adfestimation: Estimation of the Angular Dependence Function (ADF)

In ReturnCurves: Estimation of Return Curves

Estimation of the Angular Dependence Function (ADF)

Description

\loadmathjax

Estimation of the angular dependence function \mjeqn\lambda(\omega) introduced by \insertCiteWadsworthTawn2013;textualReturnCurves.

Usage

adf_est(
  margdata,
  w = NULL,
  method = c("hill", "cl"),
  q = 0.95,
  qalphas = rep(0.95, 2),
  k = 7,
  constrained = FALSE,
  tol = 1e-04,
  par_init = rep(0, k - 1)
)

Arguments

margdata	An S4 object of class margtransf.class. See margtransf for more details.
w	Sequence of rays between 0 and 1. Default is NULL, where a pre-defined grid is used.
method	String that indicates which method is used for the estimation of the angular dependence function. Must either be "hill", to use the Hill estimator \insertCiteHill1975ReturnCurves, or "cl" to use the smooth estimator based on Bernstein-Bezier polynomials estimated by composite maximum likelihood.
q	Marginal quantile used to define the threshold \mjeqnu_\omega of the min-projection variable \mjeqnT^1 at ray \mjeqn\omega \mjeqn\left(t^1_\omega = t_\omega - u_\omega | t_\omega > u_\omega\right), and/or Hill estimator \insertCiteHill1975ReturnCurves. Default is 0.95.
qalphas	A vector containing the marginal quantile used for the Heffernan and Tawn conditional extremes model \insertCiteHeffernanTawn2004ReturnCurves for each variable, if constrained = TRUE. Default is rep(0.95, 2).
k	Polynomial degree for the Bernstein-Bezier polynomials used for the estimation of the angular dependence function with the composite likelihood method \insertCiteMurphyBarltropetal2024ReturnCurves. Default is 7.
constrained	Logical. If FALSE (default) no knowledge of the conditional extremes parameters is incorporated in the angular dependence function estimation.
tol	Convergence tolerance for the composite maximum likelihood procedure. Success is declared when the difference of log-likelihood values between iterations does not exceed this value. Default is 0.0001.
par_init	Initial values for the parameters \mjeqn\beta of the Bernstein-Bezier polynomials used for estimation of the angular dependence function with the composite likelihood method \insertCiteMurphyBarltropetal2024ReturnCurves. Default is rep(0, k-1).

Details

The angular dependence function \mjeqn\lambda(\omega) can be estimated through a pointwise estimator, obtained with the Hill estimator, or via a smoother approach, obtained using Bernstein-Bezier polynomials and estimated via composite likelihood methods.

Knowledge of the conditional extremes framework introduced by \insertCiteHeffernanTawn2004;textualReturnCurves can be incorporated by setting constrained = TRUE. Let \mjeqn\alpha^1_x\mid y=\alpha_x\mid y / (1+\alpha_x\mid y) and \mjeqn\alpha^1_y\mid x=1 /(1+\alpha_y\mid x) with \mjeqn\alpha_x\mid y and \mjeqn\alpha_y\mid x being the conditional extremes parameters. After obtaining \mjeqn\hat\alpha_x\mid y and \mjeqn\hat\alpha_y\mid x via maximum likelihood estimation, \mjeqn\lambda(\omega)=\max\lbrace \omega, 1-\omega\rbrace for \mjeqn\omega \in [0, \hat\alpha^1_x\mid y)\cup (\hat\alpha^1_y\mid x, 1] and is estimated as before for \mjeqn\omega \in [\hat\alpha^1_x\mid y,\hat\alpha^1_y\mid x]. For more details see \insertCiteMurphyBarltropetal2024;textualReturnCurves.

Value

An object of S4 class adf_est.class. This object returns the arguments of the function and two extra slots:

interval:	A vector containing the maximum likelihood estimates from the conditional extremes model, \mjeqn\hat\alpha^1_x\mid y and \mjeqn\hat\alpha^1_y\mid x, if constrained = TRUE. If constrained = FALSE, then c(0, 1) is returned; we note that this has no meaningful interpretation as the estimation is performed in an unconstrained interval.
adf:	A vector containing the estimates of the angular dependence function.

References

\insertAllCited

Examples

library(ReturnCurves)

data(airdata)

n <- dim(airdata)[1]

margdata <- margtransf(airdata)

lambda <- adf_est(margdata = margdata, method = "hill")

plot(lambda)

# To see the the S4 object's slots
str(lambda)

# To access the estimates of the ADF
lambda@adf

# If constrained = T, the MLE estimates for the conditional extremes model
# can be accessed as
lambda@interval

ReturnCurves documentation built on April 4, 2025, 5:36 a.m.