# dist-DistributionFits: Parameter Fit of a Distribution In fBasics: Rmetrics - Markets and Basic Statistics

## 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

 ``` 1 2 3 4 5 6 7 8 9 10 11``` ```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

 ```1 2 3 4 5 6 7 8 9``` ``` ## 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) ```

fBasics documentation built on Nov. 17, 2017, 2:14 p.m.