Package 'evd'

Parametric Dependence Functions of Bivariate Extreme Value Models

Description

Calculate or plot the dependence function $A$ for nine parametric bivariate extreme value models.

Usage

abvevd(x = 0.5, dep, asy = c(1,1), alpha, beta, model = c("log", "alog",
    "hr", "neglog", "aneglog", "bilog", "negbilog", "ct", "amix"),
     rev = FALSE, plot = FALSE, add = FALSE, lty = 1, lwd = 1, col = 1,
     blty = 3, blwd = 1, xlim = c(0,1), ylim = c(0.5,1), xlab = "t",
     ylab = "A(t)", ...)

Arguments

x	A vector of values at which the dependence function is evaluated (ignored if plot or add is TRUE). $A(1/2)$ is returned by default since it is often a useful summary of dependence.
dep	Dependence parameter for the logistic, asymmetric logistic, Husler-Reiss, negative logistic and asymmetric negative logistic models.
asy	A vector of length two, containing the two asymmetry parameters for the asymmetric logistic and asymmetric negative logistic models.
alpha, beta	Alpha and beta parameters for the bilogistic, negative bilogistic, Coles-Tawn and asymmetric mixed models.
model	The specified model; a character string. Must be either "log" (the default), "alog", "hr", "neglog", "aneglog", "bilog", "negbilog", "ct" or "amix" (or any unique partial match), for the logistic, asymmetric logistic, Husler-Reiss, negative logistic, asymmetric negative logistic, bilogistic, negative bilogistic, Coles-Tawn and asymmetric mixed models respectively. The definition of each model is given in rbvevd. If parameter arguments are given that do not correspond to the specified model those arguments are ignored, with a warning.
rev	Logical; reverse the dependence function? This is equivalent to evaluating the function at 1-x.
plot	Logical; if TRUE the function is plotted. The x and y values used to create the plot are returned invisibly. If plot and add are FALSE (the default), the arguments following add are ignored.
add	Logical; add to an existing plot? The existing plot should have been created using either abvevd or abvnonpar, the latter of which plots (or calculates) a non-parametric estimate of the dependence function.
lty, blty	Function and border line types. Set blty to zero to omit the border.
lwd, blwd	Function an border line widths.
col	Line colour.
xlim, ylim	x and y-axis limits.
xlab, ylab	x and y-axis labels.
...	Other high-level graphics parameters to be passed to plot.

Details

Any bivariate extreme value distribution can be written as

$G(z_1,z_2) = \exp\left[-(y_1+y_2)A\left( \frac{y_1}{y_1+y_2}\right)\right]$

for some function $A(\cdot)$ defined on $[0,1]$, where

$y_i = \{1+s_i(z_i-a_i)/b_i\}^{-1/s_i}$

for $1+s_i(z_i-a_i)/b_i > 0$ and $i = 1,2$, with the (generalized extreme value) marginal parameters given by $(a_i,b_i,s_i)$, $b_i > 0$. If $s_i = 0$ then $y_i$ is defined by continuity.

$A(\cdot)$ is called (by some authors) the dependence function. It follows that $A(0)=A(1)=1$, and that $A(\cdot)$ is a convex function with $\max(x,1-x) \leq A(x)\leq 1$ for all $0\leq x\leq1$. The lower and upper limits of $A$ are obtained under complete dependence and independence respectively. $A(\cdot)$ does not depend on the marginal parameters.

Some authors take B(x) = A(1-x) as the dependence function. If the argument rev = TRUE, then B(x) is plotted/evaluated.

Value

abvevd calculates or plots the dependence function for one of nine parametric bivariate extreme value models, at specified parameter values.

See Also

abvnonpar, fbvevd, rbvevd, amvevd

Examples

abvevd(dep = 2.7, model = "hr")
abvevd(seq(0,1,0.25), dep = 0.3, asy = c(.7,.9), model = "alog")
abvevd(alpha = 0.3, beta = 1.2, model = "negbi", plot = TRUE)

bvdata <- rbvevd(100, dep = 0.7, model = "log")
M1 <- fitted(fbvevd(bvdata, model = "log"))
abvevd(dep = M1["dep"], model = "log", plot = TRUE)
abvnonpar(data = bvdata, add = TRUE, lty = 2)

---

Non-parametric Estimates for Dependence Functions of the Bivariate Extreme Value Distribution

Description

Calculate or plot non-parametric estimates for the dependence function $A$ of the bivariate extreme value distribution.

Usage

abvnonpar(x = 0.5, data, epmar = FALSE, nsloc1 = NULL,
    nsloc2 = NULL, method = c("cfg", "pickands", "tdo", "pot"),
     k = nrow(data)/4, convex = FALSE, rev = FALSE, madj = 0,
    kmar = NULL, plot = FALSE, add = FALSE, lty = 1, lwd = 1,
    col = 1, blty = 3, blwd = 1, xlim = c(0, 1), ylim = c(0.5, 1),
    xlab = "t", ylab = "A(t)", ...)

Arguments

x	A vector of values at which the dependence function is evaluated (ignored if plot or add is TRUE). $A(1/2)$ is returned by default since it is often a useful summary of dependence.
data	A matrix or data frame with two columns, which may contain missing values.
epmar	If TRUE, an empirical transformation of the marginals is performed in preference to marginal parametric GEV estimation, and the nsloc arguments are ignored.
nsloc1, nsloc2	A data frame with the same number of rows as data, for linear modelling of the location parameter on the first/second margin. The data frames are treated as covariate matrices, excluding the intercept. A numeric vector can be given as an alternative to a single column data frame.
method	The estimation method (see Details). Typically either "cfg" (the default) or "pickands". The method "tdo" performs poorly and is not recommended. The method "pot" is for peaks over threshold modelling where only large data values are used for estimation.
k	An integer parameter for the "pot" method. Only the largest k values are used, as described in bvtcplot.
convex	Logical; take the convex minorant?
rev	Logical; reverse the dependence function? This is equivalent to evaluating the function at 1-x.
madj	Performs marginal adjustments for the "pickands" method (see Details).
kmar	In the rare case that the marginal distributions are known, specifies the GEV parameters to be used instead of maximum likelihood estimates.
plot	Logical; if TRUE the function is plotted. The x and y values used to create the plot are returned invisibly. If plot and add are FALSE (the default), the arguments following add are ignored.
add	Logical; add to an existing plot? The existing plot should have been created using either abvnonpar or abvevd, the latter of which plots (or calculates) the dependence function for a number of parametric models.
lty, blty	Function and border line types. Set blty to zero to omit the border.
lwd, blwd	Function and border line widths.
col	Line colour.
xlim, ylim	x and y-axis limits.
xlab, ylab	x and y-axis labels.
...	Other high-level graphics parameters to be passed to plot.

Details

The dependence function $A(\cdot)$ of the bivariate extreme value distribution is defined in abvevd. Non-parametric estimates are constructed as follows. Suppose $(z_{i1},z_{i2})$ for $i=1,\ldots,n$ are $n$ bivariate observations that are passed using the data argument. If epmar is FALSE (the default), then the marginal parameters of the GEV margins are estimated (under the assumption of independence) and the data is transformed using

$y_{i1} = \{1+\hat{s}_1(z_{i1}-\hat{a}_1)/ \hat{b}_1\}_{+}^{-1/\hat{s}_1}$

and

$y_{i2} = \{1+\hat{s}_2(z_{i2}-\hat{a}_2)/ \hat{b}_2\}_{+}^{-1/\hat{s}_2}$

for $i = 1,\ldots,n$, where $(\hat{a}_1,\hat{b}_1,\hat{s}_1)$ and $(\hat{a}_2,\hat{b}_2,\hat{s}_2)$ are the maximum likelihood estimates for the location, scale and shape parameters on the first and second margins. If nsloc1 or nsloc2 are given, the location parameters may depend on $i$ (see fgev).

Two different estimators of the dependence function can be implemented. They are defined (on $0 \leq w \leq 1$) as follows.

method = "cfg" (Caperaa, Fougeres and Genest, 1997)

$\log(A_c(w)) = \frac{1}{n} \left\{ \sum_{i=1}^n \log(\max[(1-w)y_{i1}, wy_{i1}]) - (1-w)\sum_{i=1}^n y_{i1} - w \sum_{i=1}^n y_{i2} \right\}$

method = "pickands" (Pickands, 1981)

$A_p(w) = n\left\{\sum_{i=1}^n \min\left(\frac{y_{i1}}{w}, \frac{y_{i2}}{1-w}\right)\right\}^{-1}$

Two variations on the estimator $A_p(\cdot)$ are also implemented. If the argument madj = 1, an adjustment given in Deheuvels (1991) is applied. If the argument madj = 2, an adjustment given in Hall and Tajvidi (2000) is applied. These are marginal adjustments; they are only useful when empirical marginal estimation is used.

Let $A_n(\cdot)$ be any estimator of $A(\cdot)$. None of the estimators satisfy $\max(w,1-w) \leq A_n(w) \leq 1$ for all $0\leq w \leq1$. An obvious modification is

$A_n^{'}(w) = \min(1, \max\{A_n(w), w, 1-w\}).$

This modification is always implemented.

Convex estimators can be derived by taking the convex minorant, which can be achieved by setting convex to TRUE.

Value

abvnonpar calculates or plots a non-parametric estimate of the dependence function of the bivariate extreme value distribution.

Note

I have been asked to point out that Hall and Tajvidi (2000) suggest putting a constrained smoothing spline on their modified Pickands estimator, but this is not done here.

References

Caperaa, P. Fougeres, A.-L. and Genest, C. (1997) A non-parametric estimation procedure for bivariate extreme value copulas. Biometrika, 84, 567–577.

Pickands, J. (1981) Multivariate extreme value distributions. Proc. 43rd Sess. Int. Statist. Inst., 49, 859–878.

Deheuvels, P. (1991) On the limiting behaviour of the Pickands estimator for bivariate extreme-value distributions. Statist. Probab. Letters, 12, 429–439.

Hall, P. and Tajvidi, N. (2000) Distribution and dependence-function estimation for bivariate extreme-value distributions. Bernoulli, 6, 835–844.

See Also

abvevd, amvnonpar, bvtcplot, fgev

Examples

bvdata <- rbvevd(100, dep = 0.7, model = "log")
abvnonpar(seq(0, 1, length = 10), data = bvdata, convex = TRUE)
abvnonpar(data = bvdata, method = "pick", plot = TRUE)

M1 <- fitted(fbvevd(bvdata, model = "log"))
abvevd(dep = M1["dep"], model = "log", plot = TRUE)
abvnonpar(data = bvdata, add = TRUE, lty = 2)

---

Parametric Dependence Functions of Multivariate Extreme Value Models

Description

Calculate the dependence function $A$ for the multivariate logistic and multivariate asymmetric logistic models; plot the estimated function in the trivariate case.

Usage

amvevd(x = rep(1/d,d), dep, asy, model = c("log", "alog"), d = 3, plot =
    FALSE, col = heat.colors(12), blty = 0, grid = if(blty) 150 else 50,
    lower = 1/3, ord = 1:3, lab = as.character(1:3), lcex = 1)

Arguments

x	A vector of length d or a matrix with d columns, in which case the dependence function is evaluated across the rows (ignored if plot is TRUE). The elements/rows of the vector/matrix should be positive and should sum to one, or else they should have a positive sum, in which case the rows are rescaled and a warning is given. $A(1/d,\dots,1/d)$ is returned by default since it is often a useful summary of dependence.
dep	The dependence parameter(s). For the logistic model, should be a single value. For the asymmetric logistic model, should be a vector of length $2^d-d-1$, or a single value, in which case the value is used for each of the $2^d-d-1$ parameters (see rmvevd).
asy	The asymmetry parameters for the asymmetric logistic model. Should be a list with $2^d-1$ vector elements containing the asymmetry parameters for each separate component (see rmvevd and Examples).
model	The specified model; a character string. Must be either "log" (the default) or "alog" (or any unique partial match), for the logistic and asymmetric logistic models respectively. The definition of each model is given in rmvevd.
d	The dimension; an integer greater than or equal to two. The trivariate case d = 3 is the default.
plot	Logical; if TRUE, and the dimension d is three (the default dimension), the dependence function of a trivariate model is plotted. For plotting in the bivariate case, use abvevd. If FALSE (the default), the following arguments are ignored.
col	A list of colours (see image). The first colours in the list represent smaller values, and hence stronger dependence. Each colour represents an equally spaced interval between lower and one.
blty	The border line type, for the border that surrounds the triangular image. By default blty is zero, so no border is plotted. Plotting a border leads to (by default) an increase in grid (and hence computation time), to ensure that the image fits within it.
grid	For plotting, the function is evaluated at grid^2 points.
lower	The minimum value for which colours are plotted. By defualt $\code{lower} = 1/3$ as this is the theoretical minimum of the dependence function of the trivariate extreme value distribution.
ord	A vector of length three, which should be a permutation of the set $\{1,2,3\}$. The points $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$ (the vertices of the simplex) are depicted clockwise from the top in the order defined by ord.The argument alters the way in which the function is plotted; it does not change the function definition.
lab	A character vector of length three, in which case the ith margin is labelled using the ith component, or NULL, in which case no labels are given. The actual location of the margins, and hence the labels, is defined by ord.
lcex	A numerical value giving the amount by which the labels should be scaled relative to the default. Ignored if lab is NULL.

Details

Let $z = (z_1,\dots,z_d)$ and $w = (w_1,\dots,w_d)$. Any multivariate extreme value distribution can be written as

$G(z) = \exp\left\{- \left\{\sum\nolimits_{j=1}^{d} y_j \right\} A\left(\frac{y_1}{\sum\nolimits_{j=1}^{d} y_j}, \dots, \frac{y_d}{\sum\nolimits_{j=1}^{d} y_j}\right)\right\}$

for some function $A$ defined on the simplex $S_d = \{w \in R^d_+ : \sum\nolimits_{j=1}^{d} w_j = 1\}$, where

$y_i = \{1+s_i(z_i-a_i)/b_i\}^{-1/s_i}$

for $1+s_i(z_i-a_i)/b_i > 0$ and $i = 1,\dots,d$, and where the (generalized extreme value) marginal parameters are given by $(a_i,b_i,s_i)$, $b_i > 0$. If $s_i = 0$ then $y_i$ is defined by continuity.

$A$ is called (by some authors) the dependence function. It follows that $A(w) = 1$ when $w$ is one of the $d$ vertices of $S_d$, and that $A$ is a convex function with $\max(w_1,\dots,w_d) \leq A(w)\leq 1$ for all $w$ in $S_d$. The lower and upper limits of $A$ are obtained under complete dependence and mutual independence respectively. $A$ does not depend on the marginal parameters.

Value

A numeric vector of values. If plotting, the smallest evaluated function value is returned invisibly.

See Also

amvnonpar, abvevd, rmvevd, image

Examples

amvevd(dep = 0.5, model = "log")
s3pts <- matrix(rexp(30), nrow = 10, ncol = 3)
s3pts <- s3pts/rowSums(s3pts)
amvevd(s3pts, dep = 0.5, model = "log")
## Not run: amvevd(dep = 0.05, model = "log", plot = TRUE, blty = 1)
amvevd(dep = 0.95, model = "log", plot = TRUE, lower = 0.94)

asy <- list(.4, .1, .6, c(.3,.2), c(.1,.1), c(.4,.1), c(.2,.3,.2))
amvevd(s3pts, dep = 0.15, asy = asy, model = "alog")
amvevd(dep = 0.15, asy = asy, model = "al", plot = TRUE, lower = 0.7)

---

Non-parametric Estimates for Dependence Functions of the Multivariate Extreme Value Distribution

Description

Calculate non-parametric estimates for the dependence function $A$ of the multivariate extreme value distribution and plot the estimated function in the trivariate case.

Usage

amvnonpar(x = rep(1/d,d), data, d = 3, epmar = FALSE, nsloc = NULL,
    madj = 0, kmar = NULL, plot = FALSE, col = heat.colors(12),
    blty = 0, grid = if(blty) 150 else 50, lower = 1/3, ord = 1:3,
    lab = as.character(1:3), lcex = 1)

Arguments

x	A vector of length d or a matrix with d columns, in which case the dependence function is evaluated across the rows (ignored if plot is TRUE). The elements/rows of the vector/matrix should be positive and should sum to one, or else they should have a positive sum, in which case the rows are rescaled and a warning is given. $A(1/d,\dots,1/d)$ is returned by default since it is often a useful summary of dependence.
data	A matrix or data frame with d columns, which may contain missing values.
d	The dimension; an integer greater than or equal to two. The trivariate case d = 3 is the default.
epmar	If TRUE, an empirical transformation of the marginals is performed in preference to marginal parametric GEV estimation, and the nsloc argument is ignored.
nsloc	A data frame with the same number of rows as data, or a list containing d elements of this type, for linear modelling of the marginal location parameters. In the former case, the argument is applied to all margins. The data frames are treated as covariate matrices, excluding the intercept. Numeric vectors can be given as alternatives to single column data frames. A list can contain NULL elements for stationary modelling of selected margins.
madj	Performs marginal adjustments. See abvnonpar.
kmar	In the rare case that the marginal distributions are known, specifies the GEV parameters to be used instead of maximum likelihood estimates.
plot	Logical; if TRUE, and the dimension d is three (the default dimension), the dependence function of a trivariate extreme value distribution is plotted. For plotting in the bivariate case, use abvnonpar. If FALSE (the default), the following arguments are ignored.
col	A list of colours (see image). The first colours in the list represent smaller values, and hence stronger dependence. Each colour represents an equally spaced interval between lower and one.
blty	The border line type, for the border that surrounds the triangular image. By default blty is zero, so no border is plotted. Plotting a border leads to (by default) an increase in grid (and hence computation time), to ensure that the image fits within it.
grid	For plotting, the function is evaluated at grid^2 points.
lower	The minimum value for which colours are plotted. By default $\code{lower} = 1/3$ as this is the theoretical minimum of the dependence function of the trivariate extreme value distribution.
ord	A vector of length three, which should be a permutation of the set $\{1,2,3\}$. The points $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$ (the vertices of the simplex) are depicted clockwise from the top in the order defined by ord. The argument alters the way in which the function is plotted; it does not change the function definition.
lab	A character vector of length three, in which case the ith margin is labelled using the ith component, or NULL, in which case no labels are given. By default, lab is as.character(1:3). The actual location of the margins, and hence the labels, is defined by ord.
lcex	A numerical value giving the amount by which the labels should be scaled relative to the default. Ignored if lab is NULL.

Value

A numeric vector of estimates. If plotting, the smallest evaluated estimate is returned invisibly.

Note

The rows of data that contain missing values are not used in the estimation of the dependence structure, but every non-missing value is used in estimating the margins.

The dependence function of the multivariate extreme value distribution is defined in amvevd. The function amvevd calculates and plots dependence functions of multivariate logistic and multivariate asymmetric logistic models.

The estimator plotted or calculated is a multivariate extension of the bivariate Pickands estimator defined in abvnonpar.

See Also

amvevd, abvnonpar, fgev

Examples

s5pts <- matrix(rexp(50), nrow = 10, ncol = 5)
s5pts <- s5pts/rowSums(s5pts)
sdat <- rmvevd(100, dep = 0.6, model = "log", d = 5)
amvnonpar(s5pts, sdat, d = 5)

## Not run: amvnonpar(data = sdat, plot = TRUE)
## Not run: amvnonpar(data = sdat, plot = TRUE, ord = c(2,3,1), lab = LETTERS[1:3])
## Not run: amvevd(dep = 0.6, model = "log", plot = TRUE)
## Not run: amvevd(dep = 0.6, model = "log", plot = TRUE, blty = 1)

---

Compare Nested EVD Objects

Description

Compute an analysis of deviance table for two or more nested evd objects.

Usage

## S3 method for class 'evd'
anova(object, object2, ..., half = FALSE)

Arguments

object	An object of class "evd".
object2	An object of class "evd" that represents a model nested within object.
...	Further successively nested objects.
half	For some non-regular tesing problems the deviance difference is known to be one half of a chi-squared random variable. Set half to TRUE in these cases.

Value

An object of class c("anova", "data.frame"), with one row for each model, and the following five columns

M.Df	The number of parameters.
Deviance	The deviance.
Df	The number of parameters of the model in the previous row minus the number of parameters.
Chisq	The deviance minus the deviance of the model in the previous row (or twice this if half is TRUE).
Pr(>chisq)	The p-value calculated by comparing the quantile Chisq with a chi-squared distribution on Df degrees of freedom.

Warning

Circumstances may arise such that the asymptotic distribution of the test statistic is not chi-squared. In particular, this occurs when the smaller model is constrained at the edge of the parameter space. It is up to the user recognize this, and to interpret the output correctly.

In some cases the asymptotic distribution is known to be one half of a chi-squared; you can set half = TRUE in these cases.

See Also

fbvevd, fextreme, fgev, forder

Examples

uvdata <- rgev(100, loc = 0.13, scale = 1.1, shape = 0.2)
trend <- (-49:50)/100
M1 <- fgev(uvdata, nsloc = trend)
M2 <- fgev(uvdata)
M3 <- fgev(uvdata, shape = 0)
anova(M1, M2, M3)

bvdata <- rbvevd(100, dep = 0.75, model = "log")
M1 <- fbvevd(bvdata, model = "log")
M2 <- fbvevd(bvdata, model = "log", dep = 0.75)
M3 <- fbvevd(bvdata, model = "log", dep = 1)
anova(M1, M2)
anova(M1, M3, half = TRUE)

---

Parametric Bivariate Extreme Value Distributions

Description

Density function, distribution function and random generation for nine parametric bivariate extreme value models.

Usage

dbvevd(x, dep, asy = c(1, 1), alpha, beta, model = c("log", "alog",
    "hr", "neglog", "aneglog", "bilog", "negbilog", "ct", "amix"),
    mar1 = c(0, 1, 0), mar2 = mar1, log = FALSE) 
pbvevd(q, dep, asy = c(1, 1), alpha, beta, model = c("log", "alog",
    "hr", "neglog", "aneglog", "bilog", "negbilog", "ct", "amix"),
    mar1 = c(0, 1, 0), mar2 = mar1, lower.tail = TRUE) 
rbvevd(n, dep, asy = c(1, 1), alpha, beta, model = c("log", "alog",
    "hr", "neglog", "aneglog", "bilog", "negbilog", "ct", "amix"),
    mar1 = c(0, 1, 0), mar2 = mar1)

Arguments

x, q	A vector of length two or a matrix with two columns, in which case the density/distribution is evaluated across the rows.
n	Number of observations.
dep	Dependence parameter for the logistic, asymmetric logistic, Husler-Reiss, negative logistic and asymmetric negative logistic models.
asy	A vector of length two, containing the two asymmetry parameters for the asymmetric logistic and asymmetric negative logistic models.
alpha, beta	Alpha and beta parameters for the bilogistic, negative bilogistic, Coles-Tawn and asymmetric mixed models.
model	The specified model; a character string. Must be either "log" (the default), "alog", "hr", "neglog", "aneglog", "bilog", "negbilog", "ct" or "amix" (or any unique partial match), for the logistic, asymmetric logistic, Husler-Reiss, negative logistic, asymmetric negative logistic, bilogistic, negative bilogistic, Coles-Tawn and asymmetric mixed models respectively. If parameter arguments are given that do not correspond to the specified model those arguments are ignored, with a warning.
mar1, mar2	Vectors of length three containing marginal parameters, or matrices with three columns where each column represents a vector of values to be passed to the corresponding marginal parameter.
log	Logical; if TRUE, the log density is returned.
lower.tail	Logical; if TRUE (default), the distribution function is returned; the survivor function is returned otherwise.

Details

Define

$y_i = y_i(z_i) = \{1+s_i(z_i-a_i)/b_i\}^{-1/s_i}$

for $1+s_i(z_i-a_i)/b_i > 0$ and $i = 1,2$, where the marginal parameters are given by $\code{mari} = (a_i,b_i,s_i)$, $b_i > 0$. If $s_i = 0$ then $y_i$ is defined by continuity.

In each of the bivariate distributions functions $G(z_1,z_2)$ given below, the univariate margins are generalized extreme value, so that $G(z_i) = \exp(-y_i)$ for $i = 1,2$. If $1+s_i(z_i-a_i)/b_i \leq 0$ for some $i = 1,2$, the value $z_i$ is either greater than the upper end point (if $s_i < 0$), or less than the lower end point (if $s_i > 0$), of the $i$th univariate marginal distribution.

model = "log" (Gumbel, 1960)

The bivariate logistic distribution function with parameter $\code{dep} = r$ is

$G(z_1,z_2) = \exp\left[-(y_1^{1/r}+y_2^{1/r})^r\right]$

where $0 < r \leq 1$. This is a special case of the bivariate asymmetric logistic model. Complete dependence is obtained in the limit as $r$ approaches zero. Independence is obtained when $r = 1$.

model = "alog" (Tawn, 1988)

The bivariate asymmetric logistic distribution function with parameters $\code{dep} = r$ and $\code{asy} = (t_1,t_2)$ is

$G(z_1,z_2) = \exp\left\{-(1-t_1)y_1-(1-t_2)y_2- [(t_1y_1)^{1/r}+(t_2y_2)^{1/r}]^r\right\}$

where $0 < r \leq 1$ and $0 \leq t_1,t_2 \leq 1$. When $t_1 = t_2 = 1$ the asymmetric logistic model is equivalent to the logistic model. Independence is obtained when either $r = 1$, $t_1 = 0$ or $t_2 = 0$. Complete dependence is obtained in the limit when $t_1 = t_2 = 1$ and $r$ approaches zero. Different limits occur when $t_1$ and $t_2$ are fixed and $r$ approaches zero.

model = "hr" (Husler and Reiss, 1989)

The Husler-Reiss distribution function with parameter $\code{dep} = r$ is

$G(z_1,z_2) = \exp\left(-y_1\Phi\{r^{-1}+{\textstyle\frac{1}{2}} r[\log(y_1/y_2)]\} - y_2\Phi\{r^{-1}+{\textstyle\frac{1}{2}}r [\log(y_2/y_1)]\}\right)$

where $\Phi(\cdot)$ is the standard normal distribution function and $r > 0$. Independence is obtained in the limit as $r$ approaches zero. Complete dependence is obtained as $r$ tends to infinity.

model = "neglog" (Galambos, 1975)

The bivariate negative logistic distribution function with parameter $\code{dep} = r$ is

$G(z_1,z_2) = \exp\left\{-y_1-y_2+ [y_1^{-r}+y_2^{-r}]^{-1/r}\right\}$

where $r > 0$. This is a special case of the bivariate asymmetric negative logistic model. Independence is obtained in the limit as $r$ approaches zero. Complete dependence is obtained as $r$ tends to infinity. The earliest reference to this model appears to be Galambos (1975, Section 4).

model = "aneglog" (Joe, 1990)

The bivariate asymmetric negative logistic distribution function with parameters parameters $\code{dep} = r$ and $\code{asy} = (t_1,t_2)$ is

$G(z_1,z_2) = \exp\left\{-y_1-y_2+ [(t_1y_1)^{-r}+(t_2y_2)^{-r}]^{-1/r}\right\}$

where $r > 0$ and $0 < t_1,t_2 \leq 1$. When $t_1 = t_2 = 1$ the asymmetric negative logistic model is equivalent to the negative logistic model. Independence is obtained in the limit as either $r$, $t_1$ or $t_2$ approaches zero. Complete dependence is obtained in the limit when $t_1 = t_2 = 1$ and $r$ tends to infinity. Different limits occur when $t_1$ and $t_2$ are fixed and $r$ tends to infinity. The earliest reference to this model appears to be Joe (1990), who introduces a multivariate extreme value distribution which reduces to $G(z_1,z_2)$ in the bivariate case.

model = "bilog" (Smith, 1990)

The bilogistic distribution function with parameters $\code{alpha} = \alpha$ and $\code{beta} = \beta$ is

$G(z_1,z_2) = \exp\left\{-y_1 q^{1-\alpha} - y_2 (1-q)^{1-\beta}\right\}$

where $q = q(y_1,y_2;\alpha,\beta)$ is the root of the equation

$(1-\alpha) y_1 (1-q)^\beta - (1-\beta) y_2 q^\alpha = 0,$

$0 < \alpha,\beta < 1$. When $\alpha = \beta$ the bilogistic model is equivalent to the logistic model with dependence parameter $\code{dep} = \alpha = \beta$. Complete dependence is obtained in the limit as $\alpha = \beta$ approaches zero. Independence is obtained as $\alpha = \beta$ approaches one, and when one of $\alpha,\beta$ is fixed and the other approaches one. Different limits occur when one of $\alpha,\beta$ is fixed and the other approaches zero. A bilogistic model is fitted in Smith (1990), where it appears to have been first introduced.

model = "negbilog" (Coles and Tawn, 1994)

The negative bilogistic distribution function with parameters $\code{alpha} = \alpha$ and $\code{beta} = \beta$ is

$G(z_1,z_2) = \exp\left\{- y_1 - y_2 + y_1 q^{1+\alpha} + y_2 (1-q)^{1+\beta}\right\}$

where $q = q(y_1,y_2;\alpha,\beta)$ is the root of the equation

$(1+\alpha) y_1 q^\alpha - (1+\beta) y_2 (1-q)^\beta = 0,$

$\alpha > 0$ and $\beta > 0$. When $\alpha = \beta$ the negative bilogistic model is equivalent to the negative logistic model with dependence parameter $\code{dep} = 1/\alpha = 1/\beta$. Complete dependence is obtained in the limit as $\alpha = \beta$ approaches zero. Independence is obtained as $\alpha = \beta$ tends to infinity, and when one of $\alpha,\beta$ is fixed and the other tends to infinity. Different limits occur when one of $\alpha,\beta$ is fixed and the other approaches zero.

model = "ct" (Coles and Tawn, 1991)

The Coles-Tawn distribution function with parameters $\code{alpha} = \alpha > 0$ and $\code{beta} = \beta > 0$ is

$G(z_1,z_2) = \exp\left\{-y_1 [1 - \mbox{Be}(q;\alpha+1,\beta)] - y_2 \mbox{Be}(q;\alpha,\beta+1) \right\}$

where $q = \alpha y_2 / (\alpha y_2 + \beta y_1)$ and $\mbox{Be}(q;\alpha,\beta)$ is the beta distribution function evaluated at $q$ with $\code{shape1} = \alpha$ and $\code{shape2} = \beta$. Complete dependence is obtained in the limit as $\alpha = \beta$ tends to infinity. Independence is obtained as $\alpha = \beta$ approaches zero, and when one of $\alpha,\beta$ is fixed and the other approaches zero. Different limits occur when one of $\alpha,\beta$ is fixed and the other tends to infinity.

model = "amix" (Tawn, 1988)

The asymmetric mixed distribution function with parameters $\code{alpha} = \alpha$ and $\code{beta} = \beta$ has a dependence function with the following cubic polynomial form.

$A(t) = 1 - (\alpha +\beta)t + \alpha t^2 + \beta t^3$

where $\alpha$ and $\alpha + 3\beta$ are non-negative, and where $\alpha + \beta$ and $\alpha + 2\beta$ are less than or equal to one. These constraints imply that beta lies in the interval [-0.5,0.5] and that alpha lies in the interval [0,1.5], though alpha can only be greater than one if beta is negative. The strength of dependence increases for increasing alpha (for fixed beta). Complete dependence cannot be obtained. Independence is obtained when both parameters are zero. For the definition of a dependence function, see abvevd.

Value

dbvevd gives the density function, pbvevd gives the distribution function and rbvevd generates random deviates, for one of nine parametric bivariate extreme value models.

Note

The logistic and asymmetric logistic models respectively are simulated using bivariate versions of Algorithms 1.1 and 1.2 in Stephenson(2003). All other models are simulated using a root finding algorithm to simulate from the conditional distributions.

The simulation of the bilogistic and negative bilogistic models requires a root finding algorithm to evaluate $q$ within the root finding algorithm used to simulate from the conditional distributions. The generation of bilogistic and negative bilogistic random deviates is therefore relatively slow (about 2.8 seconds per 1000 random vectors on a 450MHz PIII, 512Mb RAM).

The bilogistic and negative bilogistic models can be represented under a single model, using the integral of the maximum of two beta distributions (Joe, 1997).

The Coles-Tawn model is called the Dirichelet model in Coles and Tawn (1991).

References

Coles, S. G. and Tawn, J. A. (1991) Modelling extreme multivariate events. J. Roy. Statist. Soc., B, 53, 377–392.

Coles, S. G. and Tawn, J. A. (1994) Statistical methods for multivariate extremes: an application to structural design (with discussion). Appl. Statist., 43, 1–48.

Galambos, J. (1975) Order statistics of samples from multivariate distributions. J. Amer. Statist. Assoc., 70, 674–680.

Gumbel, E. J. (1960) Distributions des valeurs extremes en plusieurs dimensions. Publ. Inst. Statist. Univ. Paris, 9, 171–173.

Husler, J. and Reiss, R.-D. (1989) Maxima of normal random vectors: between independence and complete dependence. Statist. Probab. Letters, 7, 283–286.

Joe, H. (1990) Families of min-stable multivariate exponential and multivariate extreme value distributions. Statist. Probab. Letters, 9, 75–81.

Joe, H. (1997) Multivariate Models and Dependence Concepts, London: Chapman & Hall.

Smith, R. L. (1990) Extreme value theory. In Handbook of Applicable Mathematics (ed. W. Ledermann), vol. 7. Chichester: John Wiley, pp. 437–471.

Stephenson, A. G. (2003) Simulating multivariate extreme value distributions of logistic type. Extremes, 6(1), 49–60.

Tawn, J. A. (1988) Bivariate extreme value theory: models and estimation. Biometrika, 75, 397–415.

See Also

abvevd, rgev, rmvevd

Examples

pbvevd(matrix(rep(0:4,2), ncol=2), dep = 0.7, model = "log")
pbvevd(c(2,2), dep = 0.7, asy = c(0.6,0.8), model = "alog")
pbvevd(c(1,1), dep = 1.7, model = "hr")

margins <- cbind(0, 1, seq(-0.5,0.5,0.1))
rbvevd(11, dep = 1.7, model = "hr", mar1 = margins)
rbvevd(10, dep = 1.2, model = "neglog", mar1 = c(10, 1, 1))
rbvevd(10, alpha = 0.7, beta = 0.52, model = "bilog")

dbvevd(c(0,0), dep = 1.2, asy = c(0.5,0.9), model = "aneglog")
dbvevd(c(0,0), alpha = 0.75, beta = 0.5, model = "ct", log = TRUE)
dbvevd(c(0,0), alpha = 0.7, beta = 1.52, model = "negbilog")

---

Bivariate Threshold Choice Plot

Description

Produces a diagnostic plot to assist with threshold choice for bivariate data.

Usage

bvtcplot(x, spectral = FALSE, xlab, ylab, ...)

Arguments

x	A matrix or data frame, ordinarily with two columns, which may contain missing values.
spectral	If TRUE, an estimate of the spectral measure is plotted instead of the diagnostic plot.
ylab, xlab	Graphics parameters.
...	Other arguments to be passed to the plotting function.

Details

If spectral is FALSE (the default), produces a threshold choice plot as illustrated in Beirlant et al. (2004). With $n$ non-missing bivariate observations, the integers $k = 1,\dots,n-1$ are plotted against the values $(k/n)r_{(n-k)}$, where $r_{(n-k)}$ is the $(n-k)$th order statistic of the sum of the margins following empirical transformation to standard Frechet.

A vertical line is drawn at k0, where k0 is the largest integer for which the y-axis is above the value two. If spectral is FALSE, the largest k0 data points are used to plot an estimate of the spectal measure $H([0, w])$ versus $w$.

Value

A list is invisibly returned giving k0 and the values used to produce the plot.

References

Beirlant, J., Goegebeur, Y., Segers, J. and Teugels, J. L. (2004) Statistics of Extremes: Theory and Applications., Chichester, England: John Wiley and Sons.

See Also

fbvpot, tcplot

Examples

## Not run: bvtcplot(lossalae)
## Not run: bvtcplot(lossalae, spectral = TRUE)

---

Calculate Conditional Copulas for Parametric Bivariate Extreme Value Distributions

Description

Conditional copula functions, conditioning on either margin, for nine parametric bivariate extreme value models.

Usage

ccbvevd(x, mar = 2, dep, asy = c(1, 1), alpha, beta, model = c("log", 
    "alog", "hr", "neglog", "aneglog", "bilog", "negbilog", "ct", 
    "amix"), lower.tail = TRUE)

Arguments

x	A matrix or data frame, ordinarily with two columns, which may contain missing values. A data frame may also contain a third column of mode logical, which itself may contain missing values (see Details).
mar	One or two; conditions on this margin.
dep	Dependence parameter for the logistic, asymmetric logistic, Husler-Reiss, negative logistic and asymmetric negative logistic models.
asy	A vector of length two, containing the two asymmetry parameters for the asymmetric logistic and asymmetric negative logistic models.
alpha, beta	Alpha and beta parameters for the bilogistic, negative bilogistic, Coles-Tawn and asymmetric mixed models.
model	The specified model; a character string. Must be either "log" (the default), "alog", "hr", "neglog", "aneglog", "bilog", "negbilog", "ct" or "amix" (or any unique partial match), for the logistic, asymmetric logistic, Husler-Reiss, negative logistic, asymmetric negative logistic, bilogistic, negative bilogistic, Coles-Tawn and asymmetric mixed models respectively. If parameter arguments are given that do not correspond to the specified model those arguments are ignored, with a warning.
lower.tail	Logical; if TRUE (default), the conditional distribution function is returned; the conditional survivor function is returned otherwise.

Details

The function calculates $P(U_1 < x_1|U_2 = x_2)$, where $(U_1,U_2)$ is a random vector with Uniform(0,1) margins and with a dependence structure given by the specified parametric model. By default, the values of $x_1$ and $x_1$ are given by the first and second columns of the argument x. If mar = 1 then this is reversed.

If x has a third column $x_3$ of mode logical, then the function returns $P(U_1 < x_1|U_2 = x_2,I = x_3)$, according to inference proceedures derived by Stephenson and Tawn (2004). See fbvevd. This requires numerical integration, and hence will be slower.

This function is mainly for internal use. It is used by plot.bvevd to calculate the conditional P-P plotting diagnostics.

Value

A numeric vector of probabilities.

References

Stephenson, A. G. and Tawn, J. A. (2004) Exploiting Occurence Times in Likelihood Inference for Componentwise Maxima. Biometrika 92(1), 213–217.

See Also

rbvevd, fbvevd, plot.bvevd

---

Dependence Measure Plots

Description

Plots of estimates of the dependence measures chi and chi-bar for bivariate data.

Usage

chiplot(data, nq = 100, qlim = NULL, which = 1:2, conf = 0.95, trunc =
    TRUE, spcases = FALSE, lty = 1, cilty = 2, col = 1, cicol = 1,
    xlim = c(0,1), ylim1 = c(-1,1), ylim2 = c(-1,1), main1 = "Chi Plot",
    main2 = "Chi Bar Plot", xlab = "Quantile", ylab1 = "Chi", ylab2 =
    "Chi Bar", ask = nb.fig < length(which) && dev.interactive(), ...)

Arguments

data	A matrix or data frame with two columns. Rows (observations) with missing values are stripped from the data before any computations are performed.
nq	The number of quantiles at which the measures are evaluated.
qlim	The limits of the quantiles at which the measures are evaluated (see Details).
which	If only one plot is required, specify 1 for chi and 2 for chi-bar.
conf	The confidence coefficient of the plotted confidence intervals.
trunc	Logical; truncate the estimates at their theoretical upper and lower bounds?
spcases	If TRUE, plots greyed lines corresponding to the special cases of perfect positive/negative dependence and exact independence.
lty, cilty	Line types for the estimates of the measures and for the confidence intervals respectively. Use zero to supress.
col, cicol	Colour types for the estimates of the measures and for the confidence intervals respectively.
xlim, xlab	Limits and labels for the x-axis; they apply to both plots.
ylim1	Limits for the y-axis of the chi plot. If this is NULL (the default) the upper limit is one, and the lower limit is the minimum of zero and the smallest plotted value.
ylim2	Limits for the y-axis of the chi-bar plot.
main1, main2	The plot titles for the chi and chi-bar plots respectively.
ylab1, ylab2	The y-axis labels for the chi and chi-bar plots respectively.
ask	Logical; if TRUE, the user is asked before each plot.
...	Other arguments to be passed to matplot.

Details

These measures are explained in full detail in Coles, Heffernan and Tawn (1999). A brief treatment is also given in Section 8.4 of Coles(2001). A short summary is given as follows. We assume that the data are iid random vectors with common bivariate distribution function $G$, and we define the random vector $(X,Y)$ to be distributed according to $G$.

The chi plot is a plot of $q$ against empirical estimates of

$\chi(q) = 2 - \log(\Pr(F_X(X) < q, F_Y(Y) < q)) / \log(q)$

where $F_X$ and $F_Y$ are the marginal distribution functions, and where $q$ is in the interval (0,1). The quantity $\chi(q)$ is bounded by

$2 - \log(2u - 1)/\log(u) \leq \chi(q) \leq 1$

where the lower bound is interpreted as -Inf for $q \leq 1/2$ and zero for $q = 1$. These bounds are reflected in the corresponding estimates.

The chi bar plot is a plot of $q$ against empirical estimates of

$\bar{\chi}(q) = 2\log(1-q)/\log(\Pr(F_X(X) > q, F_Y(Y) > q)) - 1$

where $F_X$ and $F_Y$ are the marginal distribution functions, and where $q$ is in the interval (0,1). The quantity $\bar{\chi}(q)$ is bounded by $-1 \leq \bar{\chi}(q) \leq 1$ and these bounds are reflected in the corresponding estimates.

Note that the empirical estimators for $\chi(q)$ and $\bar{\chi}(q)$ are undefined near $q=0$ and $q=1$. By default the function takes the limits of $q$ so that the plots depicts all values at which the estimators are defined. This can be overridden by the argument qlim, which must represent a subset of the default values (and these can be determined using the component quantile of the invisibly returned list; see Value).

The confidence intervals within the plot assume that observations are independent, and that the marginal distributions are estimated exactly. The intervals are constructed using the delta method; this may lead to poor interval estimates near $q=0$ and $q=1$.

The function $\chi(q)$ can be interpreted as a quantile dependent measure of dependence. In particular, the sign of $\chi(q)$ determines whether the variables are positively or negatively associated at quantile level $q$. By definition, variables are said to be asymptotically independent when $\chi(1)$ (defined in the limit) is zero. For independent variables, $\chi(q) = 0$ for all $q$ in (0,1). For perfectly dependent variables, $\chi(q) = 1$ for all $q$ in (0,1). For bivariate extreme value distributions, $\chi(q) = 2(1-A(1/2))$ for all $q$ in (0,1), where $A$ is the dependence function, as defined in abvevd. If a bivariate threshold model is to be fitted (using fbvpot), this plot can therefore act as a threshold identification plot, since e.g. the use of 95% marginal quantiles as threshold values implies that $\chi(q)$ should be approximately constant above $q = 0.95$.

The function $\bar{\chi}(q)$ can again be interpreted as a quantile dependent measure of dependence; it is most useful within the class of asymptotically independent variables. For asymptotically dependent variables (i.e. those for which $\chi(1) < 1$), we have $\bar{\chi}(1) = 1$, where $\bar{\chi}(1)$ is again defined in the limit. For asymptotically independent variables, $\bar{\chi}(1)$ provides a measure that increases with dependence strength. For independent variables $\bar{\chi}(q) = 0$ for all $q$ in (0,1), and hence $\bar{\chi}(1) = 0$.

Value

A list with components quantile, chi (if 1 is in which) and chibar (if 2 is in which) is invisibly returned. The components quantile and chi contain those objects that were passed to the formal arguments x and y of matplot in order to create the chi plot. The components quantile and chibar contain those objects that were passed to the formal arguments x and y of matplot in order to create the chi-bar plot.

Author(s)

Jan Heffernan and Alec Stephenson

References

Coles, S. G., Heffernan, J. and Tawn, J. A. (1999) Dependence measures for extreme value analyses. Extremes, 2, 339–365.

Coles, S. G. (2001) An Introduction to Statistical Modelling of Extreme Values, London: Springer–Verlag.

See Also

fbvevd, fbvpot, matplot

Examples

par(mfrow = c(1,2))
smdat1 <- rbvevd(1000, dep = 0.6, model = "log")
smdat2 <- rbvevd(1000, dep = 1, model = "log")
chiplot(smdat1)
chiplot(smdat2)

---

Identify Clusters of Exceedences

Description

Identify clusters of exceedences.

Usage

clusters(data, u, r = 1, ulow = -Inf, rlow = 1, cmax = FALSE, keep.names
    = TRUE, plot = FALSE, xdata = seq(along = data), lvals = TRUE, lty =
    1, lwd = 1, pch = par("pch"), col = if(n > 250) NULL else "grey",
    xlab = "Index", ylab = "Data", ...)

Arguments

data	A numeric vector, which may contain missing values.
u	A single value giving the threshold, unless a time varying threshold is used, in which case u should be a vector of thresholds, typically with the same length as data (or else the usual recycling rules are applied).
r	A postive integer denoting the clustering interval length. By default the interval length is one.
ulow	A single value giving the lower threshold, unless a time varying lower threshold is used, in which case ulow should be a vector of lower thresholds, typically with the same length as data (or else the usual recycling rules are applied). By default there is no lower threshold (or equivalently, the lower threshold is -Inf).
rlow	A postive integer denoting the lower clustering interval length. By default the interval length is one.
cmax	Logical; if FALSE (the default), a list containing the clusters of exceedences is returned. If TRUE a numeric vector containing the cluster maxima is returned.
keep.names	Logical; if FALSE, the function makes no attempt to retain the names/indices of the observations within the returned object. If data contains a large number of observations, this can make the function run much faster. The argument is mainly designed for internal use.
plot	Logical; if TRUE a plot is given that depicts the identified clusters, and the clusters (if cmax is FALSE) or cluster maxima (if cmax is TRUE) are returned invisibly. If FALSE (the default), the following arguments are ignored.
xdata	A numeric vector with the same length as data, giving the values to be plotted on the x-axis.
lvals	Logical; should the values below the threshold and the line depicting the lower threshold be plotted?
lty, lwd	Line type and width for the lines depicting the threshold and the lower threshold.
pch	Plotting character.
col	Strips of colour col are used to identify the clusters. An observation is contained in the cluster if the centre of the corresponding plotting character is contained in the coloured strip. If col is NULL the strips are omitted. By default the strips are coloured "grey", but are omitted whenever data contains more than 250 observations.
xlab, ylab	Labels for the x and y axis.
...	Other graphics parameters.

Details

The clusters of exceedences are identified as follows. The first exceedence of the threshold initiates the first cluster. The first cluster then remains active until either r consecutive values fall below (or are equal to) the threshold, or until rlow consecutive values fall below (or are equal to) the lower threshold. The next exceedence of the threshold (if it exists) then initiates the second cluster, and so on. Missing values are allowed, in which case they are treated as falling below (or equal to) the threshold, but falling above the lower threshold.

Value

If cmax is FALSE (the default), a list with one component for each identified cluster. If cmax is TRUE, a numeric vector containing the cluster maxima. In any case, the returned object has an attribute acs, giving the average cluster size (where the cluster size is defined as the number of exceedences within a cluster), which will be NaN if there are no values above the threshold (and hence no clusters).

If plot is TRUE, the list of clusters, or vector of cluster maxima, is returned invisibly.

See Also

exi, exiplot

Examples

clusters(portpirie, 4.2, 3)
clusters(portpirie, 4.2, 3, cmax = TRUE)
clusters(portpirie, 4.2, 3, 3.8, plot = TRUE)
clusters(portpirie, 4.2, 3, 3.8, plot = TRUE, lvals = FALSE)
tvu <- c(rep(4.2, 20), rep(4.1, 25), rep(4.2, 20))
clusters(portpirie, tvu, 3, plot = TRUE)

---

Calculate Confidence Intervals

Description

Calculate profile and Wald confidence intervals of parameters in fitted models.

Usage

## S3 method for class 'evd'
confint(object, parm, level = 0.95, ...)
## S3 method for class 'profile.evd'
confint(object, parm, level = 0.95, ...)

Arguments

object	Either a fitted model object (of class evd) for Wald confidence intervals, or a profile trace (of class profile.evd) for profile likelihood confidence intervals.
parm	A character vector of parameters; a confidence interval is calculated for each parameter. If missing, then intervals are returned for all parameters in the fitted model or profile trace.
level	A single number giving the confidence level.
...	Not used.

Value

A matrix with two columns giving lower and upper confidence limits.

For profile confidence intervals, this function assumes that the profile trace is unimodal. If the profile trace is not unimodal then the function will give spurious results.

See Also

profile.evd

Examples

m1 <- fgev(portpirie)
confint(m1)
## Not run: pm1 <- profile(m1)
## Not run: plot(pm1)
## Not run: confint(pm1)

---

Perform Hypothesis Test Of Independence

Description

Perform score and likelihood ratio tests of independence for bivariate data, assuming a logistic dependence model as the alternative.

Usage

evind.test(x, method = c("ratio", "score"), verbose = FALSE)

Arguments

x	A matrix or data frame, ordinarily with two columns, which may contain missing values.
method	The test methodology; either "ratio" for the likelihood ratio test or "score" for the score test.
verbose	If TRUE, shows estimates of the marginal parameters in addition to the dependence parameter.

Details

This simple function fits a stationary bivariate logistic model to the data and performs a hypothesis test of $\code{dep} = 1$ versus $\code{dep} < 1$ using the methodology in Tawn (1988). The null distributions for the printed test statistics are chi-squared on one df for the likelihood ratio test, and standard normal for the score test.

Value

An object of class "htest".

References

Tawn, J. A. (1988) Bivariate extreme value theory: models and estimation. Biometrika, 75, 397–415.

See Also

fbvevd, t.test

Examples

evind.test(sealevel)
evind.test(sealevel, method = "score")

---

Simulate Markov Chains With Extreme Value Dependence Structures

Description

Simulation of first order Markov chains, such that each pair of consecutive values has the dependence structure of one of nine parametric bivariate extreme value distributions.

Usage

evmc(n, dep, asy = c(1,1), alpha, beta, model = c("log", "alog",
    "hr", "neglog", "aneglog", "bilog", "negbilog", "ct", "amix"),
    margins = c("uniform","rweibull","frechet","gumbel"))

Arguments

n	Number of observations.
dep	Dependence parameter for the logistic, asymmetric logistic, Husler-Reiss, negative logistic and asymmetric negative logistic models.
asy	A vector of length two, containing the two asymmetry parameters for the asymmetric logistic and asymmetric negative logistic models.
alpha, beta	Alpha and beta parameters for the bilogistic, negative bilogistic, Coles-Tawn and asymmetric mixed models.
model	The specified model; a character string. Must be either "log" (the default), "alog", "hr", "neglog", "aneglog", "bilog", "negbilog", "ct" or "amix" (or any unique partial match), for the logistic, asymmetric logistic, Husler-Reiss, negative logistic, asymmetric negative logistic, bilogistic, negative bilogistic, Coles-Tawn and asymmetric mixed models respectively. The definition of each model is given in rbvevd. If parameter arguments are given that do not correspond to the specified model those arguments are ignored, with a warning.
margins	The marginal distribution of each value; a character string. Must be either "uniform" (the default), "rweibull", "frechet" or "gumbel" (or any unique partial match), for the uniform, standard reverse Weibull, standard Gumbel and standard Frechet distributions respectively.

Value

A numeric vector of length n.

See Also

marma, rbvevd

Examples

evmc(100, alpha = 0.1, beta = 0.1, model = "bilog")
evmc(100, dep = 10, model = "hr", margins = "gum")

---

Estimates of the Extremal Index

Description

Estimates of the extremal index.

Usage

exi(data, u, r = 1, ulow = -Inf, rlow = 1)

Arguments

data	A numeric vector, which may contain missing values.
u	A single value giving the threshold, unless a time varying threshold is used, in which case u should be a vector of thresholds, typically with the same length as data (or else the usual recycling rules are applied).
r	Either a postive integer denoting the clustering interval length, or zero, in which case the intervals estimator of Ferro and Segers (2003) is used and following arguments are ignored. By default the interval length is one.
ulow	A single value giving the lower threshold, unless a time varying lower threshold is used, in which case ulow should be a vector of lower thresholds, typically with the same length as data (or else the usual recycling rules are applied). By default there is no lower threshold (or equivalently, the lower threshold is -Inf).
rlow	A postive integer denoting the lower clustering interval length. By default the interval length is one.

Details

If r is a positive integer the extremal index is estimated using the inverse of the average cluster size, using the clusters of exceedences derived from clusters. If r is zero, an estimate based on inter-exceedance times is used (Ferro and Segers, 2003).

If there are no exceedances of the threshold, the estimate is NaN. If there is only one exceedance, the estimate is one.

Value

A single value estimating the extremal index.

References

Ferro, C. A. T. and Segers, J. (2003) Inference for clusters of extreme values. JRSS B, 65, 545–556.

See Also

clusters, exiplot

Examples

exi(portpirie, 4.2, r = 3, ulow = 3.8)
tvu <- c(rep(4.2, 20), rep(4.1, 25), rep(4.2, 20))
exi(portpirie, tvu, r = 1)
exi(portpirie, tvu, r = 0)

---

Plot Estimates of the Extremal Index

Description

Plots estimates of the extremal index.

Usage

exiplot(data, tlim, r = 1, ulow = -Inf, rlow = 1, add = FALSE, 
    nt = 100, lty = 1, xlab = "Threshold", ylab = "Ext. Index",
    ylim = c(0,1), ...)

Arguments

data	A numeric vector, which may contain missing values.
tlim	A numeric vector of length two, giving the limits for the (time invariant) thresholds at which the estimates are evaluated.
r, ulow, rlow	The estimation method. See exi.
add	Add to an existing plot?
nt	The number of thresholds at which the estimates are evaluated.
lty	Line type.
xlab, ylab	x and y axis labels.
ylim	y axis limits.
...	Other arguments passed to plot or lines.

Details

The estimates are calculated using the function exi.

Value

A list with components x and y is invisibly returned. The first component contains the thresholds, the second contains the estimates.

See Also

clusters, exi

Examples

sdat <- mar(100, psi = 0.5)
tlim <- quantile(sdat, probs = c(0.4,0.9))
exiplot(sdat, tlim)
exiplot(sdat, tlim, r = 4, add = TRUE, lty = 2)
exiplot(sdat, tlim, r = 0, add = TRUE, lty = 4)

---

Distributions of Maxima and Minima

Description

Density function, distribution function, quantile function and random generation for the maximum/minimum of a given number of independent variables from a specified distribution.

Usage

dextreme(x, densfun, distnfun, ..., distn, mlen = 1, largest = TRUE,
    log = FALSE)
pextreme(q, distnfun, ..., distn, mlen = 1, largest = TRUE,
    lower.tail = TRUE) 
qextreme(p, quantfun, ..., distn, mlen = 1, largest = TRUE,
    lower.tail = TRUE) 
rextreme(n, quantfun, ..., distn, mlen = 1, largest = TRUE)

Arguments

x, q	Vector of quantiles.
p	Vector of probabilities.
n	Number of observations.
densfun, distnfun, quantfun	Density, distribution and quantile function of the specified distribution. The density function must have a log argument (a simple wrapper can always be constructed to achieve this).
...	Parameters of the specified distribution.
distn	A character string, optionally given as an alternative to densfun, distnfun and quantfun such that the density, distribution and quantile functions are formed upon the addition of the prefixes d, p and q respectively.
mlen	The number of independent variables.
largest	Logical; if TRUE (default) use maxima, otherwise minima.
log	Logical; if TRUE, the log density is returned.
lower.tail	Logical; if TRUE (default) probabilities are P[X <= x], otherwise P[X > x].

Value

dextreme gives the density function, pextreme gives the distribution function and qextreme gives the quantile function of the maximum/minimum of mlen independent variables from a specified distibution. rextreme generates random deviates.

See Also

rgev, rorder

Examples

dextreme(2:4, dnorm, pnorm, mean = 0.5, sd = 1.2, mlen = 5)
dextreme(2:4, distn = "norm", mean = 0.5, sd = 1.2, mlen = 5)
dextreme(2:4, distn = "exp", mlen = 2, largest = FALSE)
pextreme(2:4, distn = "exp", rate = 1.2, mlen = 2)
qextreme(seq(0.9, 0.6, -0.1), distn = "exp", rate = 1.2, mlen = 2)
rextreme(5, qgamma, shape = 1, mlen = 10)
p <- (1:9)/10
pexp(qextreme(p, distn = "exp", rate = 1.2, mlen = 1), rate = 1.2)
## [1] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

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Failure Times

Description

Failure times.

Usage

failure

Format

A vector containing 24 observations.

Source

van Montfort, M. A. J. and Otten, A. (1978) On testing a shape parameter in the presence of a scale parameter. Math. Operations Forsch. Statist., Ser. Statistics, 9, 91–104.

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Maximum-likelihood Fitting of Bivariate Extreme Value Distributions

Description

Fit models for one of nine parametric bivariate extreme value distributions, including linear modelling of the marginal location parameters, and allowing any of the parameters to be held fixed if desired.

Usage

fbvevd(x, model = c("log", "alog", "hr", "neglog", "aneglog", "bilog",
    "negbilog", "ct", "amix"), start, ..., sym = FALSE,
    nsloc1 = NULL, nsloc2 = NULL, cshape = cscale, cscale = cloc,
    cloc = FALSE, std.err = TRUE, corr = FALSE, method = "BFGS",
    warn.inf = TRUE)

Arguments

x	A matrix or data frame, ordinarily with two columns, which may contain missing values. A data frame may also contain a third column of mode logical, which itself may contain missing values (see More Details).
model	The specified model; a character string. Must be either "log" (the default), "alog", "hr", "neglog", "aneglog", "bilog", "negbilog", "ct" or "amix" (or any unique partial match), for the logistic, asymmetric logistic, Husler-Reiss, negative logistic, asymmetric negative logistic, bilogistic, negative bilogistic, Coles-Tawn and asymmetric mixed models respectively. The definition of each model is given in rbvevd.
start	A named list giving the initial values for the parameters over which the likelihood is to be maximized. If start is omitted the routine attempts to find good starting values using marginal maximum likelihood estimators.
...	Additional parameters, either for the bivariate extreme value model or for the optimization function optim. If parameters of the model are included they will be held fixed at the values given (see Examples).
sym	Logical; if TRUE, the dependence structure of the models "alog", "aneglog" or "ct" are constrained to be symmetric (see Details). For all other models, the argument is ignored (and a warning is given).
nsloc1, nsloc2	A data frame with the same number of rows as x, for linear modelling of the location parameter on the first/second margin (see Details). The data frames are treated as covariate matrices, excluding the intercept. A numeric vector can be given as an alternative to a single column data frame.
cshape	Logical; if TRUE, a common shape parameter is fitted to each margin.
cscale	Logical; if TRUE, a common scale parameter is fitted to each margin, and the default value of cshape is then TRUE, so that under this default common scale and shape parameters are fitted.
cloc	Logical; if TRUE, a common location parameter is fitted to each margin, and the default values of cshape and cscale are then TRUE, so that under these defaults common marginal parameters are fitted.
std.err	Logical; if TRUE (the default), the standard errors are returned.
corr	Logical; if TRUE, the correlation matrix is returned.
method	The optimization method (see optim for details).
warn.inf	Logical; if TRUE (the default), a warning is given if the negative log-likelihood is infinite when evaluated at the starting values.

Details

The dependence parameter names are one or more of dep, asy1, asy2, alpha and beta, depending on the model selected (see rbvevd). The marginal parameter names are loc1, scale1 and shape1 for the first margin, and loc2, scale2 and shape2 for the second margin. If nsloc1 is not NULL, so that a linear model is implemented for the first marginal location parameter, the parameter names for the first margin are loc1, loc1x1, ..., loc1xn, scale and shape, where x1, ..., xn are the column names of nsloc1, so that loc1 is the intercept of the linear model, and loc1x1, ..., loc1xn are the ncol(nsloc1) coefficients. When nsloc2 is not NULL, the parameter names for the second margin are constructed similarly.

It is recommended that the covariates within the linear models for the location parameters are (at least approximately) centered and scaled (i.e. that the columns of nsloc1 and nsloc2 are centered and scaled), particularly if automatic starting values are used, since the starting values for the associated parameters are then zero. If cloc is TRUE, both nsloc1 and nsloc2 must be identical, since a common linear model is then implemented on both margins.

If cshape is true, the models are constrained so that shape2 = shape1. The parameter shape2 is then taken to be specified, so that e.g. the common shape parameter can only be fixed at zero using shape1 = 0, since using shape2 = 0 gives an error. Similar comments apply for cscale and cloc.

If sym is TRUE, the asymmetric logistic and asymmetric negative logistic models are constrained so that asy2 = asy1, and the Coles-Tawn model is constrained so that beta = alpha. The parameter asy2 or beta is then taken to be specified, so that e.g. the parameters asy1 and asy2 can only be fixed at 0.8 using asy1 = 0.8, since using asy2 = 0.8 gives an error.

Bilogistic and negative bilogistic models constrained to symmetry are logistic and negative logistic models respectively. The (symmetric) mixed model (e.g. Tawn, 1998) can be obtained as a special case of the asymmetric logistic or asymmetric mixed models (see Examples).

The value Dependence given in the printed output is $2(1-A(1/2))$, where $A$ is the estimated dependence function (see abvevd). It measures the strength of dependence, and lies in the interval [0,1]; at independence and complete dependence it is zero and one respectively (Coles, Heffernan and Tawn, 1999). See chiplot for further information.

Value

Returns an object of class c("bvevd","evd").

The generic accessor functions fitted (or fitted.values), std.errors, deviance, logLik and AIC extract various features of the returned object.

The functions profile and profile2d can be used to obtain deviance profiles. The function anova compares nested models, and the function AIC compares non-nested models. The function plot produces diagnostic plots.

An object of class c("bvevd","evd") is a list containing the following components

estimate	A vector containing the maximum likelihood estimates.
std.err	A vector containing the standard errors.
fixed	A vector containing the parameters that have been fixed at specific values within the optimization.
fixed2	A vector containing the parameters that have been set to be equal to other model parameters.
param	A vector containing all parameters (those optimized, those fixed to specific values, and those set to be equal to other model parameters).
deviance	The deviance at the maximum likelihood estimates.
dep.summary	The estimate of $2(1-A(1/2))$.
corr	The correlation matrix.
var.cov	The variance covariance matrix.
convergence, counts, message	Components taken from the list returned by optim.
data	The data passed to the argument x.
tdata	The data, transformed to stationarity (for non-stationary models).
nsloc1, nsloc2	The arguments nsloc1 and nsloc2.
n	The number of rows in x.
sym	The argument sym.
cmar	The vector c(cloc, cscale, cshape).
model	The argument model.
call	The call of the current function.

More Details

If x is a data frame with a third column of mode logical, then the model is fitted using the likelihood derived by Stephenson and Tawn (2004). This is appropriate when each bivariate data point comprises componentwise maxima from some underlying bivariate process, and where the corresponding logical value denotes whether or not the maxima were caused by the same event within that process.

Under this scheme the diagnostic plots that are produced using plot are somewhat different to those described in plot.bvevd: the density, dependence function and quantile curves plots contain fitted functions for observations where the logical case is unknown, and the conditional P-P plots condition on both the logical case and the given margin (which requires numerical integration at each data point).

Artificial Constraints

For numerical reasons parameters are subject to artificial constraints. Specifically, these constraints are: marginal scale parameters not less than 0.01; dep not less than [0.1] [0.2] [0.05] in [logistic] [Husler-Reiss] [negative logistic] models; dep not greater than [10] [5] in [Husler-Reiss] [negative logistic] models; asy1 and asy2 not less than 0.001; alpha and beta not less than [0.1] [0.1] [0.001] in [bilogistic] [negative bilogistic] [Coles-Tawn] models; alpha and beta not greater than [0.999] [20] [30] in [bilogistic] [negative bilogistic] [Coles-Tawn] models.

Warning

The standard errors and the correlation matrix in the returned object are taken from the observed information, calculated by a numerical approximation. They must be interpreted with caution when either of the marginal shape parameters are less than $-0.5$, because the usual asymptotic properties of maximum likelihood estimators do not then hold (Smith, 1985).

References

Coles, S. G., Heffernan, J. and Tawn, J. A. (1999) Dependence measures for extreme value analyses. Extremes, 2, 339–365.

Smith, R. L. (1985) Maximum likelihood estimation in a class of non-regular cases. Biometrika, 72, 67–90.

Stephenson, A. G. and Tawn, J. A. (2004) Exploiting Occurence Times in Likelihood Inference for Componentwise Maxima. Biometrika 92(1), 213–217.

Tawn, J. A. (1988) Bivariate extreme value theory: models and estimation. Biometrika, 75, 397–415.

See Also

anova.evd, optim, plot.bvevd, profile.evd, profile2d.evd, rbvevd

Examples

bvdata <- rbvevd(100, dep = 0.6, model = "log", mar1 = c(1.2,1.4,0.4))
M1 <- fbvevd(bvdata, model = "log")
M2 <- fbvevd(bvdata, model = "log", dep = 0.75)
anova(M1, M2)
par(mfrow = c(2,2))
plot(M1)
plot(M1, mar = 1)
plot(M1, mar = 2)
## Not run: par(mfrow = c(1,1))
## Not run: M1P <- profile(M1, which = "dep")
## Not run: plot(M1P)

trend <- (-49:50)/100
rnd <- runif(100, min = -.5, max = .5)
fbvevd(bvdata, model = "log", nsloc1 = trend)
fbvevd(bvdata, model = "log", nsloc1 = trend, nsloc2 = data.frame(trend
= trend,  random = rnd))
fbvevd(bvdata, model = "log", nsloc1 = trend, nsloc2 = data.frame(trend
= trend, random = rnd), loc2random = 0)

bvdata <- rbvevd(100, dep = 1, asy = c(0.5,0.5), model = "anegl")
anlog <- fbvevd(bvdata, model = "anegl")
mixed <- fbvevd(bvdata, model = "anegl", dep = 1, sym = TRUE)
anova(anlog, mixed)
amixed <- fbvevd(bvdata, model = "amix")
mixed <- fbvevd(bvdata, model = "amix", beta = 0)
anova(amixed, mixed)

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