Prob function

Weissman estimator of small exceedance probabilities and large return periods

Compute estimates of a small exceedance probability $P(X>q)$ or large return period $1/P(X>q)$ using the approach of Weissman (1978).

Prob(data, gamma, q, plot = FALSE, add = FALSE, 
     main = "Estimates of small exceedance probability", ...)
          
Return(data, gamma, q, plot = FALSE, add = FALSE, 
       main = "Estimates of large return period", ...)
        
Weissman.p(data, gamma, q, plot = FALSE, add = FALSE, 
           main = "Estimates of small exceedance probability", ...)
          
Weissman.r(data, gamma, q, plot = FALSE, add = FALSE, 
           main = "Estimates of large return period", ...)

Arguments

• data: Vector of $n$ observations.

• gamma: Vector of $n-1$ estimates for the EVI, typically Hill estimates are used.

• q: The used large quantile (we estimate $P(X>q)$ or $1/P(X>q)$ for $q$ large).

• plot: Logical indicating if the estimates should be plotted as a function of $k$, default is FALSE.

• add: Logical indicating if the estimates should be added to an existing plot, default is FALSE.

• main: Title for the plot, default is "Estimates of extreme quantile" for Prob

  and "Estimates of large return period" for Return.

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

Details

See Section 4.2.1 of Albrecher et al. (2017) for more details.

Weissman.p and Weissman.r are the same functions as Prob and Return but with a different name for compatibility with the old S-Plus code.

Returns

A list with following components: - k: Vector of the values of the tail parameter $k$.

• P: Vector of the corresponding probability estimates, only returned for Prob.

• R: Vector of the corresponding estimates for the return period, only returned for Return.

• q: The used large quantile.

References

Albrecher, H., Beirlant, J. and Teugels, J. (2017). Reinsurance: Actuarial and Statistical Aspects, Wiley, Chichester.

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

Weissman, I. (1978). "Estimation of Parameters and Large Quantiles Based on the k Largest Observations." Journal of the American Statistical Association, 73, 812--815.

Author(s)

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

See Also

Quant

Examples

data(soa)

# Look at last 500 observations of SOA data
SOAdata <- sort(soa$size)[length(soa$size)-(0:499)]

# Hill estimator
H <- Hill(SOAdata)

# Exceedance probability
q <- 10^6
# Weissman estimator
Prob(SOAdata,gamma=H$gamma,q=q,plot=TRUE)

# Return period
q <- 10^6
# Weissman estimator
Return(SOAdata,gamma=H$gamma,q=q,plot=TRUE)