The Semi-Parametric Distribution (spd) package

Description

The Semi-Parametric Distribution is a piecewise distribution constructed by parametrically modelling the tails of the distribution using an appropriate distribution (e.g. generalized pareto) and the interior by kernel methods. The package implements fit, distribution, density, quantile and random number generation. Currently, only the generalized pareto distribution is implemented for modelling the tails, but the package can easily be extended.

Details

The main functionality of the package is contained in the SPD class, created by calling spdfit. Methods for density dspd, distribution pspd, quantile qspd and random number generation rspd exist and take 2 main arguments, the input value and the fitted object.
The spd package uses the "bkde" function from the package KernSmooth for the kernel interior fit, while for the tail fit borrows from the fExtremes package and implements a locally modified copy of the gpd functionality and methods.

Author(s)

Alec Stephenson for the functions from R\'s "evd-package",
Alec Stephenson for the functions from R\'s "evir-package",
Alexander McNeil for the EVIS functions underlying the "evir-package",
Diethelm Wuertz for the functions from R\'s "fExtremes-package",
M.P.Wand and M.C.Jones for the functions from R\'s "KernSmooth-package",
Alexios Ghalanos for this package.

References

Carmona, R. and J. Morrisson (2001). Heavy Tails and Copulas with Evanesce, ORFE Tech. Report, Princeton University
Carmona, R. (2001). Statistical Analysis of Financial Data, with an implementation in Splus
Embrechts, P., Klueppelberg, C., Mikosch, T. (1997); Modelling Extremal Events, Springer

Examples

## Not run: 
library(MASS)
x<-SP500/100
fit<-spdfit(x)
show(fit)

## End(Not run)

Class: Generalized Pareto Distribution

Description

Locally implemented and slightly modified class for the generalized pareto distribution (gpd) fit, borrowed from package fExtremes. Created on modelling the tails of the data by spdfit

Objects from the Class

Objects of this class cannot be created by user as the methods are not exported.

Slots

call:

Object of class "call"

method:

Object of class "character"

parameter:

Object of class "list"

data:

Object of class "list"

fit:

Object of class "list"

residuals:

Object of class "numeric"

title:

Object of class "character"

description:

Object of class "character"

Note

S3 plot method exists which provides for visual inspection of the fit and is called by the higher level S3 plot method of the SPD class

Author(s)

Alec Stephenson for the functions from R\'s "evd-package", Alec Stephenson for the functions from R\'s "evir-package", Alexander McNeil for the EVIS functions underlying the "evir-package", Diethelm Wuertz for the functions from R\'s "fExtremes-package", M.P.Wand and M.C.Jones for the functions from R\'s "KernSmooth-package", Alexios Ghalanos for this package.

References

Carmona, R. and J. Morrisson (2001). Heavy Tails and Copulas with Evanesce, ORFE Tech. Report, Princeton University
Carmona, R. (2001). Statistical Analysis of Financial Data, with an implementation in Splus
Embrechts, P., Klueppelberg, C., Mikosch, T. (1997); Modelling Extremal Events, Springer

Class: GPDTAILS

Description

Class: SPD with Generalized Pareto Distribution (GPD) Tails

Objects from the Class

Objects can be created by calling spdfit. The main implemented class of the spd package holding the details of the fitted object with gpd tails.

Slots

call:

...

method:

the gpd fitting method

kernel:

the kernel type

data:

the original dataset

threshold:

the upper and lower thresholds fitted

ptails:

the upper and lower cutoff points.

fit:

the upper and lower gpd fit and the interior kernel fit objects.

title:

optional title of project

description:

optional description

Extends

Class SPD, directly.

Author(s)

Alexios Ghalanos

References

Carmona, R. and J. Morrisson (2001). Heavy Tails and Copulas with Evanesce, ORFE Tech. Report, Princeton University
Carmona, R. (2001). Statistical Analysis of Financial Data, with an implementation in Splus

Class: Semi-Parametric Distribution

Description

Virtual Class for holding the tail fit and kernel interior objects.

Methods

dspd

signature(x = "numeric", fit = "SPD", linear = "logical"): density function

pspd

signature(q = "numeric", fit = "SPD", linear = "logical"): distribution function

qspd

signature(p = "numeric", fit = "SPD", linear = "logical"): quantile function

rspd

signature(n = "numeric", fit = "SPD", linear = "logical"): random number generation function

show

signature(object = "SPD"): show method

Objects from the Class

A virtual Class: No objects may be created from it.

Author(s)

Alexios Ghalanos

References

Carmona, R. and J. Morrisson (2001). Heavy Tails and Copulas with Evanesce, ORFE Tech. Report, Princeton University
Carmona, R. (2001). Statistical Analysis of Financial Data, with an implementation in Splus

Examples

showClass("SPD")

Method: Plotting (S4) for implemented S4 classes

Description

Locally implemented and modified methods for plotting the fit of the GPDFIT object (taken from package fExtremes), and the overall fit of the GPDTAILS object.

Usage

plot(x,y,...)

Arguments

Examples

## Not run: 
library(MASS)
x<-SP500/100
fit<-spdfit(x)
plot(fit,which=1)
# this in fact exctracts the GPDFIT object (from GPDTAILS) for which plot 
# methods exist.
plot(fit,which=3)

## End(Not run)

Method: Semi-Parametric Distribution

Description

Density, Distribution, Quantile and Random Number Generation methods for the Semi-Parametric Distribution.

Usage

dspd(x, fit, linear)
pspd(q, fit, linear)
qspd(p, fit, linear)
rspd(n, fit, linear)

Arguments

Value

All values are numeric vectors:
d* returns the density (pdf),
p* returns the probability (cdf),
q* returns the quantiles (inverse cdf), and
r* generates random deviates.

Note

The density is computed using the generalized pareto distribution in the tails, while for the middle, the density is computed by using a smooth gradient approach. Interpolation is used to splice together the ends with the middle portion, providing for an approximate piecewise constant density function. As such, caution should be used when interpreting results obtained by use of this function.

Author(s)

Alec Stephenson for the functions from R\'s evd package,
Alec Stephenson for the functions from R\'s evir package,
Alexander McNeil for the EVIS functions underlying the evir package,
Diethelm Wuetrz for the fExtremes Implementation of the gpd,
Alexios Ghalanos for the SPD Implementation,

References

Embrechts, P., Klueppelberg, C., Mikosch, T. (1997); Modelling Extremal Events, Springer.
Carmona, R. (2004);Statistical Anlaysis of Financial Data in Splus, Springer.

Examples

## Not run: 
library(MASS)
x = SP500/100
fit=spdfit(x, upper=0.9, lower=0.1)
## rspd  -
   par(mfrow = c(2, 2), cex = 0.7)
   r = rspd(n = 1000, fit)
   hist(r, n = 100, probability = TRUE, xlab = "r", 
   col = "steelblue", border = "white",main = "Density")
   box()
## dspd -
   # Plot empirical density and compare with true density:
   r = rspd(n = 1000, fit)
   hist(r, n = 100, probability = TRUE, xlab = "r", 
   col = "steelblue", border = "white",main = "Density")
   box()
   x = seq(-0.3, 0.3, length.out = 1000)
   lines(x, dspd(x, fit), col = "darkorange",lwd=2)
   
## pspd -
   # Plot df and compare with true df:
   plot(sort(r), (1:length(r)/length(r)), 
   ylim = c(0, 1), pch = 19, 
   cex = 0.5, ylab = "p", xlab = "q", main = "CDF")
   grid()
   q = seq(-0.3, 0.3, length.out = 1000)
   lines(q, pspd(q, fit), col = "darkorange",lwd=2)

## End(Not run)

Method: Fitting the Semi-Parametric Distribution

Description

The semi-parametric distribution fitting method.

Usage

spdfit(data, upper = 0.9, lower = 0.1, tailfit="GPD", type = c("mle", "pwm"), 
kernelfit = c("normal", "box", "epanech", "biweight", "triweight"), 
information = c("observed", "expected"), title = NULL, description = NULL, ...)

Arguments

Value

Returns an object of class SPD.

Examples

## Not run: 
library(MASS)
x<-SP500/100
fit<-spdfit(x)
show(fit)
#plot(fit,which="all")

## End(Not run)