The characteristic functions for selected probability distributions supported
by R. All base distributions are included, with the exception of
wilcox and signedrank. For more resources please see the
References, and for complete details and formulas see the charfunc
vignette, which can be accessed by vignette("charfunc") at the
command prompt. Only the simplest formulas are listed below.
cfbeta(t, shape1, shape2, ncp = 0)
cfbinom(t, size, prob)
cfcauchy(t, location = 0, scale = 1)
cfchisq(t, df, ncp = 0)
cfexp(t, rate = 1)
cff(t, df1, df2, ncp, kmax = 10)
cfgamma(t, shape, rate = 1, scale = 1/rate)
cfgeom(t, prob)
cfhyper(t, m, n, k)
cflnorm(t, meanlog = 0, sdlog = 1)
cflogis(t, location = 0, scale = 1)
cfnbinom(t, size, prob, mu)
cfnorm(t, mean = 0, sd = 1)
cfpois(t, lambda)
cfsignrank(t, n)
cft(t, df, ncp)
cfunif(t, min=0, max=1)
cfweibull(t, shape, scale = 1)
cfwilcox(t, m, n)numeric value. Some of the above are vectorized functions.
degrees of freedom (\(> 0\), maybe non-integer)
degrees of freedom (\(> 0\), maybe non-integer)
the number of balls drawn from the urn.
upper limit of summation.
vector of (positive) means.
location and scale parameters; scale must be positive.
the number of white balls in the urn.
mean and standard deviation of the distribution
on the log scale with default values of 0 and 1 respectively.
vector of means.
(unif) lower and upper limits of the distribution. Must be finite and in the correct order.
(nbinom) alternative parametrization via mean
the number of black balls in the urn.
non-centrality parameter \(\delta\)
probability of success in each trial.
an alternative way to specify the scale; must be positive.
vector of standard deviations.
shape parameter, must be positive (gamma, weibull)
shape parameters (beta).
number of trials (binom) or target for number of successful trials (nbinom).
a complex number in rectangular (cartesian) coordinates.
For the probability density function, see dbeta.
The characteristic function for central Beta is given by
$$ \phi(t) = _{1}F_{1}(\alpha; \alpha + \beta, it) $$
where \(F\) is the confluent hypergeometric function calculated with
kummerM in the fAsianOptions package.
As of the time of this writing, we must calculate the characteristic function of the noncentral Beta with numerical integration according to the definition.
For the probability mass function, see dbinom.
The characteristic function is given by $$ \phi(t) = [p e^{it} + (1-p)]^{n} $$
For the probability density function, see dcauchy.
The characteristic function is given by $$ \phi(t) = e^(it\theta - \sigma|t|) $$
For the probability density function, see dchisq.
The characteristic function is given by $$ \phi(t) = \frac{\exp(\frac{i\delta t}{1 - 2it})}{(1 - 2it)^{df/2}} $$
For the probability density function, see dexp.
This is the special case of gamma when \(\alpha = 1\).
For the probability density function, see df.
For the central \(F\) we use confluent hypergeometric function of the second kind,
also known as kummerU, from the
fAsianOptions package.
For noncentral \(F\) we use confluent hypergeometric function of the first kind. See the vignette for details.
For the probability density function, see dgamma.
The characteristic function is given by $$ \phi(t) = (1 - \beta it)^(-\alpha) $$
For the probability mass function, see dgeom.
This is the special case of negative binomial when \(r = 1\).
For the probability mass function, see dhyper.
The formula for the characteristic function is based on the Gaussian
hypergeometric series, calculated with hypergeo in
package hypergeo. It is too complicated to be included here; please see
the vignette.
For the probability density function, see dlogis.
The characteristic function is given by $$ \phi(t) = \pi t / \sinh(\pi t) $$
For the probability density function, see dlnorm.
This characteristic function is uniquely complicated and delicate, but there is a recent numerical algorithm for computation due to Beaulieu (2008). See the vignette and the References.
For the probability mass function, see dnbinom.
The characteristic function is given by $$ \phi(t) = (p/(1-(1-p)*e^{it}))^{r} $$
For the probability density function, see dnorm.
The characteristic function is $$ \phi (t) = e^{i\mu t + t^{2} \sigma^{2} /2} $$
For the probability mass function, see dpois.
The characteristic function is $$ \phi (t) = e^{\lambda (e^{it} - 1)} $$
For the probability density function, see dsignrank.
The characteristic function is calculated according to the definition.
For the probability density function, see dt.
See the vignette for a formula for the characteristic function for central t.
As of the time of this writing, we must calculate the characteristic function of the noncentral t with numerical integration according to the definition.
For the probability density function, see dunif.
The characteristic function is $$ \phi (t) = \frac{e^{itb} - e^{ita}}{(b - a)it} $$
For the probability density function, see dweibull.
We must at the time of this writing calculate the characteristic function with numerical integration according to the definition.
For the probability density function, see dwilcox.
The characteristic function is calculated according to the definition.
The characteristic function \(\phi\) of a random variable \(X\) is defined by $$ \phi(t) = E e^{itX} $$ for all \(-\infty < t < \infty\).
Every random variable has a characteristic function, and every characteristic function uniquely determines the distribution of its associated random variable. For more details on characteristic functions and their properties, see Lukacs (1970).
Abramowitz, M. and Stegun, I. A. (1972) Handbook of Mathematical Functions. New York: Dover.
Beaulieu, N.C. (2008) Fast convenient numerical computation of lognormal characteristic functions, IEEE Transactions on Communications, Volume 56, Issue 3, 331--333.
Hurst, S. (1995) The Characteristic Function of the Student-t Distribution, Financial Mathematics Research Report No. FMRR006-95, Statistics Research Report No. SRR044-95.
Johnson, N. L., Kotz, S., and Kemp, A. W. (1992) Univariate Discrete Distributions, Second Edition. New York: Wiley.
Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, volume 1. New York: Wiley.
Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, volume 2. New York: Wiley.
Lukacs, E. (1970) Characteristic Functions, Second Edition. London: Griffin.