R: Parameter Fit of a Distribution

DistributionFits {fBasics} | R Documentation

## 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,
`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)

    ## S4 method for signature 'fDISTFIT'
    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.

### 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`.

### 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.

### 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)

[Package fBasics version 3042.89 Index]