MLE of continuous univariate distributions defined on the real line {MLE}

MLE of continuous univariate distributions defined on the real line

Description

MLE of continuous univariate distributions defined on the real line.

Usage

real.mle(x, distr = "normal", v = 5, tol = 1e-7)

Arguments

x

A numerical vector with data.

distr

The distribution to fit, "normal" stands for the normal distribution, "gumbel" for the Gumbel, "cauchy" for the Cauchy, "logistic" for the logistic distribution, "ct" for the (central) t distribution, "t" for the (non-central) t distribution, "wigner" is the Wigner semicircle distribution and "laplace" is the Laplace distribution. "cauchy0" and "gnormal0" are the Cauchy and generalised normal distributions, respectively, with zero location. The generalised normal distribution is also known as the exponential power distribution or the generalized error distribution.

v

The degrees of freedom of the t distribution.

tol

The tolerance level up to which the maximisation stops set to 1e-07 by default.

Details

Instead of maximising the log-likelihood via a numerical optimiser we have used a Newton-Raphson algorithm which is faster. See wikipedia for the equation to be solved. For the t distribution we need the degrees of freedom and estimate the location and scatter parameters.

The Cauchy is the t distribution with 1 degree of freedom. The Laplace distribution is also called double exponential distribution.

Value

Usually a list with three elements, but this is not for all cases.

iters

The number of iterations required for the Newton-Raphson to converge.

scale

The estimated scale parameter of the Cauchy distribution.

loglik

The value of the maximised log-likelihood.

param

The vector of the parameters.

Author(s)

Michail Tsagris and Sofia Piperaki.

R implementation and documentation: Michail Tsagris [e-mail] and Sofia Piperaki [e-mail].

References

Johnson, Norman L. Kemp, Adrianne W. Kotz, Samuel (2005). Univariate Discrete Distributions (third edition). Hoboken, NJ: Wiley-Interscience.

https://en.wikipedia.org/wiki/Wigner_semicircle_distribution

Do M.N. and Vetterli M. (2002). Wavelet-based Texture Retrieval Using Generalised Gaussian Density and Kullback-Leibler Distance. Transaction on Image Processing. 11(2): 146–158.

See Also

positive.mle, circ.mle, disc.mle

Examples

x <- rnorm(1000, 10, 2)
a <- real.mle(x, distr = "normal")