fitSplicedBayes: Parameter Estimation for Spliced Distributions
In OpVaR: Statistical Methods for Modelling Operational Risk

Description Usage Arguments Details Value Author(s) References See Also Examples

MCMC based parameter estimation for spliced distribution functions, where the body is modeled by a Weibull, gamma or log-normal distribution or Kernel density estimation for a log-normal distribution and the tail is fitted with a GPD.

fitSplicedBayesWeibullGPD(cell, prior, burnin=10, niter=100, 
proposal_scale=evmix::fweibullgpd(cell,method="Nelder-Mead")$se, 
start=evmix::fweibullgpd(cell,method="Nelder-Mead")$optim$par)

fitSplicedBayesGammaGPD(cell, prior, burnin=10, niter=100, 
proposal_scale=evmix::fgammagpd(cell,method="Nelder-Mead")$se, 
start=evmix::fgammagpd(cell,method="Nelder-Mead")$optim$par)

fitSplicedBayesLognormGPD(cell, prior, burnin=10, niter=100, 
proposal_scale=evmix::flognormgpd(cell,method="Nelder-Mead")$se, 
start=evmix::flognormgpd(cell,method="Nelder-Mead")$optim$par)

fitSplicedBayesKDEGPD(cell, prior, burnin=10, niter=100, 
proposal_scale=evmix::flognormgpd(cell,method="Nelder-Mead")$se, 
start=evmix::flognormgpd(cell,method="Nelder-Mead")$optim$par)

`cell`	database for the parameter estimation
`prior`	parameters of the particular prior distribution
`burnin`	length of the burn-in-phase, standard value = 10
`niter`	MCMC sample size, standard value = 100
`proposal_scale`	respective scale values used for determining the prior distribution for each parameter
`start`	starting values (shapes) for the MH-algorithm steps

The input contains the dataset, whose compound distribution should be fitted with the function. The second part of the input is a list with the parameters of the prior distribution. It is necessary to indicate the prior as a list containing the full parameter vector in the order bodypar_1=c(),bodypar_2=c(), threshold=c(), beta=c(), xi=c(), where bodypar_1 and bodypar_2 are the parameters of the body (Weibull: wshape and wscale as shape and scale parameter, log-norm: mu and sigma, gamma: gshape and gscale as shape and scale parameter).threshold is the threshold seperating the distributions.beta and xi are the scale and shape parameters of the generalized Pareto distribution. Using the MH-algorithm the true parameters of the body and tail distribution should be approximated.

Sevdist object containing the parameters of the target distribution.

Kristina Dehler, Nicole Derfuss

Behrens, C. N.; Lopes, H. F.; Gamerman, D. (2004). Bayesian analysis of extreme events with threshold estimation. Statistical Modelling 4, 227-244.

Dehler, K. (2017). Bayesianische Methoden im operationellen Risiko. Master's thesis.

Ergashev, B.; Mittnik, S.; Sekeris, E. (2013). A Bayesian Approach to Extreme Value Estimation in Operational Risk Modeling. The Journal of Operational Risk 8(4), 55-81.

MacDonald, A.; Scarrott, C.J.; Lee, D.; Darlow, B.; Reale, M.; Russell, G. (2011). A flexible extreme value mixture model. Computational Statistics and Data Analysis 55(6), 2137-2157.

Read buildSplicedSevdist for further information about building sevdist objects with type "spliced". The initial parameter values are fitted using the methods from the package 'evmix', i.e. for the WeibullGPD the function fweibullgpd. For the algorithm also the package 'truncnorm' is required.

### Example for estimating the parameters of the spliced Weibull-GPD

## Not run: 

data("lossdat")
data=lossdat[[1]]$Loss

## starting values - method of moments

# Weibull distribution (shape, scale)
model <- function(x) c(F1 = x[2]*gamma(1+1/x[1])-mean(data),
	F2 = x[2]^2*(gamma(1+2/x[1])-(gamma(1+1/x[1]))^2)-sd(data)^2)
weibullpar <- rootSolve::multiroot(f = model,start = c((sd(data)/mean(data))^(-1.086),
	mean(data)/(gamma(1+1/((sd(data)/mean(data))^(-1.086))))))$root

# generalized Pareto distribution (tresh, beta, xi)
thresh=quantile(data,.9) # threshold
data2=data[which(data>thresh)]
model=function(x) c(F1=thresh+x[1]/(1-x[2])-mean(data2),
	F2=x[1]^2/(1-x[2])^2/(1-2*x[2])-sd(data2)^2)
gpdpar=c(thresh,rootSolve::multiroot(f=model,start=c(mean(data2),1/mean(data2)))$root)

## parameters of the prior distribution

prior <- list(xi = c(gpdpar[3],sd(data)),tau = c(quantile(data,.9),sd(data)/10),
	beta = c(gpdpar[2],sd(data)),wscale = c(weibullpar[2],sd(data)),
	wshape = c(weibullpar[1],sd(data)))

## estimation of (shape, scale, thresh, beta, xi)

fitSplicedBayesWeibullGPD(data, prior = prior, proposal_scale = evmix::fweibullgpd(data, 
	method = "Nelder-Mead", pvector = c(weibullpar[1], weibullpar[2], gpdpar[1], 
	gpdpar[2], gpdpar[3]))$se, start = evmix::fweibullgpd(data, 
	method = "Nelder-Mead", pvector = c(weibullpar[1], weibullpar[2], gpdpar[1], 
	gpdpar[2], gpdpar[3]))$optim$par)

## End(Not run)