returncurve: Estimation of the Return Curve

In ReturnCurves: Estimation of Return Curves

Estimation of the Return Curve

Description

\loadmathjax

Estimation of the \mjeqnpp-probability return curve following \insertCiteMurphyBarltropetal2023;textualReturnCurves.

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 \mjeqnp < 1-qp < 1-q and \mjeqnp < 1-q_\alphap < 1-qalphas.
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

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

This method focuses on estimation of \mjeqnRC(p)RC(p) for small \mjeqnpp near \mjeqn00, so that \mjeqn(X,Y) are in the tail of the distribution.

\mjeqn

Pr(X>x, Y>y) is estimated using the angular dependence function \mjeqn\lambda(\omega) introduced by \insertCiteWadsworthTawn2013;textualReturnCurves. More details on how to estimate \mjeqn\lambda(\omega) can be found in adf_est.

The return curve estimation \mjeqn\hatRC(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, \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.
rc:	A matrix with the estimates of the Return Curve.

References

\insertAllCited

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

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