Estimates three of the parameters of the Pareto(IV) distribution by maximum likelihood estimation. Some special cases of this distribution are also handled.
paretoIV(location = 0, lscale = "loglink", linequality = "loglink",
lshape = "loglink", iscale = 1, iinequality = 1, ishape = NULL,
imethod = 1)
paretoIII(location = 0, lscale = "loglink", linequality = "loglink",
iscale = NULL, iinequality = NULL)
paretoII(location = 0, lscale = "loglink", lshape = "loglink",
iscale = NULL, ishape = NULL)Location parameter, called \(a\) below. It is assumed known.
Parameter link functions for the
scale parameter (called \(b\) below),
inequality parameter (called \(g\) below), and
shape parameter (called \(s\) below).
See Links for more choices.
A log link is the default for all because all these parameters are
positive.
Initial values for the parameters.
A NULL value means that it is obtained internally.
If convergence failure occurs, use these arguments to input
some alternative initial values.
Method of initialization for the shape parameter. Currently only values 1 and 2 are available. Try the other value if convergence failure occurs.
An object of class "vglmff" (see vglmff-class).
The object is used by modelling functions such as vglm,
and vgam.
The Pareto(IV) distribution is very general,
for example, special cases include the Pareto(I), Pareto(II),
Pareto(III), and Burr family of distributions.
[Johnson et al. (1994) says on p.19 that fitting Type IV by ML is
very difficult and rarely attempted].
Consequently, reasonably good initial values are recommended,
and convergence to a local solution may occur. For this
reason setting trace=TRUE is a good idea for monitoring
the convergence. Large samples are ideally required to get
reasonable results.
The Pareto(IV) distribution, which is used in actuarial science, economics, finance and telecommunications, has a cumulative distribution function that can be written $$F(y) = 1 - [1 + ((y-a)/b)^{1/g}]^{-s}$$ for \(y > a\), \(b>0\), \(g>0\) and \(s>0\). The \(a\) is called the location parameter, \(b\) the scale parameter, \(g\) the inequality parameter, and \(s\) the shape parameter.
The location parameter is assumed known otherwise the Pareto(IV) distribution will not be a regular family. This assumption is not too restrictive in modelling because in typical applications this parameter is known, e.g., in insurance and reinsurance it is pre-defined by a contract and can be represented as a deductible or a retention level.
The inequality parameter is so-called because of its interpretation in the economics context. If we choose a unit shape parameter value and a zero location parameter value then the inequality parameter is the Gini index of inequality, provided \(g \leq 1\).
The fitted values are currently the median, e.g.,
qparetoIV is used for paretoIV().
There are a number of special cases of the Pareto(IV) distribution.
These include the Pareto(I), Pareto(II), Pareto(III), and Burr family
of distributions.
Denoting \(PIV(a,b,g,s)\) as the Pareto(IV) distribution,
the Burr distribution \(Burr(b,g,s)\) is \(PIV(a=0,b,1/g,s)\),
the Pareto(III) distribution \(PIII(a,b,g)\) is \(PIV(a,b,g,s=1)\),
the Pareto(II) distribution \(PII(a,b,s)\) is \(PIV(a,b,g=1,s)\),
and
the Pareto(I) distribution \(PI(b,s)\) is \(PIV(b,b,g=1,s)\).
Thus the Burr distribution can be fitted using the
negloglink link
function and using the default location=0 argument.
The Pareto(I) distribution can be fitted using paretoff
but there is a slight change in notation: \(s=k\) and
\(b=\alpha\).
Johnson N. L., Kotz S., and Balakrishnan N. (1994) Continuous Univariate Distributions, Volume 1, 2nd ed. New York: Wiley.
Brazauskas, V. (2003) Information matrix for Pareto(IV), Burr, and related distributions. Comm. Statist. Theory and Methods 32, 315--325.
Arnold, B. C. (1983) Pareto Distributions. Fairland, Maryland: International Cooperative Publishing House.
# NOT RUN {
pdata <- data.frame(y = rparetoIV(2000, scale = exp(1),
ineq = exp(-0.3), shape = exp(1)))
# }
# NOT RUN {
par(mfrow = c(2, 1))
with(pdata, hist(y)); with(pdata, hist(log(y)))
# }
# NOT RUN {
fit <- vglm(y ~ 1, paretoIV, data = pdata, trace = TRUE)
head(fitted(fit))
summary(pdata)
coef(fit, matrix = TRUE)
Coef(fit)
summary(fit)
# }
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