# Characteristic functions

### Description

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.

### Usage

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 | ```
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)
``` |

### Arguments

`t` |
numeric value. Some of the above are vectorized functions. |

`df` |
degrees of freedom ( |

`df1, df2` |
numerator and denominator degrees of freedom, must be positive |

`k` |
the number of balls drawn from the urn. |

`kmax` |
the number of terms in the series. |

`lambda` |
vector of (positive) means. |

`location, scale` |
location and scale parameters; scale must be positive. |

`m` |
the number of white balls in the urn. |

`meanlog, sdlog` |
mean and standard deviation of the distribution
on the log scale with default values of |

`mean` |
vector of means. |

`min, max` |
(unif) lower and upper limits of the distribution. Must be finite and in the correct order. |

`mu` |
(nbinom) alternative parametrization via mean |

`n` |
the number of black balls in the urn. |

`ncp` |
non-centrality parameter |

`prob` |
probability of success in each trial. |

`rate` |
an alternative way to specify the scale; must be positive. |

`sd` |
vector of standard deviations. |

`size` |
number of trials (binom) or target for number of successful trials (nbinom). |

`shape` |
shape parameter, must be positive (gamma, weibull) |

`shape1, shape2` |
positive parameters of the Beta distribution. |

### Details

The characteristic function *φ* of a random variable *X* is defined by

*phi(t) = E e^(-itX)*

for all *-∞ < t < ∞*.

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).

### Value

a complex number in rectangular (cartesian) coordinates.

### Beta distribution

For the probability density function, see `dbeta`

.

The characteristic function for central Beta is given by

*
phi(t) = _{1}F_{1}(shape1; shape1 + shape2, 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.

### Binomial distribution

For the probability mass function, see `dbinom`

.

The characteristic function is given by

*
phi(t) = [pe^(it) + (1-p)]^n
*

### Cauchy Distribution

For the probability density function, see `dcauchy`

.

The characteristic function is given by

*
phi(t) = e^(i*t*theta - sigma*|t|)
*

### Chi-square Distribution

For the probability density function, see `dchisq`

.

The characteristic function is given by

*
phi(t) = \frac{\exp(\frac{iδ t}{1 - 2it})}{(1 - 2it)^{df/2}} *

### Exponential Distribution

For the probability density function, see `dexp`

.

This is the special case of gamma when *α = 1*.

### F Distribution

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.

### Gamma Distribution

For the probability density function, see `dgamma`

.

The characteristic function is given by

*
phi(t) = (1 - beta*i*t)^(-alpha)
*

### Geometric Distribution

For the probability mass function, see `dgeom`

.

This is the special case of negative binomial when *r = 1*.

### Hypergeometric Distribution

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.

### Logistic Distribution

For the probability density function, see `dlogis`

.

The characteristic function is given by

*
phi(t) = π t \cosech π t
*

### Lognormal Distribution

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.

### Negative Binomial Distribution

For the probability mass function, see `dnbinom`

.

The characteristic function is given by

*
phi(t) = [p/(1-(1-p)*e^(it))]^r
*

### Normal Distribution

For the probability density function, see `dnorm`

.

The characteristic function is

*
phi(t) = e^(i*mu*t + t^2 * sigma^2 /2) *

### Poisson Distribution

For the probability mass function, see `dpois`

.

The characteristic function is

*
phi(t) = e^(lambda*(e^it - 1))
*

### Wilcoxon Sign Rank Distribution

For the probability density function, see `dsignrank`

.

The characteristic function is calculated according to the definition.

### Student's t Distribution

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.

### Continuous Uniform Distribution

For the probability density function, see `dunif`

.

The characteristic function is

*
phi(t) = \frac{e^(itb) - e^(ita)}{(b - a)it}
*

### Weibull Distribution

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.

### Wilcoxon Rank Sum Distribution

For the probability density function, see `dwilcox`

.

The characteristic function is calculated according to the definition.

### Author(s)

G. Jay Kerns gkerns@ysu.edu.

### Source

For `clnorm`

a fast numerical algorithm is used that originated with and was
published and communicated to me by N. C. Beaulieu: see

### References

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, available online: http://wwwmaths.anu.edu.au/research.reports/srr/95/044/

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.

### See Also

`besselK`

`kummerM`

`kummerU`

`hypergeo`