dig {ig}R Documentation

Probability density function (pdf) of the inverse Gaussian type distribution

Description

Probability density function for the inverse Gaussian type distribution with mean parameter mu, scale parameter lambda and associated kernel g. The IGTD is a generalization of the inverse Gaussian type distribution; for details see Sanhueza, Leiva and Balakrishnan (2007). The g function corresponds to the kernel of the pdf of the associated symmetrical distribution. In the ig package, the IGTD can be obtained from the following kernels: Laplace, logistic, normal (classical case) and Student-t. All these kernels are implemented in the R software. The Laplace or double exponential distribution is obtained from the normalp package developed by Mineo (2005).

Usage

dig(x, mu = 1, lambda = 1, kernel = "normal", parameter.nu = 1, log = FALSE)

Arguments

x Vector of observations.
mu Mean.
lambda Scale parameter.
kernel Kernel of the pdf of the associated symmetrical distribution by means of which the IGTD is obtained. The kernels: "Laplace", "logistic", "normal" and "t" are available.
parameter.nu Additional parameter of the IGTD when the t kernel is used. This parameter corresponds to a shape parameter known as "degree of freedom". For default nu=1, in which case the Cauchy distribution is obtained. The Student-t distribution has always degrees of kurtosis greater than normal distribution. This aspect is transferred to the IGTD and produces robust parameter estimates for the IGTD.
log Logical; if TRUE, probabilities p are given as log(p).

Details

If mu, lambda or g are not specified, then they assume the default values 0, 1 and "normal", respectively. The IGTD has pdf given by

f_T(t)=f_Z(a_{t}) sqrt{λ}/sqrt{t^{3}},

with t > 0, μ>0 and λ>0, where f_Z(cdot)= c,g(cdot) is the pdf of the associated symmetrical about zero distribution and a_{t} = a_{t}(μ,λ) = sqrt{λ/μ} [sqrt{t/μ} - sqrt{μ/t}].

Value

dig() gives the pdf of an IGTD.

Author(s)

Víctor Leiva <victor.leiva@uv.cl>, Hugo Hernández <hugo.hernandez@msn.com>, and Antonio Sanhueza <asanhue@ufro.cl>.

References

Mineo, A. (2003). A new package for the general error distribution. R News 3, 13-16.

Sanhueza, A., Leiva, V. and Balakrishnan, N. (2007). A new class of inverse Gaussian type distributions. Metrika (in press).

Examples

## Compute the pdf of the IGTD with g="normal" for a vector x with mu=1, lambda=1
## At the end we have the graph of this pdf
x <- seq(0, 4,by=0.01)
fx <- dig(x,mu=1.0,lambda=1.0,kernel="normal")
print(fx)
plot(x, fx, main = "pdf of the IGTD (classical case)", ylab="f(x)")

[Package ig version 1.0 Index]