ExtremalDep (version 0.0.4-4)

pExtDep: Parametric and non-parametric distribution function of Extremal Dependence

Description

This function evaluates the distribution function of parametric multivariate extreme distributions and the angular probability distribution represented through Bernstein polynomials.

Usage

pExtDep(q, type, method="Parametric", model, par, plot=TRUE, 
        main, xlab, cex.lab, cex.axis, lwd,...)

Value

When par is a vector: if q is a matrix the function returns a vector of length nrow(q), otherwise a scalar.

When par is a matrix: if q is a matrix the function returns a matrix with nrow(par) rows and nrow(q) columns, otherwise a vector of length nrow(par).

Arguments

q

A vector or matrix of quantiles.

type

A character string taking value "lower", "inv.lower" or "upper". Required when method="Parametric".

method

A character string taking value "Parametric" or "NonParametric"

model

A character string with the name of the model: "HR" (Husler-Reiss), "ET" (Extremal-t) or "EST" (Extremal Skew-t). Required when method="Parametric".

par

A vector or a matrix representing the parameters of the (parametric or non-parametric) model. When in matrix format, rows indicate different sets of parameter values.

plot

A logical value; if TRUE (default) a plot is displayed. See details.

main, xlab, cex.lab, cex.axis, lwd

Arguments of the hist() function.

...

Additional graphical parameter when plot=TRUE.

Author

Simone Padoan, , https://faculty.unibocconi.it/simonepadoan/; Boris Beranger, https://www.borisberanger.com;

Details

Note that when method="Parametric", the distribution function is only available in 2 and 3 dimensions. Refer to the dim_ExtDep function for the appropriate length of the parameter vector.
When type="lower", the cumulative distribution function is computed, i.e., $$G(\boldsymbol{x}) = P\left(\boldsymbol{X} \leq \boldsymbol{x}\right), x \in R^d, d=2,3.$$
When type="inv.lower", the survival function is computed, i.e., $$1-G(\boldsymbol{x}) = 1-P\left(\boldsymbol{X} \leq \boldsymbol{x}\right).$$ This corresponds to the probability of at least one component of \(X\) is greater than its corresponding element in \(x\).
When type="upper", the joint probability of exceedance is computed, i.e., $$P\left(\boldsymbol{X} \geq \boldsymbol{x}\right).$$
Finally, when method="NonParametric", the distribution function is only available in 2 dimensions.

The argument plot is only applicable when par is a matrix. Typically its main use should be when par corresponds to some posterior sample (e.g. from fExtDep with moethod="BayesianPPP"). A histogram of the probabilities evaluated at each set of parameters is displayed, as well as a kernel density estimator, \(2.5\%, 50\%, 97.5\%\) quantiles (crosses) and mean (dot). The argument ... is used to specify additional parameters in the hist() function.

References

Beranger, B. and Padoan, S. A. (2015). Extreme dependence models, chapater of the book Extreme Value Modeling and Risk Analysis: Methods and Applications, Chapman Hall/CRC.

Beranger, B., Padoan, S. A. and Sisson, S. A. (2017). Models for extremal dependence derived from skew-symmetric families. Scandinavian Journal of Statistics, 44(1), 21-45.

Husler, J. and Reiss, R.-D. (1989), Maxima of normal random vectors: between independence and complete dependence, Statistics and Probability Letters, 7, 283--286.

Marcon, G., Padoan, S.A., Naveau, P., Muliere, P. and Segers, J. (2017) Multivariate Nonparametric Estimation of the Pickands Dependence Function using Bernstein Polynomials. Journal of Statistical Planning and Inference, 183, 1-17.

See Also

dExtDep, rExtDep, fExtDep, fExtDep.np

Examples

# Example using the trivariate Extremal Skew-t
pExtDep(q=c(1,1.2, 0.6), type="lower", method="Parametric", 
        model="EST", par=c(0.2, 0.4, 0.6,2,2,2,1))

# Example using the bivariate Extremal-t
pExtDep(q=rbind(c(1.2, 0.6), c(1.1, 1.3)), type="inv.lower", 
        method="Parametric", model="ET", par=c(0.2, 1))
pExtDep(q=rbind(c(1.2, 0.6), c(1.1, 1.3)), type="inv.lower", 
        method="Parametric", model="EST", par=c(0.2, 0, 0, 1))

# Example of non-parametric angular density
beta <- c(1.0000000, 0.8714286, 0.7671560, 0.7569398, 
          0.7771908, 0.8031573, 0.8857143, 1.0000000)
pExtDep(q=rbind(c(0.1,0.9),c(0.2,0.8)), method="NonParametric", par=beta)