SpliceFitGPD {ReIns}

Splicing of mixed Erlang and GPD using POT-MLE

Description

Fit spliced distribution of a mixed Erlang distribution and a Generalised Pareto Distribution (GPD). The parameters of the GPD are determined using the POT-MLE approach.

Usage

SpliceFitGPD(X, const = NULL, tsplice = NULL, M = 3, s = 1:10, trunclower = 0, 
             ncores = NULL, criterium = c("BIC","AIC"), reduceM = TRUE,
             eps = 10^(-3), beta_tol = 10^(-5), maxiter = Inf)

Arguments

X

Data used for fitting the distribution.

const

The probability of the quantile where the ME distribution will be spliced with the GPD distribution. Default is NULL meaning the input from tsplice is used.

tsplice

The point where the ME distribution will be spliced with the GPD distribution. Default is NULL meaning the input from const is used.

M

Initial number of Erlang mixtures, default is 3. This number can change when determining an optimal mixed Erlang fit using an information criterion.

s

Vector of spread factors for the EM algorithm, default is 1:10. We loop over these factors when determining an optimal mixed Erlang fit using an information criterion, see Verbelen et al. (2016).

trunclower

Lower truncation point. Default is 0.

ncores

Number of cores to use when determining an optimal mixed Erlang fit using an information criterion. When NULL (default), max(nc-1,1) cores are used where nc is the number of cores as determined by detectCores.

criterium

Information criterion used to select the number of components of the ME fit and s. One of "AIC" and "BIC" (default).

reduceM

Logical indicating if M should be reduced based on the information criterion, default is TRUE.

eps

Covergence threshold used in the EM algorithm (ME part). Default is 10^(-3).

beta_tol

Threshold for the mixing weights below which the corresponding shape parameter vector is considered neglectable (ME part). Default is 10^(-5).

maxiter

Maximum number of iterations in a single EM algorithm execution (ME part). Default is Inf meaning no maximum number of iterations.

Details

See Reynkens et al. (2017), Section 4.3.1 of Albrecher et al. (2017) and Verbelen et al. (2015) for details. The code follows the notation of the latter. Initial values follow from Verbelen et al. (2016).

Value

A SpliceFit object.

Author(s)

Tom Reynkens with R code from Roel Verbelen for fitting the mixed Erlang distribution.

References

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.

Verbelen, R., Antonio, K. and Claeskens, G. (2016). "Multivariate Mixtures of Erlangs for Density Estimation Under Censoring." Lifetime Data Analysis, 22, 429–455.

See Also

SpliceFitPareto, SpliceFiticPareto, Splice, GPDfit

Examples

## Not run: 

# GPD random sample
X <- rgpd(1000, gamma = 0.5, sigma = 2)

# Splice ME and GPD
splicefit <- SpliceFitGPD(X, 0.6)



x <- seq(0, 20, 0.01)

# Plot of spliced CDF
plot(x, pSplice(x, splicefit), type="l", xlab="x", ylab="F(x)")

# Plot of spliced PDF
plot(x, dSplice(x, splicefit), type="l", xlab="x", ylab="f(x)")



# Fitted survival function and empirical survival function 
SpliceECDF(x, X, splicefit)

# Log-log plot with empirical survival function and fitted survival function
SpliceLL(x, X, splicefit)

# PP-plot of empirical survival function and fitted survival function
SplicePP(X, splicefit)

# PP-plot of empirical survival function and 
# fitted survival function with log-scales
SplicePP(X, splicefit, log=TRUE)

# Splicing QQ-plot
SpliceQQ(X, splicefit)

## End(Not run)