Nonparametric Bootstrap Confidence Intervals

Description

Compute nonparametric bootstrap (1-α)\% confidence bands for the Pickands dependence function.

Usage

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  beed.confband(data, x, d = 3, est = c('ht','md','cfg'), 
  margin = c('emp','Gev'), k = 13, nboot = 200, y=NULL, conf = 0.95, 
  matrix = FALSE, plot = FALSE, print = FALSE)

Arguments

data

A (n x d) matrix of component-wise maxima.

x

A (m x d) design matrix (see Details).

d

A postive integer (greater than or equal to two) indicating the number of variables. The trivariate case d = 3 is the default.

est

A string denoting the estimation method (see Details).

margin

A string denoting the type marginal distributions (see Details).

k

A postive integer denoting the order of the Bernstein polynomial. k = 13 is set by default.

nboot

A postive integer indicating the number of bootstrap replicates.

y

A numeric vector (of size m) with an initial estimate of the Pickands function. If NULL, The initial estimation is performed by using the estimation method chosen in est.

conf

A real value in (0,1) denoting the confidence level of the interval. The value conf = 0.95 is the default.

matrix

Logical; FALSE by default. If TRUE, and the dimension d is three (the default dimension), the value of A is collected in a square matrix.

plot

Logical; FALSE by default. If TRUE, the confidence bands are plotted.

print

Logical; FALSE by default. If TRUE, the number of the iteration is printed.

Details

Two methods for computing bootstrap (1-α)\% point-wise and simultaneous confident bands for the Pickands dependence function are used.

The first method derives the confidence bands computing the point-wise α/2 and 1-α/2 quantiles of the bootstrap sample distribution of the Pickands dependence Bernstein based estimator.

The second method derives the confidence bands, first computing the point-wise α/2 and 1-α/2 quantiles of the bootstrap sample distribution of polynomial coefficients estimators, and then the Pickands dependence is computed using the Bernstein polynomial representation. See Marcon et al. (2014) for details.

Most of the settings are the same as in the function beed.

Value

A

Estimate of the Pickands dependence function.

bootA

A matrix with nboot columns that reports the estimates of the Pickands function for each data resampling.

A.up.beta/A.low.beta

Vectors of upper and lower bands of the Pickands dependence function obtained using the bootstrap sampling distribution of the polynomial coefficients estimator.

A.up.pointwise/A.low.pointwise

Vectors of upper and lower bands of the Pickands dependence function obtained using the bootstrap sampling distribution of the Pickands dependence function estimator.

up.beta/low.beta

Vectors of upper and lower bounds of the bootstrap sampling distribution of the polynomial coefficients estimator.

Note

This routine relies on the bootstrap routine (see beed.boot).

Author(s)

Simone Padoan, simone.padoan@unibocconi.it, faculty.bocconi.it/simonepadoan; Giulia Marcon

References

Marcon, G., Padoan, S.A., Naveau, P., Muliere, P. and Segers, J. (2016) Multivariate Nonparametric Estimation of the Pickands Dependence Function using Bernstein Polynomials. Journal of Statistical Planning and Inference, To appear.

See Also

beed, beed.boot.

Examples

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## Not run: 
w <- seq(0, 1, length = 100)  
data <- rbvevd(50, dep = 0.4, model = 'log', mar1 = c(1,1,1))

# Note you should consider 500 bootstrap replications. 
# In order to obtain fastest results we used 50!
cb <- beed.confband(data, cbind(w, 1-w), 2, 'md', 'emp', 20, 50) 
                    
plot(w, w, type='n', xlab="w", ylab="A(w)", ylim=c(.5,1))
polygon(c(0, 0.5, 1), c(1, 0.5, 1), lty=1, lwd=1, border='grey')
lines(w, cb$A, lty=1)
lines(w, cb$A.up.beta, lty=2)
lines(w, cb$A.low.beta, lty=2)


## End(Not run)

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