Prob: Weissman estimator of small exceedance probabilities and... In ReIns: Functions from "Reinsurance: Actuarial and Statistical Aspects"

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Weissman estimator of small exceedance probabilities and large return periods

Description

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

Usage

``````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.

Value

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.

Author(s)

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

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.

`Quant`

Examples

``````data(soa)

# Look at last 500 observations of SOA data

# Hill estimator

# Exceedance probability
q <- 10^6
# Weissman estimator