Empirical-Likelihood Based Inference for the Angular Measure

Description

This function computes empirical-likelihood based estimators for the angular distribution function of a bivariate extreme value distribution.

Usage

angcdf(Y, tau = 0.95, method = "euclidean", raw = TRUE)

Arguments

Details

method = "euclidean" implements the maximum Euclidean likelihood spectral distribution function as introduced by de Carvalho et al (2013). method = "empirical" implements the maximum Empirical likelihood spectral distribution function as introduced by Einmahl and Segers (2009).

Value

The plot method depicts the empirical likelihood-based angular distribution function.

Author(s)

Miguel de Carvalho

References

de Carvalho, M., Oumow, B., Segers, J. and Warchol, M. (2013) A Euclidean likelihood estimator for bivariate tail dependence. Communications in Statistics—Theory and Methods, 42, 1176–1192.

Einmahl, J. H. J., and Segers, J. (2009) Maximum empirical likelihood estimation of the spectral measure of an extreme-value distribution. The Annals of Statistics, 37, 2953–2989.

Examples

## de Carvalho et al (2013, Fig. 7)
data(beatenberg)
attach(beatenberg)
fit <- angcdf(beatenberg, tau = 0.98, raw = FALSE)
plot(fit)
rug(fit$w)

Empirical-Likelihood Based Inference for the Angular Density

Description

This function computes empirical-likelihood based estimators for the angular distribution function of a bivariate extreme value distribution.

Usage

angdensity(Y, tau = 0.95, nu, grid = seq(0.01, 0.99, length = 2^8),
	   method = "euclidean", raw = TRUE)

Arguments

Details

The smooth angular density was introduced in by de Carvalho et al (2013). method = "euclidean" implements the version of the method based on Euclidean likelihood weights, whereas method = "empirical" uses Empirical likelihood weights.

Value

The plot method depicts the smooth angular density.

Author(s)

Miguel de Carvalho

References

de Carvalho, M., Oumow, B., Segers, J. and Warchol, M. (2013) A Euclidean likelihood estimator for bivariate tail dependence. Communications in Statistics—Theory and Methods, 42, 1176–1192.

Examples

## de Carvalho et al (2013, Fig. 7)
data(beatenberg)
attach(beatenberg)
fit <- angdensity(beatenberg, tau = 0.98, nu = 163, raw = FALSE)
plot(fit)
rug(fit$w)

Smooth Empirical-Likelihood Based Inference for the Angular Measure

Description

This function computes smooth empirical-likelihood based estimators for the angular distribution function of a bivariate extreme value distribution.

Usage

angscdf(Y, tau = 0.95, nu, grid = seq(0.01, 0.99, length = 2^8),
	method = "euclidean", raw = TRUE)

Arguments

Details

method = "euclidean" implements the maximum Euclidean likelihood spectral distribution function as introduced by de Carvalho et al (2013). method = "empirical" implements the maximum Empirical likelihood spectral distribution function as introduced by Einmahl and Segers (2009).

Value

The plot method depicts the empirical likelihood-based angular distribution function.

Author(s)

Miguel de Carvalho

References

de Carvalho, M., Oumow, B., Segers, J. and Warchol, M. (2013) A Euclidean likelihood estimator for bivariate tail dependence. Communications in Statistics—Theory and Methods, 42, 1176–1192.

Einmahl, J. H. J., and Segers, J. (2009) Maximum empirical likelihood estimation of the spectral measure of an extreme-value distribution. The Annals of Statistics, 37, 2953–2989.

Examples

## de Carvalho et al (2013, Fig. 7)
data(beatenberg)
attach(beatenberg)
fit <- angscdf(beatenberg, tau = 0.98, nu = 163, raw = FALSE)
plot(fit)
rug(fit$w)

Beatenberg

Description

Preprocessed pairs of temperatures in unit Fréchet scale from Beatenberg forest, registered under forest cover and in the open field. Preprocessing is conducted as described in Ferrez et al (2011).

Usage

beatenberg

Format

The beatenberg data frame has 2839 rows and 2 columns.

References

Ferrez, J., A. C. Davison, and Rebetez., M. (2011) Extreme temperature analysis under forest cover compared to an open field. Agricultural and Forest Meteorology, 151, 992–1001.

Kernel Smoothed Scedasis Density

Description

This function computes a kernel scedasis density estimate.

Usage

cdensity(Y, threshold = quantile(Y[, 2], 0.95), ...)

Arguments

Details

Kernel smoothing for the scedasis density was introduced by Einmahl et al (2016).

Value

The plot method depicts the smooth scedasis density.

Author(s)

Miguel de Carvalho

References

Einmahl, J. H., Haan, L., and Zhou, C. (2016) Statistics of heteroscedastic extremes. Journal of the Royal Statistical Society: Ser. B, 78(1), 31–51.

Examples

data(lse)
attach(lse)
Y <- data.frame(DATE[-1], -diff(log(ROYAL.DUTCH.SHELL.B)))
T <- dim(Y)[1]
k <- floor((0.4258597) * T / (log(T)))
fit <- cdensity(Y, kernel = "biweight", bw = 0.1 / sqrt(7), 
                threshold = sort(Y[, 2])[T - k])
plot(fit)
plot(fit, original = FALSE)

Empirical Scedasis Distribution Function

Description

This function computes the empirical scedasis distribution function.

Usage

cdf(Y, threshold = quantile(Y[, 2], 0.95))

Arguments

Details

The empirical scedasis distribution function was introduced by Einmahl et al (2016).

Value

The plot method depicts the empirical cumulative scedasis function, and the reference line for the case of constant frequency of extremes over time (if uniform = TRUE).

Author(s)

Miguel de Carvalho

References

Einmahl, J. H., Haan, L., and Zhou, C. (2016) Statistics of heteroscedastic extremes. Journal of the Royal Statistical Society: Ser. B, 78(1), 31–51.

Examples

data(sp500)
attach(sp500)
Y <- data.frame(date[-1], -diff(log(close)))
fit <- cdf(Y)
plot(fit)
plot(fit, original = FALSE)

Mode Mass Function

Description

This function computes the mode mass function.

Usage

cmodes(Y, thresholds = apply(Y[, -1], 2, quantile, probs =
                 0.95), nu = 100, ...)

Arguments

Details

The scedasis functions on which the mode mass function is based are computed using the default "nrd0" option for bandwidth.

Value

The plot method depicts the smooth mode mass function along with the smooth scedasis densities.

Author(s)

Miguel de Carvalho

References

Rubio, R., de Carvalho, M., and Huser, R. (2018) Similarity-Based Clustering of Extreme Losses from the London Stock Exchange. Submitted.

Examples

data(lse)
attach(lse)
nlr <- -apply(log(lse[, -1]), 2, diff)
Y <- data.frame(DATE[-1], nlr)
T <- dim(Y)[1]
k <- floor((0.4258597) * T / (log(T)))
fit <- cmodes(Y, thresholds = as.numeric(apply(nlr, 2, sort)[T - k, ]),  
              kernel = "biweight", bw = 0.1 / sqrt(7), nu = 100)
plot(fit)

K-Geometric Means Algorithm for Value-at-Risk

Description

This function performs k-geometric means for time-varying value-at-risk.

Usage

  kgvar(y, centers, iter.max = 10, conf.level = 0.95)

Arguments

Details

The intermediate sequence \kappa_T is chosen proportional to T/\log T.

Value

kgvar returns an object of class "kgvar" which has a fitted method. It is a list with at least the following components:

The plot method depicts the k-geometric means algorithm for time-varying value-at-risk. If c.c is TRUE, the method displays the cluster means.

Author(s)

Miguel de Carvalho, Rodrigo Rubio.

References

Rubio, R., de Carvalho, M. and Huser, R. (2018) Similarity-Based Clustering of Extreme Losses from the London Stock Exchange. Submitted.

Examples

## Not run: 
## Example (Overlapping version of Fig. 8 in Supplementary Materials)
data(lse)
attach(lse)
y <- -apply(log(lse[, -1]), 2, diff) 
fit <- kgvar(y, centers = 3)
plot(fit, c.c = TRUE, ylim = c(0, 0.1))

## End(Not run)

K-Means Clustering for Heteroscedastic Extremes

Description

This function performs k-means clustering for heteroscedastic extremes.

Usage

  khetmeans(y, centers, iter.max = 10, alpha = 0.5)

Arguments

Details

The intermediate sequence \kappa_T is chosen proportional to T/\log T.

Value

khetmeans returns an object of class "khetmeans" which has a fitted method. It is a list with at least the following components:

The plot method depicts the k-means clustering for heteroscedastic extremes. If c.c is TRUE, the method displays the cluster means.

Author(s)

Miguel de Carvalho, Rodrigo Rubio.

References

Rubio, R., de Carvalho, M. and Huser, R. (2018) Similarity-Based Clustering of Extreme Losses from the London Stock Exchange. Submitted.

Examples

## Not run: 
## Example 1  (Scenario B, T = 5000)
## This example requires package evd 
require(evd)
set.seed(12)
T <- 5000
n <- 30
b <- 0.1
gamma1 <- 0.7
gamma2 <- 1
grid <- seq(0, 1, length = 100)
c2 <- function(s)
    dbeta(s, 2, 5)
c3 <- function(s)
    dbeta(s, 5, 2)
X <- matrix(0, ncol = T, nrow = n)
for(i in 1:5)
  for(j in 1:T)
    X[i,  j] <- rgev(1, c2(j / T), c2(j / T), gamma1)
for(i in 6:15)
  for(j in 1:T)
    X[i,  j] <- rgev(1, c2(j / T), c2(j / T), gamma2)
for(i in 16:20)
  for(j in 1:T)
    X[i,  j] <- rgev(1, c3(j / T), c3(j / T), gamma1)
for(i in 21:30)
  for(j in 1:T)
    X[i,  j] <- rgev(1, c3(j / T), c3(j / T), gamma2)
Y <- t(X)
fit <- khetmeans(Y, centers = 4)
plot(fit, c.c = TRUE)
lines(grid, c2(grid), type = 'l', lwd = 8, col = 'black')
lines(grid, c3(grid), type = 'l', lwd = 8, col = 'black')

## End(Not run)

## Not run: 
## Example 2 (Overlapping version of Fig. 5 in Supplementary Materials)
data(lse)
attach(lse)
y <- -apply(log(lse[, -1]), 2, diff)
fit <- khetmeans(y, centers = 3)
plot(fit, c.c = TRUE, ylim = c(0, 3))

## End(Not run)

Selected Stocks from the London Stock Exchange

Description

Prices at close from 26 selected stocks from the London stock exchange from 1989 till 2016.

Usage

lse

Format

The lse data frame has 6894 rows and 27 columns.

References

Rubio, R., de Carvalho, M., and Huser (2018) Similarity-based clustering of extreme losses from the London stock exchange.

Unit Fréchet Scatterplot in Log-log Scale

Description

This function depicts a scatterplot of bivariate data transformed to unit Fréchet scale.

Usage

plotFrechet(Y, tau = 0.95, raw = TRUE, ...)

Arguments

Details

The solid line corresponds to the boundary threshold in the log-log scale, with both axes being logarithmic.

Author(s)

Miguel de Carvalho

Examples

## de Carvalho et al (2013, Fig. 5)
data(beatenberg)
plotFrechet(beatenberg, xlab = "Forest Cover", ylab = "Open Field",
            raw = FALSE)

Standard & Poor 500

Description

Daily Standard and Poor’s index at close from 1988 till 2007.

Usage

sp500

Format

The sp500 data frame has 5043 rows and 2 columns.

References

de Carvalho, M. (2016) Statistics of extremes: Challenges and opportunities. In: Handbook of EVT and its Applications to Finance and Insurance. Eds F. Longin. Hoboken: Wiley.

Einmahl, J. H., Haan, L., and Zhou, C. (2016) Statistics of heteroscedastic extremes. Journal of the Royal Statistical Society: Ser. B, 78(1), 31–51.