adf_est {ReturnCurves}

Estimation of the Angular Dependence Function (ADF)

Description

Estimation of the angular dependence function \(\lambda(\omega)\) introduced by Wadsworth and Tawn (2013).

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 (Hill 1975), 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 \(u_\omega\) of the min-projection variable \(T^1\) at ray \(\omega\) \(\left(t^1_\omega = t_\omega - u_\omega | t_\omega > u_\omega\right)\), and/or Hill estimator (Hill 1975). Default is 0.95.

qalphas

A vector containing the marginal quantile used for the Heffernan and Tawn conditional extremes model (Heffernan and Tawn 2004) 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 (Murphy-Barltrop et al. 2024). 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 \(\beta\) of the Bernstein-Bezier polynomials used for estimation of the angular dependence function with the composite likelihood method (Murphy-Barltrop et al. 2024). Default is rep(0, k-1).

Details

The angular dependence function \(\lambda(\omega)\) 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 Heffernan and Tawn (2004) can be incorporated by setting constrained = TRUE. Let \(\alpha^1_{x\mid y}=\alpha_{x\mid y} / (1+\alpha_{x\mid y})\) and \(\alpha^1_{y\mid x}=1 /(1+\alpha_{y\mid x})\) with \(\alpha_{x\mid y}\) and \(\alpha_{y\mid x}\) being the conditional extremes parameters. After obtaining \(\hat{\alpha}_{x\mid y}\) and \(\hat{\alpha}_{y\mid x}\) via maximum likelihood estimation, \(\lambda(\omega)=\max\lbrace \omega, 1-\omega\rbrace\) for \(\omega \in [0, \hat{\alpha}^1_{x\mid y})\cup (\hat{\alpha}^1_{y\mid x}, 1]\) and is estimated as before for \(\omega \in [\hat{\alpha}^1_{x\mid y},\hat{\alpha}^1_{y\mid x}]\). For more details see Murphy-Barltrop et al. (2024).

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, \(\hat{\alpha}^1_{x\mid y}\) and \(\hat{\alpha}^1_{y\mid x}\), 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

Heffernan JE, Tawn JA (2004). “A conditional approach for multivariate extreme values (with discussion).” Journal of the Royal Statistical Society: Series B (Statistical Methodology), 66(3), 497-546. doi: 10.1111/j.1467-9868.2004.02050.x, https://rss.onlinelibrary.wiley.com/doi/pdf/10.1111/j.1467-9868.2004.02050.x.

Hill BM (1975). “A Simple General Approach to Inference About the Tail of a Distribution.” The Annals of Statistics, 3(5), 1163 – 1174. doi: 10.1214/aos/1176343247.

Murphy-Barltrop CJR, Wadsworth JL, Eastoe EF (2024). “Improving estimation for asymptotically independent bivariate extremes via global estimators for the angular dependence function.” 2303.13237.

Wadsworth JL, Tawn JA (2013). “A new representation for multivariate tail probabilities.” Bernoulli, 19(5B), 2689-2714. ISSN 13507265, doi: 10.3150/12-BEJ471.

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