ReIns (version 1.0.7)

LStail: Least Squares tail estimator

Description

Computes the Least Squares (LS) estimates of the EVI based on the last \(k\) observations of the generalised QQ-plot.

Usage

LStail(data, rho = -1, lambda = 0.5, logk = FALSE, plot = FALSE, add = FALSE, 
       main = "LS estimates of the EVI", ...)
            
TSfraction(data, rho = -1, lambda = 0.5, logk = FALSE, plot = FALSE, add = FALSE, 
           main = "LS estimates of the EVI", ...)

Arguments

data

Vector of \(n\) observations.

rho

Estimate for \(\rho\), or NULL when \(\rho\) needs to be estimated using the method of Beirlant et al. (2002). Default is -1.

lambda

Parameter used in the method of Beirlant et al. (2002), only used when rho=NULL. Default is 0.5.

logk

Logical indicating if the estimates are plotted as a function of \(\log(k)\) (logk=TRUE) or as a function of \(k\). Default is FALSE.

plot

Logical indicating if the estimates of \(\gamma\) should be plotted as a function of \(k\), default is FALSE.

add

Logical indicating if the estimates of \(\gamma\) should be added to an existing plot, default is FALSE.

main

Title for the plot, default is "LS estimates of the EVI".

…

Additional arguments for the plot function, see plot for more details.

Value

k

Vector of the values of the tail parameter \(k\).

gamma

Vector of the corresponding LS estimates for the EVI.

b

Vector of the corresponding LS estimates for b.

rho

Vector of the estimates for \(\rho\) when rho=NULL or the given input for rho otherwise.

Details

We estimate \(\gamma\) (EVI) and \(b\) using least squares on the following regression model (Beirlant et al., 2005): \(Z_j = \gamma + b(n/k) (j/k)^{-\rho} + \epsilon_j\) with \(Z_j = (j+1) \log(UH_{j,n}/UH_{j+1,n})\) and \(UH_{j,n}=X_{n-j,n}H_{j,n}\), where \(H_{j,n}\) is the Hill estimator with threshold \(X_{n-j,n}\).

See Section 5.8 of Beirlant et al. (2004) for more details.

The function TSfraction is included for compatibility with the old S-Plus code.

References

Beirlant, J., Dierckx, G. and Guillou, A. (2005). "Estimation of the Extreme Value Index and Regression on Generalized Quantile Plots." Bernoulli, 11, 949--970.

Beirlant, J., Dierckx, G., Guillou, A. and Starica, C. (2002). "On Exponential Representations of Log-spacing of Extreme Order Statistics." Extremes, 5, 157--180.

Beirlant J., Goegebeur Y., Segers, J. and Teugels, J. (2004). Statistics of Extremes: Theory and Applications, Wiley Series in Probability, Wiley, Chichester.

See Also

genQQ

Examples

# NOT RUN {
data(soa)

# LS tail estimator
LStail(soa$size, plot=TRUE, ylim=c(0,0.5))
# }