fBasics (version 3011.87)

DistributionFits: Parameter Fit of a Distribution

Description

A collection and description of moment and maximum likelihood estimators to fit the parameters of a distribution.

The functions are:

nFit

MLE parameter fit for a normal distribution,

tFit

MLE parameter fit for a Student t-distribution,

Usage

nFit(x, doplot = TRUE, span = "auto", title = NULL, description = NULL, ...)
tFit(x, df = 4, doplot = TRUE, span = "auto", trace = FALSE, title = NULL,  description = NULL, ...) 
stableFit(x, alpha = 1.75, beta = 0, gamma = 1, delta = 0,  type = c("q", "mle"), doplot = TRUE, control = list(), trace = FALSE, title = NULL, description = NULL)  
"show"(object)

Arguments

control

[stableFit] - a list of control parameters, see function nlminb.

alpha, beta, gamma, delta

[stable] - The parameters are alpha, beta, gamma, and delta: value of the index parameter alpha with alpha = (0,2]; skewness parameter beta, in the range [-1, 1]; scale parameter gamma; and shift parameter delta.

description

a character string which allows for a brief description.

df

the number of degrees of freedom for the Student distribution, df > 2, maybe non-integer. By default a value of 4 is assumed.

object

[show] - an S4 class object as returned from the fitting functions.

doplot

a logical flag. Should a plot be displayed?

span

x-coordinates for the plot, by default 100 values automatically selected and ranging between the 0.001, and 0.999 quantiles. Alternatively, you can specify the range by an expression like span=seq(min, max, times = n), where, min and max are the left and right endpoints of the range, and n gives the number of the intermediate points.

title

a character string which allows for a project title.

trace

a logical flag. Should the parameter estimation process be traced?

type

a character string which allows to select the method for parameter estimation: "mle", the maximum log likelihood approach, or "qm", McCulloch's quantile method.

x

a numeric vector.

...

parameters to be parsed.

Value

The functions tFit, hypFit and nigFit return a list with the following components:

estimate

the point at which the maximum value of the log liklihood function is obtained.

minimum

the value of the estimated maximum, i.e. the value of the log liklihood function.

code

an integer indicating why the optimization process terminated.

gradient

the gradient at the estimated maximum.

Remark: The parameter estimation for the stable distribution via the maximum Log-Likelihood approach may take a quite long time.

Details

Stable Parameter Estimation: Estimation techniques based on the quantiles of an empirical sample were first suggested by Fama and Roll [1971]. However their technique was limited to symmetric distributions and suffered from a small asymptotic bias. McCulloch [1986] developed a technique that uses five quantiles from a sample to estimate alpha and beta without asymptotic bias. Unfortunately, the estimators provided by McCulloch have restriction alpha>0.6.

Examples

## nFit -
   # Simulate random normal variates N(0.5, 2.0):
   set.seed(1953)
   s = rnorm(n = 1000, 0.5, 2) 

## nigFit -  
   # Fit Parameters:
   nFit(s, doplot = TRUE)