LStail function

Least Squares tail estimator

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

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.

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.

Returns

• 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.

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.

Author(s)

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

See Also

genQQ

Examples

data(soa)

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