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\name{DistributionFits}
\alias{DistributionFits}
\alias{nFit}
\alias{tFit}
\alias{stableFit}
\title{Fit normal, Student-t and stable distributions}
\description{
A collection of moment and maximum likelihood estimators to fit the
parameters of a distribution.
\cr
The functions are:
\tabular{ll}{
\code{nFit} \tab MLE parameter fit for a normal distribution, \cr
\code{tFit} \tab MLE parameter fit for a Student t-distribution, \cr
\code{stableFit} \tab MLE and Quantile Method stable parameter fit. }
}
\usage{
nFit(x, doplot = TRUE, span = "auto", title = NULL, description = NULL, \dots)
tFit(x, df = 4, doplot = TRUE, span = "auto", trace = FALSE, title = NULL,
description = NULL, \dots)
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{
\item{x}{
a numeric vector.
}
\item{doplot}{
a logical flag. Should a plot be displayed?
}
\item{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 \code{span=seq(min, max,
times = n)}, where, \code{min} and \code{max} are the
left and right endpoints of the range, and \code{n} gives
the number of the intermediate points.
}
\item{control}{
a list of control parameters, see function \code{nlminb}.
}
\item{alpha, beta, gamma, delta}{
The parameters are \code{alpha}, \code{beta}, \code{gamma},
and \code{delta}:\cr
value of the index parameter \code{alpha} with \code{alpha = (0,2]};
skewness parameter \code{beta}, in the range [-1, 1];
scale parameter \code{gamma}; and
shift parameter \code{delta}.
}
\item{description}{
a character string which allows for a brief description.
}
\item{df}{
the number of degrees of freedom for the Student distribution,
\code{df > 2}, maybe non-integer. By default a value of 4 is
assumed.
}
\item{title}{
a character string which allows for a project title.
}
\item{trace}{
a logical flag. Should the parameter estimation process be
traced?
}
\item{type}{
a character string which allows to select the method for
parameter estimation: \code{"mle"}, the maximum log likelihood
approach, or \code{"qm"}, McCulloch's quantile method.
}
\item{\dots}{
parameters to be parsed.
}
}
\value{
an object from class \code{"\linkS4class{fDISTFIT}"}.
Slot \code{fit} has components \code{estimate}, \code{minimum}, \code{code}
and \code{gradient} (but for \code{nFit} \code{code} is \code{NA} and
\code{gradient} is missing).
}
\details{
\bold{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 \code{alpha} and \code{beta}
without asymptotic bias. Unfortunately, the estimators provided by
McCulloch have restriction \code{alpha>0.6}.
\emph{Remark:} The parameter estimation for the stable distribution
via the maximum Log-Likelihood approach may take a quite long time.
}
\examples{
set.seed(1953)
s <- rnorm(n = 1000, 0.5, 2)
nFit(s, doplot = TRUE)
}
\keyword{distribution}
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