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\name{mde}
\alias{Mde}
\alias{mde}
\title{Minimum Distance Estimation}
\description{
Minimum distance fitting of univariate distributions, allowing
parameters to be held fixed if desired.
}
\usage{
mde(x, fun, start, measure = c("CvM", "chi-square", "LAS"),
weights = NULL, ...)
}
\arguments{
\item{x}{a vector or an object of class \code{"grouped data"} (in
which case only the first column of frequencies is used).}
\item{fun}{function returning a cumulative distribution (for
\code{measure = "CvM"} and \code{measure = "chi-square"}) or a
limited expected value (for \code{measure = "LAS"}) evaluated at
its first argument.}
\item{start}{a named list giving the parameters to be optimized with
initial values}
\item{measure}{either \code{"CvM"} for the Cramer-von Mises method,
\code{"chi-square"} for the modified chi-square method, or \code{"LAS"}
for the layer average severity method.}
\item{weights}{weights; see Details.}
\item{\dots}{Additional parameters, either for \code{fun} or for
\code{optim}. In particular, it can be used to specify bounds via
\code{lower} or \code{upper} or both. If arguments of \code{fun}
are included they will be held fixed.}
}
\details{
The Cramer-von Mises method (\code{"CvM"}) minimizes the squared
difference between the theoretical cdf and the empirical cdf at the
data points (for individual data) or the ogive at the knots (for
grouped data).
The modified chi-square method (\code{"chi-square"}) minimizes the
modified chi-square statistic for grouped data, that is the squared
difference between the expected and observed frequency within each
group.
The layer average severity method (\code{"LAS"}) minimizes the
squared difference between the theoretical and empirical limited
expected value within each group for grouped data.
All sum of squares can be weighted. If arguments \code{weights} is
missing, weights default to 1 for \code{measure = "CvM"} and
\code{measure = "LAS"}; for \code{measure = "chi-square"}, weights
default to \eqn{1/n_j}{1/n[j]}, where \eqn{n_j}{n[j]} is the frequency
in group \eqn{j = 1, \dots, r}.
Optimization is performed using \code{\link{optim}}. For
one-dimensional problems the Nelder-Mead method is used and for
multi-dimensional problems the BFGS method, unless arguments named
\code{lower} or \code{upper} are supplied when \code{L-BFGS-B} is used
or \code{method} is supplied explicitly.
}
\value{
An object of class \code{"mde"}, a list with two components:
\item{estimate}{the parameter estimates, and}
\item{distance}{the distance.}
}
\references{
Klugman, S. A., Panjer, H. H. and Willmot, G. E. (1998),
\emph{Loss Models, From Data to Decisions}, Wiley.
}
\author{
Vincent Goulet \email{vincent.goulet@act.ulaval.ca} and
Mathieu Pigeon
}
\examples{
## Individual data example
data(dental)
mde(dental, pexp, start = list(rate = 1/200), measure = "CvM")
## Example 2.21 of Klugman et al. (1998)
data(gdental)
mde(gdental, pexp, start = list(rate = 1/200), measure = "CvM")
mde(gdental, pexp, start = list(rate = 1/200), measure = "chi-square")
mde(gdental, levexp, start = list(rate = 1/200), measure = "LAS")
## Two-parameter distribution example
try(mde(gdental, ppareto, start = list(shape = 3, scale = 600),
measure = "CvM")) # no convergence
## Working in log scale often solves the problem
pparetolog <- function(x, shape, scale)
ppareto(x, exp(shape), exp(scale))
( p <- mde(gdental, pparetolog, start = list(shape = log(3),
scale = log(600)), measure = "CvM") )
exp(p$estimate)
}
\keyword{distribution}
\keyword{htest}
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