# MDPD: The Minimum Distance Power Divergence statistics In RTDE: Robust Tail Dependence Estimation

## Description

Computes the power divergence statistics then used a minimization problem.

## Usage

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

## 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.

## Details

The Power Divergence for a density function f and observations X_1,...,X_n is defined as

Δ(f,α) = integral( f^{1+α}(x), dx, {R})-( 1+1/α ) ∑_{i=1}^n f^α(X_i)/n

for α> 0

Δ(f,0) = -∑_{i=1}^n \log f(X_i)/n

for α = 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.

## Value

`MDPD` returns the power divergence against the density function `densfun` as a numeric.

## Author(s)

Christophe Dutang

## 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 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17``` ```##### # (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) ```

RTDE documentation built on Jan. 8, 2020, 5:09 p.m.