fitBvGPD: Fitting Bivariate Peaks Over a Threshold Using Bivariate...

Description Usage Arguments Details Value Warnings Author(s) References See Also Examples

Description

Fitting a bivariate extreme value distribution to bivariate exceedances over thresholds using censored maximum likelihood procedure.

Usage

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fitbvgpd(data, threshold, model = "log", start, ..., cscale = FALSE,
cshape = FALSE, std.err.type = "observed", corr = FALSE, warn.inf = TRUE,
method = "BFGS")

Arguments

data

A matrix with two columns which gives the observation vector for margin 1 and 2 respectively. NA values are considered to fall below the threshold.

threshold

A numeric vector for the threshold (of length 2).

model

A character string which specifies the model used. Must be one of log (the default), alog, nlog, anlog, mix and amix for the logistic, asymmetric logistic, negative logistic, asymmetric negative logistic, mixed and asymmetric mixed models.

start

Optional. A list for starting values in the fitting procedure.

...

Additional parameters to be passed to the optim function or to the bivariate model. In particular, parameter of the model can be hand fixed.

cscale

Logical. Should the two scale parameters be equal?

cshape

Logical. Should the two shape parameters be equal?

std.err.type

The type of the standard error. Currently, one must specify "observed" for observed Fisher information matrix or "none" for no computations of the standard errors.

corr

Logical. Should the correlation matrix be computed?

warn.inf

Logical. Should users be warned if likelihood is not finite at starting values?

method

The optimization method, see optim.

Details

The bivariate exceedances are fitted using censored likelihood procedure. This methodology is fully described in Ledford (1996).

Most of models are described in Kluppelberg (2006).

Value

The function returns an object of class c("bvpot","pot"). As usual, one can extract several features using fitted (or fitted.values), deviance, logLik and AIC functions.

fitted.values

The maximum likelihood estimates of the bivariate extreme value distribution.

std.err

A vector containing the standard errors - only present when the observed information matrix is not singular.

var.cov

The asymptotic variance covariance matrix - only presents when the observed information matrix is not singular.

deviance

The deviance.

corr

The correlation matrix.

convergence, counts, message

Informations taken from the optim function.

threshold

The marginal thresholds.

pat

The marginal proportion above the threshold.

nat

The marginal number above the threshold.

data

The bivariate matrix of observations.

exceed1, exceed2

The marginal exceedances.

call

The call of the current function.

model

The model for the bivariate extreme value distribution.

chi

The chi statistic of Coles (1999). A value near 1 (resp. 0) indicates perfect dependence (resp. independence).

Warnings

Because of numerical problems, their exists artificial numerical constraints imposed on each model. These are:

For this purpose, users must check if estimates are near these artificial numerical constraints. Such cases may lead to substantial biases on the GP parameter estimates. One way to detect quickly if estimates are near the border constraints is to look at the standard errors for the dependence parameters. Small values (i.e. < 1e-5) often indicates that numerical constraints have been reached.

In addition, users must be aware that the mixed and asymmetric mixed models can not deal with perfect dependence.

Thus, user may want to plot the Pickands' dependence function to see if variable are near independence or dependence cases using the pickdep function.

Author(s)

Mathieu Ribatet

References

Coles, S., Heffernan, J. and Tawn, J. (1999) Dependence Measure for Extreme Value Analyses. Extremes, 2:4 339–365.

Kl\"uppelberg, C., and May A. (2006) Bivariate extreme value distributions based on polynomial dependence functions. Mathematical Methods in the Applied Sciences, 29: 1467–1480.

Ledford A., and Tawn, J. (1996) Statistics for near Independence in Multivariate Extreme Values. Biometrika, 83: 169–187.

See Also

The following usual generic functions are available print, plot and anova as well as new generic functions retlev and convassess.

See also pickdep, specdens.

For optimization in R, see optim.

Examples

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x <- rgpd(1000, 0, 1, 0.25)
y <- rgpd(1000, 3, 1, -0.25)
ind <- fitbvgpd(cbind(x, y), c(0, 3), "log")
ind
ind2 <- fitbvgpd(cbind(x, y), c(0, 3), "log", alpha = 1)
ind2
ind3 <- fitbvgpd(cbind(x, y), c(0, 3), cscale = TRUE)
ind3
##The mixed model can not deal with perfect dependent variables
##Thus, there is a substantial bias in GPD parameter estimates
dep <- fitbvgpd(cbind(x, x + 3), c(0, 3), "mix")
dep

Example output

Call: fitbvgpd(data = cbind(x, y), threshold = c(0, 3), model = "log") 
Estimator: MLE 
Dependence Model and Strength:
	Model : Logistic 
	lim_u Pr[ X_1 > u | X_2 > u] = 0.001 
Deviance: 4117.259 
     AIC: 4127.259 

Marginal Threshold: 0 3 
Marginal Number Above: 1000 1000 
Marginal Proportion Above: 1 1 
Joint Number Above: 1000 
Joint Proportion Above: 1 
Number of events such as {Y1 > u1} U {Y2 > u2}: 1000 

Estimates
 scale1   shape1   scale2   shape2    alpha  
 1.0003   0.2908   1.1230  -0.3535   0.9992  

Standard Errors
   scale1     shape1     scale2     shape2      alpha  
5.065e-02  4.015e-02  4.556e-02  2.778e-02  2.002e-06  

Asymptotic Variance Covariance
        scale1      shape1      scale2      shape2      alpha     
scale1   2.566e-03  -1.261e-03   1.897e-06  -9.146e-07   7.515e-12
shape1  -1.261e-03   1.612e-03   2.551e-07   4.774e-07  -1.598e-11
scale2   1.897e-06   2.551e-07   2.076e-03  -1.124e-03  -2.656e-12
shape2  -9.146e-07   4.774e-07  -1.124e-03   7.718e-04   8.559e-13
alpha    7.515e-12  -1.598e-11  -2.656e-12   8.559e-13   4.008e-12

Optimization Information
	Convergence: successful 
	Function Evaluations: 77 
	Gradient Evaluations: 12 


Call: fitbvgpd(data = cbind(x, y), threshold = c(0, 3), model = "log",      alpha = 1) 
Estimator: MLE 
Dependence Model and Strength:
	Model : Logistic 
	lim_u Pr[ X_1 > u | X_2 > u] = 0 
Deviance: 4117.231 
     AIC: 4125.231 

Marginal Threshold: 0 3 
Marginal Number Above: 1000 1000 
Marginal Proportion Above: 1 1 
Joint Number Above: 1000 
Joint Proportion Above: 1 
Number of events such as {Y1 > u1} U {Y2 > u2}: 1000 

Estimates
 scale1   shape1   scale2   shape2  
 0.9964   0.2990   1.1244  -0.3540  

Standard Errors
 scale1   shape1   scale2   shape2  
0.05074  0.04102  0.04568  0.02786  

Asymptotic Variance Covariance
        scale1      shape1      scale2      shape2    
scale1   2.575e-03  -1.290e-03  -2.222e-12   9.258e-13
shape1  -1.290e-03   1.683e-03   1.057e-12  -4.429e-13
scale2  -2.222e-12   1.057e-12   2.087e-03  -1.130e-03
shape2   9.258e-13  -4.429e-13  -1.130e-03   7.759e-04

Optimization Information
	Convergence: successful 
	Function Evaluations: 44 
	Gradient Evaluations: 9 


Call: fitbvgpd(data = cbind(x, y), threshold = c(0, 3), cscale = TRUE) 
Estimator: MLE 
Dependence Model and Strength:
	Model : Logistic 
	lim_u Pr[ X_1 > u | X_2 > u] = 0.004 
Deviance: 4120.646 
     AIC: 4128.646 

Marginal Threshold: 0 3 
Marginal Number Above: 1000 1000 
Marginal Proportion Above: 1 1 
Joint Number Above: 1000 
Joint Proportion Above: 1 
Number of events such as {Y1 > u1} U {Y2 > u2}: 1000 

Estimates
 scale1   shape1   shape2    alpha  
 1.0714   0.2635  -0.3244   0.9973  

Standard Errors
 scale1   shape1   shape2    alpha  
0.03427  0.03454  0.02424  0.01457  

Asymptotic Variance Covariance
        scale1      shape1      shape2      alpha     
scale1   1.175e-03  -5.209e-04  -6.798e-04  -1.314e-08
shape1  -5.209e-04   1.193e-03   3.039e-04   3.696e-06
shape2  -6.798e-04   3.039e-04   5.877e-04  -3.865e-06
alpha   -1.314e-08   3.696e-06  -3.865e-06   2.122e-04

Optimization Information
	Convergence: successful 
	Function Evaluations: 56 
	Gradient Evaluations: 12 


Call: fitbvgpd(data = cbind(x, x + 3), threshold = c(0, 3), model = "mix") 
Estimator: MLE 
Dependence Model and Strength:
	Model : Mixed 
	lim_u Pr[ X_1 > u | X_2 > u] = 0.5 
Deviance: 3637.324 
     AIC: 3647.324 

Marginal Threshold: 0 3 
Marginal Number Above: 1000 1000 
Marginal Proportion Above: 1 1 
Joint Number Above: 1000 
Joint Proportion Above: 1 
Number of events such as {Y1 > u1} U {Y2 > u2}: 1000 

Estimates
  scale1    shape1    scale2    shape2     alpha  
 1.10362  -0.02175   1.06282  -0.01522   0.99907  

Standard Errors
   scale1     shape1     scale2     shape2      alpha  
4.301e-02  7.541e-03  4.030e-02  8.280e-03  2.002e-06  

Asymptotic Variance Covariance
        scale1      shape1      scale2      shape2      alpha     
scale1   1.850e-03  -2.189e-04   1.430e-03  -1.707e-04   5.801e-10
shape1  -2.189e-04   5.687e-05  -1.808e-04   5.298e-05  -1.067e-10
scale2   1.430e-03  -1.808e-04   1.624e-03  -2.223e-04   4.834e-10
shape2  -1.707e-04   5.298e-05  -2.223e-04   6.857e-05  -9.933e-11
alpha    5.801e-10  -1.067e-10   4.834e-10  -9.933e-11   4.008e-12

Optimization Information
	Convergence: successful 
	Function Evaluations: 95 
	Gradient Evaluations: 12 

POT documentation built on May 2, 2019, 7:30 a.m.