actuar (version 3.3-4)

mde: 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, ...)

Value

An object of class "mde", a list with two components:

estimate

the parameter estimates, and

distance

the distance.

Arguments

x

a vector or an object of class "grouped data" (in which case only the first column of frequencies is used).

fun

function returning a cumulative distribution (for measure = "CvM" and measure = "chi-square") or a limited expected value (for measure = "LAS") evaluated at its first argument.

start

a named list giving the parameters to be optimized with initial values

measure

either "CvM" for the Cramer-von Mises method, "chi-square" for the modified chi-square method, or "LAS" for the layer average severity method.

weights

weights; see Details.

...

Additional parameters, either for fun or for optim. In particular, it can be used to specify bounds via lower or upper or both. If arguments of fun are included they will be held fixed.

Author

Vincent Goulet and Mathieu Pigeon

Details

The Cramer-von Mises method ("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 ("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 ("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 weights is missing, weights default to 1 for measure = "CvM" and measure = "LAS"; for measure = "chi-square", weights default to \(1/n_j\), where \(n_j\) is the frequency in group \(j = 1, \dots, r\).

Optimization is performed using optim. For one-dimensional problems the Nelder-Mead method is used and for multi-dimensional problems the BFGS method, unless arguments named lower or upper are supplied when L-BFGS-B is used or method is supplied explicitly.

References

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (1998), Loss Models, From Data to Decisions, Wiley.

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)