Minimum Distance Estimation
Minimum distance fitting of univariate distributions, allowing parameters to be held fixed if desired.
mde(x, fun, start, measure = c("CvM", "chi-square", "LAS"),
weights = NULL, ...)x |
a vector or an object of class |
fun |
function returning a cumulative distribution (for
|
start |
a named list giving the parameters to be optimized with initial values |
measure |
either |
weights |
weights; see details. |
... |
Additional parameters, either for |
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.
An object of class "mde", a list with two components:
estimate |
the parameter estimates, and |
distance |
the distance. |
Vincent Goulet vincent.goulet@act.ulaval.ca and Mathieu Pigeon
Klugman, S. A., Panjer, H. H. and Willmot, G. E. (1998), Loss Models, From Data to Decisions, Wiley.
## 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)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.