SpliceQQ function

Splicing quantile plot

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.

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

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.

Details

This QQ-plot is given by

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.

Returns

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.

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

Author(s)

Tom Reynkens

See Also

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

Examples

## Not run:

# 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)
## End(Not run)