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, stableFit MLE and Quantile Method stable parameter fit.

## 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)

## Arguments

x

a numeric vector.

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.

control

a list of control parameters, see function nlminb.

alpha, beta, gamma, delta

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.

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.

...

parameters to be parsed.

## Value

an object from class "fDISTFIT"

## 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.

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

## 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)

#>
#> Title:
#>  Normal Parameter Estimation
#>
#> Call:
#>  nFit(x = s, doplot = TRUE)
#>
#> Model:
#>  Normal Distribution
#>
#> Estimated Parameter(s):
#>      mean        sd
#> 0.4952318 2.0767958
#>