pExtDep: Parametric and Non-Parametric Distribution Function of...

In ExtremalDep: Extremal Dependence Models

Parametric and Non-Parametric Distribution Function of Extremal Dependence

Description

Evaluate the distribution function of parametric multivariate extreme-value 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, ...)

Arguments

q	A vector or matrix of quantiles.
type	A character string: "lower", "inv.lower" or "upper". Required when method = "Parametric".
method	A character string: "Parametric" or "NonParametric".
model	A character string with the model name: "HR" (Husler-Reiss), "ET" (Extremal-t), or "EST" (Extremal Skew-t). Required when method = "Parametric".
par	A vector or matrix of parameters for the model. If a matrix, rows correspond to different parameter sets.
plot	Logical; if TRUE (default), a plot is displayed. See Details.
main, xlab, cex.lab, cex.axis, lwd	Graphical arguments passed to hist().
...	Additional graphical arguments passed to hist() when plot = TRUE.

Details

When method = "Parametric", the distribution function is available in 2 or 3 dimensions only. See dim_ExtDep for the correct length of the parameter vector.

• If type = "lower", the cumulative distribution function is computed:

  G(x) = P(X \leq x), \quad x \in \mathbb{R}^d, \; d=2,3.

• If type = "inv.lower", the survival function is computed:

  1 - G(x) = P(\exists i : X_i > x_i).

• If type = "upper", the joint probability of exceedance is computed:

  P(X \geq x).

When method = "NonParametric", the distribution function is available in 2 dimensions only.

If par is a matrix and plot = TRUE, a histogram of the probabilities is displayed across parameter sets. A kernel density estimator, 2.5\%, 50\%, 97.5\% quantiles (crosses) and the mean (dot) are added.

Value

• If par is a vector: returns a scalar (if q is a vector) or a vector of length nrow(q) (if q is a matrix).

• If par is a matrix: returns a vector of length nrow(par) (if q is a vector) or a matrix with nrow(par) rows and nrow(q) columns (if q is a matrix).

Author(s)

Simone Padoan [e-mail], https://faculty.unibocconi.it/simonepadoan/
Boris Beranger [e-mail], https://www.borisberanger.com

References

Beranger, B. and Padoan, S.A. (2015). 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

# 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))

# 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))

# Bivariate Extremal Skew-t
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))

# 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)

ExtremalDep documentation built on Aug. 21, 2025, 5:57 p.m.