rc_est {ReturnCurves}

Estimation of the Return Curve

Description

Estimation of the \(p\)-probability return curve following Murphy-Barltrop et al. (2023).

Usage

rc_est(
  margdata,
  w = NULL,
  p,
  method = c("hill", "cl"),
  q = 0.95,
  qalphas = rep(0.95, 2),
  k = 7,
  constrained = FALSE,
  tol = 0.001,
  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.

p

Curve survival probability. Must be \(p < 1-q\) and \(p < 1-q_\alpha\).

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

Given a probability \(p\) and a joint survival function \(Pr(X>x, Y>y)\), the \(p\)-probability return curve is defined as \[RC(p):=\left\lbrace(x, y) \in R^2: Pr(X>x, Y>y)=p\right\rbrace.\]

This method focuses on estimation of \(RC(p)\) for small \(p\) near \(0\), so that \((X,Y)\) are in the tail of the distribution.

\(Pr(X>x, Y>y)\) is estimated using the angular dependence function \(\lambda(\omega)\) introduced by Wadsworth and Tawn (2013). More details on how to estimate \(\lambda(\omega)\) can be found in adf_est.

The return curve estimation \(\hat{RC}(p)\) is done on standard exponential margins and then back transformed onto the original margins.

Value

An object of S4 class rc_est.class. This object returns the arguments of the function and extra slot rc

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.

rc:

A matrix with the estimates of the Return Curve.

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 (2023). “New estimation methods for extremal bivariate return curves.” Environmetrics, 34(5). ISSN 1099095X, doi: 10.1002/env.2797.

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]

prob <- 10/n

margdata <- margtransf(airdata)

retcurve <- rc_est(margdata = margdata, p = prob, method = "hill")

plot(retcurve)

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

# To access the return curve estimation
retcurve@rc

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