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", ...)

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 $ρ$ when rho=NULL or the given input for rho otherwise.

Arguments

data: Vector of $n$ observations.
rho: Estimate for $ρ$ , or NULL when $ρ$ 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 $γ$ should be plotted as a function of $k$ , default is FALSE.
add: Logical indicating if the estimates of $γ$ 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.

Author

Tom Reynkens based on S-Plus code from Yuri Goegebeur.

Details

We estimate $γ$ (EVI) and $b$ using least squares on the following regression model (Beirlant et al., 2005): $Z_{j} = γ + b (n / k) (j / k)^{- ρ} + ϵ_{j}$ with $Z_{j} = (j + 1) \log (U H_{j, n} / U H_{j + 1, n})$ and $U H_{j, n} = X_{n - j, n} H_{j, n}$ , 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.

Examples

Run this code

data(soa)

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

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