ReIns (version 1.0.10)

SpliceQQ: Splicing quantile plot

Description

Computes the empirical quantiles of a data vector and the theoretical quantiles of the fitted spliced distribution. These quantiles are then plotted in a splicing QQ-plot with the theoretical quantiles on the \(x\)-axis and the empirical quantiles on the \(y\)-axis.

Usage

SpliceQQ(X, splicefit, p = NULL, plot = TRUE, main = "Splicing QQ-plot", ...)

Value

A list with following components:

sqq.the

Vector of the theoretical quantiles of the fitted spliced distribution.

sqq.emp

Vector of the empirical quantiles from the data.

Arguments

X

Vector of \(n\) observations.

splicefit

A SpliceFit object, e.g. output from SpliceFitPareto or SpliceFitGPD.

p

Vector of probabilities used in the QQ-plot. If NULL, the default, we take p equal to 1/(n+1),...,n/(n+1).

plot

Logical indicating if the quantiles should be plotted in a splicing QQ-plot, default is TRUE.

main

Title for the plot, default is "Splicing QQ-plot".

...

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

Author

Tom Reynkens

Details

This QQ-plot is given by $$(Q(p_j), \hat{Q}(p_j)),$$ for \(j=1,\ldots,n\) where \(Q\) is the quantile function of the fitted splicing model and \(\hat{Q}\) is the empirical quantile function and \(p_j=j/(n+1)\).

See Reynkens et al. (2017) and Section 4.3.1 in Albrecher et al. (2017) for more details.

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

See Also

SpliceQQ_TB, qSplice, SpliceFitPareto, SpliceFitGPD, SpliceECDF, SpliceLL, SplicePP

Examples

if (FALSE) {

# Pareto random sample
X <- rpareto(1000, shape = 2)

# Splice ME and Pareto
splicefit <- SpliceFitPareto(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)
}