Daily Log Returns on BMW Share Price

Description

These data are the daily log returns on BMW share price from Tuesday 2nd January 1973 until Tuesday 23rd July 1996. The data are contained in a numeric vector. The dates of each observation are contained in a times attribute, which is an object of class "POSIXct" (see DateTimeClasses). Note that these data form an irregular time series because no trading takes place at the weekend.

Usage

data(bmw)

Format

A numeric vector containing 6146 observations, with a times attribute which is a POSIXct object of the same length.

Danish Fire Insurance Claims

Description

These data describe large fire insurance claims in Denmark from Thursday 3rd January 1980 until Monday 31st December 1990. The data are contained in a numeric vector. The dates of each observation are contained in a times attribute, which is an object of class "POSIXct" (see DateTimeClasses). They were supplied by Mette Rytgaard of Copenhagen Re. Note that these data form an irregular time series.

Usage

data(danish)

Format

A numeric vector containing 2167 observations, with a times attribute which is a POSIXct object of the same length.

Decluster Point Process

Description

Declusters clustered point process data so that Poisson assumption is more tenable over a high threshold.

Usage

decluster(series, run = NA, picture = TRUE)

Arguments

Value

The declustered object.

References

Embrechts, P., Klueppelberg, C., Mikosch, T. (1997). Modelling Extremal Events. Springer. Chapter 8, 413–429.

See Also

pot, exindex, as.POSIXct

Examples

# decluster the 200 exceedances of a particular threshold in 
# the negative BMW data
data(bmw)
out <- pot(-bmw, ne = 200) 
decluster(out$data, 30) 

Generalized Extreme Value Distribution

Description

Cumulative probability, quantiles, density and random generation from the generalized extreme value distribution.

Usage

pgev(q, xi = 1, mu = 0, sigma = 1) 
qgev(p, xi = 1, mu = 0, sigma = 1) 
dgev(x, xi = 1, mu = 0, sigma = 1) 
rgev(n, xi = 1, mu = 0, sigma = 1)

Arguments

Value

Probability (pgev), quantile (qgev), density (dgev) or random sample (rgev) for the GEV distribution with shape xi.

See Also

dgpd, gev

Generalized Pareto Distribution

Description

Cumulative probability, quantiles, density and random generation from the generalized Pareto distribution.

Usage

pgpd(q, xi, mu = 0, beta = 1) 
qgpd(p, xi, mu = 0, beta = 1) 
dgpd(x, xi, mu = 0, beta = 1) 
rgpd(n, xi, mu = 0, beta = 1)

Arguments

Value

Probability (pgpd), quantile (qgpd), density (dgpd) or random sample (rgpd) for the GPD distribution with shape xi.

See Also

dgev, gpd

Plot of Empirical Distribution Function

Description

Plots empirical distribution function of a sample.

Usage

emplot(data, alog = "x", labels = TRUE, ...)

Arguments

Details

This is a simple explanatory function. A straight line on the double log scale indicates Pareto tail behaviour.

See Also

qplot, meplot

Examples

## Not run: data(danish)
## Not run: emplot(danish) 
# Danish fire insurance data show Pareto tail behaviour 

Estimate Extremal Index

Description

Plot estimates of extremal index using the blocks method.

Usage

exindex(data, block, start = 5, end = NA, reverse = FALSE,
    auto.scale = TRUE, labels = TRUE, ...)

Arguments

Value

A table of results is returned invisibly.

References

Embrechts, P., Klueppelberg, C., Mikosch, T. (1997). Modelling Extremal Events. Springer. Chapter 8, 413-429.

See Also

gev, hill, as.POSIXct

Examples

## Not run: data(bmw)
## Not run: exindex(bmw, 100) 
## Not run: exindex(-bmw, 100) 
# calculate extremal index for the right and left tails of the BMW 
# log returns 

Find Threshold

Description

Finds a threshold so that a given number of extremes lie above.

Usage

findthresh(data, ne)

Arguments

Details

When the data are tied a threshold is found so that at least the specified number of extremes lie above.

Value

Vector of suitable thresholds.

See Also

hill, gpd, pot

Examples

# Find threshold giving (at least) fifty exceedances for Danish data
data(danish)
findthresh(danish, 50) 

Fit Generalized Extreme Value Distribution

Description

Fits generalized extreme value distribution (GEV) to block maxima data.

Usage

gev(data, block = NA, ...)

Arguments

Value

An object of class gev describing the fit and including parameter estimates and standard errors. Fitting is carried out using maximum likelihood.

See Also

plot.gev, gumbel, optim, as.POSIXct

Examples

# Fit GEV to monthly maxima
data(bmw)
out <- gev(bmw, "month") 
# Fit GEV to maxima of blocks of 100 observations
out <- gev(bmw, 100) 
# Fit GEV to the data in nidd.annual, the annual maximum water 
# levels of the River Nidd, using the "BFGS" optimization method
data(nidd.annual) 
out <- gev(nidd.annual, method = "BFGS", control = list(maxit = 500))   

Fit Generalized Pareto Model

Description

Returns an object of class "gpd" representing the fit of a generalized Pareto model to excesses over a high threshold.

Usage

gpd(data, threshold = NA, nextremes = NA, method = c("ml", "pwm"),
    information = c("observed", "expected"), ...)

Arguments

Details

The function uses the general purpose optimization function optim when method = "ml" is chosen.

Value

An object of class "gpd" describing the fit and including parameter estimates and standard errors.

References

Parameter and quantile estimation for the generalized Pareto distribution, JRM Hosking and JR Wallis, Technometrics 29(3), pages 339-349, 1987.

See Also

plot.gpd, shape, quant, optim

Examples

data(danish)
out <- gpd(danish, 10) 
# Fits GPD to excess losses over 10 for the Danish
# fire insurance data 

Add Quantile Estimates to plot.gpd

Description

Calculates quantile estimates and confidence intervals for high quantiles above the threshold in a GPD analysis, and adds a graphical representation to an existing plot.

Usage

gpd.q(x, pp, ci.type = c("likelihood", "wald"), ci.p = 0.95,
    like.num = 50)

Arguments

Details

The GPD approximation in the tail is used to estimate quantile. The "wald" method uses the observed Fisher information matrix to calculate confidence interval. The "likelihood" method reparametrizes the likelihood in terms of the unknown quantile and uses profile likelihood arguments to construct a confidence interval.

See Also

gpd, plot.gpd, gpd.sfall, tailplot

Examples

## Not run: data(danish)
## Not run: out <- gpd(danish, 10) 
## Not run: tp <- tailplot(out) 
## Not run: gpd.q(tp, 0.999) 
# Estimates 99.9th percentile of Danish fire losses 

Add Expected Shortfall Estimates to a GPD Plot

Description

Calculates expected shortfall (tail conditional expectation) estimates and confidence intervals for high quantiles above the threshold in a GPD analysis, and adds a graphical representation to an existing plot.

Usage

gpd.sfall(x, pp, ci.p = 0.95, like.num = 50)

Arguments

Details

Expected shortfall is the expected size of the loss, given that a particular quantile of the loss distribution is exceeded. The GPD approximation in the tail is used to estimate expected shortfall. The likelihood is reparametrised in terms of the unknown expected shortfall and profile likelihood arguments are used to construct a confidence interval.

See Also

gpd, plot.gpd, tailplot, gpd.q

Examples

## Not run: data(danish)
## Not run: out <- gpd(danish, 10)
## Not run: tp <- tailplot(out)
## Not run: gpd.q(tp, 0.999) 
# Estimates 99.9th percentile of Danish fire losses 
## Not run: gpd.sfall(tp, 0.999) 
# Estimates associated expected shortfall for Danish fire losses 

Implements Bivariate POT Method

Description

Returns an object of class "gpdbiv" representing the fit of a bivariate POT (peaks over thresholds) model for joint excesses over thresholds.

Usage

gpdbiv(data1 = NA, data2 = NA, u1 = NA, u2 = NA, ne1 = NA, ne2 = NA,
    global = FALSE, method = "BFGS", ...)

Arguments

Details

This function implements a model suggested by Richard Smith (see references below). The marginal excess distributions are GPD distributions, as suggested by univariate EVT and implemented in gpd. The dependence specification is known as the logistic or Gumbel dependence structure, but it would be easy to program alternatives.

Value

An object of class "gpdbiv" representing the fit and including parameter estimates and standard errors.

References

Multivariate Threshold Methods, Richard L. Smith, in Extreme Value Theory and Applications, ed. J. Galambos, published by Kluwer, pages 225-248, 1994.

Markov Chain Models for Threshold Exceedances, R.L. Smith, J.A. Tawn, S.G. Coles, Biometrika 84, 249-268, 1997.

See Also

gpd, plot.gpdbiv, interpret.gpdbiv

Examples

data(bmw) ; data(siemens)
out <- gpdbiv(-bmw, -siemens, ne1 = 100, ne2 = 100)
interpret.gpdbiv(out, 0.05, 0.05)
## Not run: plot(out) 

Fit Gumbel Distribution

Description

Fits gumbel distribution (GEV with xi = 0) to block maxima data.

Usage

gumbel(data, block = NA, ...)

Arguments

Details

This function is primarily intended for comparison with GEV for assessing the need for a heavy-tailed Frechet (or short-tailed Weibull) to model block maxima.

Value

An object of class "gev" describing the fit and including parameter estimates and standard errors. Fitting is carried out using maximum likelihood.

See Also

plot.gev, gev, optim, as.POSIXct

Examples

# Fit Gumbel to maxima of blocks of 100 observations
data(bmw)
out <- gumbel(bmw, 100) 
# Fit Gumbel to the data in nidd.annual, the annual maximum water 
# levels of the River Nidd, using the "BFGS" optimization method
data(nidd.annual)
out <- gumbel(nidd.annual, method = "BFGS", control = list(maxit = 500))

Create Hill Plot

Description

Plot the Hill estimate of the tail index of heavy-tailed data, or of an associated quantile estimate.

Usage

hill(data, option = c("alpha","xi","quantile"), start = 15,
    end = NA, reverse = FALSE, p = NA, ci = 0.95,
    auto.scale = TRUE, labels = TRUE, ...)

Arguments

Details

This plot is usually calculated from the alpha perspective. For a generalized Pareto analysis of heavy-tailed data using the gpd function, it helps to plot the Hill estimates for xi.

See Also

shape, quant

Examples

## Not run: data(danish)
## Not run: hill(danish) 
# Hill plot of heavy-tailed Danish fire insurance data 
## Not run: hill(danish, option = "quantile", end = 500, p = 0.999) 
# Hill plot of estimated 0.999 quantile of Danish fire insurance data 

Interpret Results of Bivariate GPD Fit

Description

Interprets the results of a bivariate GPD model fitted using the bivariate POT method.

Usage

interpret.gpdbiv(out, x, y)

Arguments

Details

First marginal probabilities of exceeding the points x and y are calculated, and then joint and conditional probabilities.

Value

A vector of probabilities is invisibly returned, in printed order.

Side Effects

A simple interpretation of the fit in terms of exceedance probabilities for the point (x,y) is printed.

See Also

gpdbiv, plot.gpdbiv

Examples

data(bmw) ; data(siemens)
out <- gpdbiv(-bmw, -siemens, ne1 = 100, ne2 = 100) 
interpret.gpdbiv(out, 0.05, 0.05) 
# probabilities of 5% falls in BMW and Siemens stock prices 

Sample Mean Excess Plot

Description

Plots sample mean excesses over increasing thresholds.

Usage

meplot(data, omit = 3, labels = TRUE, ...)

Arguments

Details

An upward trend in plot shows heavy-tailed behaviour. In particular, a straight line with positive gradient above some threshold is a sign of Pareto behaviour in tail. A downward trend shows thin-tailed behaviour whereas a line with zero gradient shows an exponential tail. Because upper plotting points are the average of a handful of extreme excesses, these may be omitted for a prettier plot.

See Also

gpd, qplot

Examples

## Not run: data(danish)
## Not run: meplot(danish) 
# Sample mean excess plot of heavy-tailed Danish fire insurance data 

The River Nidd Data

Description

These data represent annual maximal levels of the River Nidd in Yorkshire. These data are suitable for analysis with gev.

Usage

data(nidd.annual)

Format

A numeric vector containing 35 observations.

The River Nidd Data

Description

These data represent high river levels of the River Nidd in Yorkshire above a threshold value of 65. These data are suitable for analysis with gpd.

Usage

data(nidd.thresh)

Format

A numeric vector containing 154 observations.

Plot Fitted GEV Model

Description

The plot method plot.gev provides two different residual plots for assessing fitted GEV model. The user selects the plot type from a menu. See the examples below.

Usage

## S3 method for class 'gev'
plot(x, ...)

Arguments

Details

Data are converted to unit exponentially distributed residuals under null hypothesis that GEV fits. Two diagnostics for iid exponential data are offered.

See Also

gev, qplot

Examples

data(bmw)
out <- gev(bmw, 100) 
## Not run: plot(out)
 
## Not run: Make a plot selection (or 0 to exit):
## Not run: 1: plot: Scatterplot of Residuals
## Not run: 2: plot: QQplot of Residuals

Plot Fitted GPD Model

Description

The plot method plot.gpd provides four different plots for assessing fitted GPD model. The user selects the plot type from a menu. See the examples below.

Usage

## S3 method for class 'gpd'
plot(x, optlog = NA, extend = 1.5, labels = TRUE, ...)

Arguments

Value

If plot 1 or 2 is selected as the final plot, a list object containing details of the plot is returned invisibly. This object should be used as the first argument of gpd.q or gpd.sfall to add quantile estimates or expected shortfall estimates to the plot.

See Also

gpd, quant, shape

Examples

data(danish)
out <- gpd(danish, 10) 
## Not run: plot(out) 

## Not run: Make a plot selection (or 0 to exit):
## Not run: 1: plot: Excess Distribution 
## Not run: 2: plot: Tail of Underlying Distribution 
## Not run: 3: plot: Scatterplot of Residuals 
## Not run: 4: plot: QQplot of Residuals 

Plot Fitted Bivariate GPD Model

Description

Provides a number of plots summarising a bivariate GPD model fitted using the bivariate POT method. See the examples below.

Usage

## S3 method for class 'gpdbiv'
plot(x, extend = 1.1, n.contours = 15, ...)

Arguments

Details

Option 1 plots the threshold exceedance data; option 2 plots contours of the fitted bivariate distribution function in the joint upper tail (above both thresholds); option 3 plots corresponding contours of the fitted joint survival function; plots 4 and 5 show the fitted tails of the marginal distributions.

See Also

gpd, gpdbiv, tailplot, interpret.gpdbiv, plot.gpd

Examples

data(bmw) ; data(siemens)
out <- gpdbiv(-bmw, -siemens, ne1 = 100, ne2 = 100) 
## Not run: plot(out)
 
## Not run: Make a plot selection (or 0 to exit): 
## Not run: 1: plot: Exceedance data
## Not run: 2: plot: Contours of Bivariate Distribution Function
## Not run: 3: plot: Contours of Bivariate Survival Function 
## Not run: 4: plot: Tail of Marginal 1 
## Not run: 5: plot: Tail of Marginal 2 

Plot Fitted POT Model

Description

The plot method plot.potd provides seven different plots for assessing fitted POT model. The user selects the plot type from a menu. See the examples below.

Usage

## S3 method for class 'potd'
plot(x, ...)

Arguments

Details

Plot 1 displays the exceedance process of the chosen threshold. Plots 2-4 assess the Poisson nature of the exceedance process by looking at the scaled gaps between exceedances, which should be iid unit exponentially distributed. Plots 5-6 assess the GPD nature of the excesses by looking at suitably defined residuals, which should again be iid unit exponentially distributed. Option 8 allows the user to call GPD plotting functions.

Value

If plot 1 or 2 from the GPD plots is selected as the final plot (i.e. option 8 is selected, followed by option 1 or 2), a list object containing details of the plot is returned invisibly. This object should be used as the first argument of gpd.q or gpd.sfall to add quantile estimates or expected shortfall estimates to the plot.

See Also

gpd, pot, plot.gpd

Examples

data(danish)
out <- pot(danish,10) 
## Not run: plot(out) 
  
## Not run: Make a plot selection (or 0 to exit): 
## Not run: 1: plot: Point Process of Exceedances 
## Not run: 2: plot: Scatterplot of Gaps 
## Not run: 3: plot: Qplot of Gaps 
## Not run: 4: plot: ACF of Gaps 
## Not run: 5: plot: Scatterplot of Residuals 
## Not run: 6: plot: Qplot of Residuals 
## Not run: 7: plot: ACF of Residuals 
## Not run: 8: plot: Go to GPD Plots   

Peaks Over Thresholds Model

Description

Fits a Poisson point process to the data, an approach sometimes known as peaks over thresholds (POT), and returns an object of class "potd".

Usage

pot(data, threshold = NA, nextremes = NA, run = NA, picture = TRUE,
    ...)

Arguments

Details

Uses optim for point process likelihood maximization.

Value

An object of class "potd" describing the fit and including parameter estimates and standard errors.

See Also

gpd, plot.potd, plot.gpd, decluster, optim, as.POSIXct

Examples

data(danish)
out <- pot(danish, 10)  
# Fits POT model to Danish fire insurance losses 

Exploratory QQplot for Extreme Value Analysis

Description

Creates a QQplot for threshold data against the exponential distribution or the generalized Pareto distribution.

Usage

qplot(data, xi = 0, trim = NA, threshold = NA, line = TRUE,
    labels = TRUE, ...)

Arguments

Details

If xi is zero the reference distribution is the exponential; if xi is non-zero the reference distribution is the generalized Pareto with that value of xi. In the case of the exponential, the plot is interpreted as follows. Concave departures from a straight line are a sign of heavy-tailed behaviour. Convex departures show thin-tailed behaviour.

See Also

gpd, meplot

Examples

## Not run: data(danish)
## Not run: qplot(danish) 
# QQplot of heavy-tailed Danish fire insurance data 

Plot of GPD Tail Estimate of a High Quantile

Description

Creates a plot showing how the estimate of a high quantile in the tail of a dataset based on the GPD approximation varies with threshold or number of extremes.

Usage

quant(data, p = 0.99, models = 30, start = 15, end = 500, reverse =
    TRUE, ci = 0.95, auto.scale = TRUE, labels = TRUE, ...)

Arguments

Details

For every model gpd is called. Evaluation may be slow. Confidence intervals by the Wald method (which is fastest).

Value

A table of results is returned invisibly.

See Also

gpd, plot.gpd, gpd.q, shape

Examples

## Not run: data(danish)
## Not run: quant(danish, 0.999) 
# Estimates of the 99.9th percentile of the Danish losses using 
# the GPD model with various thresholds 

Calculate Record Development

Description

Creates a data frame showing the development of records in a dataset and calculating the expected behaviour for iid data.

Usage

records(data, do.plot = TRUE, conf.level = 0.95, ...)

Arguments

Details

Records are counted and the observations at which they occur recorded. This is compared with the expected behaviour for iid data.

Value

A data frame.

Examples

## Not run: data(danish)
## Not run: records(danish) 
# Record fire insurance losses in Denmark 

Calculates Quantiles and Expected Shortfalls

Description

Makes a rapid calculation of point estimates of prescribed quantiles and expected shortfalls using the output of the function gpd.

Usage

riskmeasures(x, p)

Arguments

Details

This function simply calculates point estimates and (at present) makes no attempt to calculate confidence intervals for the risk measures. If confidence levels are required use gpd.q and gpd.sfall which interact with graphs of the tail of a loss distribution and are much slower.

Value

A matrix with three columns: probability level, quantile estimate, shortfall estimate.

See Also

gpd, tailplot, gpd.q, gpd.sfall

Examples

data(danish)
out <- gpd(danish, 10) 
riskmeasures(out, c(0.999, 0.9999)) 
# gives estimates of 0.999 and 0.9999 quantiles of Danish loss
# distribution as well as the associated expected shortfall estimates 

Calculate Return Levels Based on GEV Fit

Description

Calculates the k-block return level and 95% confidence interval based on a GEV model for block maxima, where k is specified by the user. The k-block return level is that level exceeded once every k blocks, on average.

Usage

rlevel.gev(out, k.blocks = 20, add = FALSE, ...)

Arguments

Details

The GEV likelihood is reparameterized in terms of the unknown return level and profile likelihood arguments are used to construct a confidence interval.

Value

Vector containing lower 95% bound of confidence interval, estimated return level and upper 95% bound.

See Also

gev, plot.gev

Examples

data(bmw)
out <- gev(bmw, "month")
# Fit GEV to monthly maxima of daily returns on BMW share price
## Not run: rlevel.gev(out, 40)
# Calculate the 40 month return level

Plot for GPD Shape Parameter

Description

Creates a plot showing how the estimate of shape varies with threshold or number of extremes.

Usage

shape(data, models = 30, start = 15, end = 500, reverse = TRUE,
    ci = 0.95, auto.scale = TRUE, labels = TRUE, ...)

Arguments

Details

For every model gpd is called. Evaluation may be slow.

Value

A table of results is returned invisibly.

See Also

gpd, plot.gpd, hill

Examples

## Not run: data(danish)
## Not run: shape(danish)
# Shape plot of heavy-tailed Danish fire insurance data 

Daily Log Returns on Siemens Share Price

Description

These data are the daily log returns on Siemens share price from Tuesday 2nd January 1973 until Tuesday 23rd July 1996. The data are contained in a numeric vector. The dates of each observation are contained in a times attribute, which is an object of class "POSIXct" (see DateTimeClasses). Note that these data form an irregular time series because no trading takes place at the weekend.

Usage

data(siemens)

Format

A numeric vector containing 6146 observations, with a times attribute which is a POSIXct object of the same length.

SP Data to June 1993

Description

The daily closing values of the S&P index from Monday 4th January 1960 until Friday 11th June 1993. The data are contained in a numeric vector. The dates of each observation are contained in a times attribute, which is an object of class "POSIXct" (see DateTimeClasses).

Usage

data(sp.raw)

Format

A numeric vector containing 8415 observations, with a times attribute which is a POSIXct object of the same length.

SP Return Data to October 1987

Description

The daily log returns on the S&P index value from Tuesday 5th January 1960 until Friday 16 October 1987. The data are contained in a numeric vector. The dates of each observation are contained in a times attribute, which is an object of class "POSIXct" (see DateTimeClasses).

Usage

data(spto87)

Format

A numeric vector containing 6985 observations, with a times attribute which is a POSIXct object of the same length.

Plot Tail Estimate From GPD Model

Description

Interacts with the output of gpd to produce a plot of the tail of the underlying distribution of the data. This is one of the options of plot.gpd, but tailplot enables the user to bypass the menu of the former.

Usage

tailplot(x, optlog = NA, extend = 1.5, labels = TRUE, ...)

Arguments

Value

A list object containing details of the plot is returned invisibly. This object should be used as the first argument of gpd.q or gpd.sfall to add quantile estimates or expected shortfall estimates to the plot.

See Also

gpd, plot.gpd, shape, quant

Examples

data(danish)
out <- gpd(danish, 10) 
## Not run: tailplot(out)