R: Minimum Distance Estimation

mde {actuar} | R Documentation

## 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

`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.

### 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, …, 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.

### Value

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

`estimate` | the parameter estimates, and
`distance` | the distance.

### Author(s)

Vincent Goulet vincent.goulet@act.ulaval.ca and Mathieu Pigeon

### 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)

[Package actuar version 3.0-0 Index]