RTDE (version 0.2-2)

MDPD: The Minimum Distance Power Divergence statistics

Description

Computes the power divergence statistics then used a minimization problem.

Usage

MDPD(theta, densfun, obs, alpha, ..., control=list())

Value

MDPD returns the power divergence against the density function densfun

as a numeric.

Arguments

theta

the parameter of the distribution given as a vector.

densfun

a function computing the theoretical density function.

obs

a numeric vector of observations

alpha

a numeric for the power divergence parameter.

...

further arguments to be passed to the density function.

control

A list of control paremeters. See section Details.

Author

Christophe Dutang

Details

The Power Divergence for a density function \(f\) and observations \(X_1,...,X_n\) is defined as $$ \Delta(f,\alpha) = \int_{R} f^{1+\alpha}(x)dx-\left ( 1+\frac{1}{\alpha} \right ) \frac{1}{n} \sum_{i=1}^n f^\alpha(X_i) $$ for \(\alpha> 0\) $$ \Delta(f,0) = -\frac{1}{n}\sum_{i=1}^n \log f(X_i) $$ for \(\alpha = 0\).

The control argument is a list that can supply any of the following components:

eps

a small positive floating-point number used when integrate stalled, default to 1e-3.

tol

the desired accuracy used in the integrate function when computing the power divergence, default to 1e-3.

lower

the lower bound of the domain of the density function, default to 1.

upper

the lower bound of the domain of the density function, default to infinity.

References

Basu, A., Harris, I.R., Hjort, N.L., Jones, M.C., (1998). Robust and efficient estimation by minimizing a density power divergence, Biometrika, 85, 549-559.

C. Dutang, Y. Goegebeur, A. Guillou (2014), Robust and bias-corrected estimation of the coefficient of tail dependence, Insurance: Mathematics and Economics

This work was supported by a research grant (VKR023480) from VILLUM FONDEN and an international project for scientific cooperation (PICS-6416).

Examples

#####
# (1) small example

omega <- 1/2
m <- 10
n <- 100
obs <- cbind(rupareto(n), rupareto(n)) + rupareto(n)

#unit Pareto transform
z <- zvalueRTDE(obs, omega, nbpoint=m, output="relexcess")

MDPD(c(1/2, 1/4), dEPD, z$Z, alpha=0, rho=-1)