Conditional Tail Expectation
Compute Conditional Tail Expectation (CTE) CTE_{1-p} of the fitted spliced distribution.
CTE(p, splicefit) ES(p, splicefit)
p |
The probability associated with the CTE (we estimate CTE_{1-p}). |
splicefit |
A |
The Conditional Tail Expectation is defined as
CTE_{1-p} = E(X | X>Q(1-p)) = E(X | X>VaR_{1-p}) = VaR_{1-p} + Π(VaR_{1-p})/p,
where Π(u)=E((X-u)_+) is the premium of the excess-loss insurance with retention u.
If the CDF is continuous in p, we have CTE_{1-p}=TVaR_{1-p}= 1/p \int_0^p VaR_{1-s} ds with TVaR the Tail Value-at-Risk.
See Reynkens et al. (2017) and Section 4.6 of Albrecher et al. (2017) for more details.
The ES function is the same function as CTE but is deprecated.
Vector with the CTE corresponding to each element of p.
Tom Reynkens with R code from Roel Verbelen for the mixed Erlang quantiles.
Albrecher, H., Beirlant, J. and Teugels, J. (2017). Reinsurance: Actuarial and Statistical Aspects, Wiley, Chichester.
Reynkens, T., Verbelen, R., Beirlant, J. and Antonio, K. (2017). "Modelling Censored Losses Using Splicing: a Global Fit Strategy With Mixed Erlang and Extreme Value Distributions". Insurance: Mathematics and Economics, 77, 65–77.
Verbelen, R., Gong, L., Antonio, K., Badescu, A. and Lin, S. (2015). "Fitting Mixtures of Erlangs to Censored and Truncated Data Using the EM Algorithm." Astin Bulletin, 45, 729–758
## Not run: # Pareto random sample X <- rpareto(1000, shape = 2) # Splice ME and Pareto splicefit <- SpliceFitPareto(X, 0.6) p <- seq(0.01, 0.99, 0.01) # Plot of CTE plot(p, CTE(p, splicefit), type="l", xlab="p", ylab=bquote(CTE[1-p])) ## End(Not run)
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