fBasics (version 280.73)

DistributionFits: Parameter Fit of a Distribution

Description

A collection and description of moment and maximum likelihood estimators to fit the parameters of a distribution. Included are estimators for the Student-t, for the stable, for the generalized hyperbolic hyperbolic, for the normal inverse Gaussian, and for empirical distributions. The functions are: ll{ 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, nigFit MLE parameter fit for a normal inverse Gaussian 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, trace = FALSE, title = NULL, 
    description = NULL)
    
nigFit(x, alpha = 1, beta = 0, delta = 1, mu = 0, 
    scale = TRUE, doplot = TRUE, span = "auto", trace = TRUE, 
    title = NULL, description = NULL, ...) 
    
## S3 method for class 'fDISTFIT':
show(object)

Arguments

alpha, beta, gamma, delta, mu

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

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?

scale

a logical flag, by default TRUE. Should the time series be scaled by its standard deviation to achieve a more stable optimization?

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 =

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. 1: relative gradient is close to zero, current iterate is probably solution; 2: successive iterates within tolerance, current iterate is probably solution; 3: last global step failed to locate a point lower than estimate. Either estimate is an approximate local minimum of the function or steptol is too small; 4: iteration limit exceeded; 5: maximum step size stepmax exceeded five consecutive times. Either the function is unbounded below, becomes asymptotic to a finite value from above in some direction or stepmax is too small.
• gradient
  the gradient at the estimated maximum.
• steps
  number of function calls.
• Remark: The parameter estimation for the stable distribution via the maximum Log-Likelihood approach may take a quite long time.

Details

The function nlm is used to minimize the "negative" maximum log-likelihood function. nlm carries out a minimization using a Newton-type algorithm. 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

## nigFit -
   # Simulate random variates HYP(1.5, 0.3, 0.5, -1.0):
   set.seed(1953)
   s = rnig(n = 1000, alpha = 1.5, beta = 0.3, delta = 0.5, mu = -1.0) 

## nigFit -  
   # Fit Parameters:
   # Note, this may take some time.
   # Starting vector (1, 0, 1, mean(s)):
   nigFit(s, alpha = 1, beta = 0, delta = 1, mu = mean(s), doplot = TRUE)