15, length.out = 1001)
plotReIm(function(t)
cfN_NegativeBinomial(t, r, p), t,
title = "CF of the Negative Binomial distribution with r = 5, p = 0.3")
## EXAMPLE2
# PDF/CDF of the compound NegativeBinomial-Exponential distribution
r <- 5
p <- 0.3
lambda <- 5
cfX <- function(t)
cfX_Exponential(t, lambda)
cf <- function(t)
cfN_NegativeBinomial(t, r, p, cfX)
x <- seq(0, 10, length.out = 101)
prob <- c(0.9, 0.95, 0.99)
options <- list()
options$isCompound <- TRUE
result <- cf2DistGP(cf, x, prob, options)
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