SpliceLL function

LL-plot with fitted and empirical survival function

This function plots the logarithm of the empirical survival function (determined using the Empirical CDF (ECDF)) versus the logarithm of the data. Moreover, the logarithm of the fitted survival function of the spliced distribution is added.

SpliceLL(x = sort(X), X, splicefit, plot = TRUE, main = "Splicing LL-plot", ...)

Arguments

• x: Vector of points to plot the fitted survival function at. By default we plot it at the data points.
• X: Data used for fitting the distribution.
• splicefit: A SpliceFit object, e.g. output from SpliceFitPareto or SpliceFitGPD.
• plot: Logical indicating if the splicing LL-plot should be made, default is TRUE.
• main: Title for the plot, default is "Splicing LL-plot".
• ...: Additional arguments for the plot function, see plot for more details.

Details

The LL-plot consists of the points

for $i=1,\ldots,n$ with $n$ the length of the data, $x_{i,n}$ the $i$-th smallest observation and $\hat{F}$ the empirical distribution function. Then, the line

with $\hat{F}_{spliced}$ the fitted spliced distribution function, is added.

Use SpliceLL_TB for censored data.

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

Returns

A list with following components: - logX: Vector of the logarithms of the sorted data.

• sll.the: Vector of the theoretical log-probabilities $\log(1-\hat{F}_{spliced}(x))$.

• logx: Vector of the logarithms of the points to plot the fitted survival function at.

• sll.emp: Vector of the empirical log-probabilities $\log(1-\hat{F}(x_{i,n}))$.

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

SpliceLL_TB, pSplice, ecdf, SpliceFitPareto, SpliceFitGPD, SpliceECDF, SplicePP, SpliceQQ

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)