ExtremalDep (version 1.0.0)

pExtDep: 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, ...)

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).

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.

Author

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

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.

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)