SpliceLL_TB: LL-plot with fitted and Turnbull survival function

Description

This function plots the logarithm of the Turnbull survival function (which is suitable for interval censored data) versus the logarithm of the data. Moreover, the logarithm of the fitted survival function of the spliced distribution is added.

Usage

SpliceLL_TB(x = sort(L), L, U = L, censored, splicefit, plot = TRUE,
            main = "Splicing LL-plot", ...)

Value

A list with following components:

logX: Vector of the logarithms of the sorted left boundaries of the intervals.
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 $\log(1-\hat{F}^{TB}(x_{i,n}))$.

Arguments

x: Vector of points to plot the fitted survival function at. By default we plot it at the points L.
L: Vector of length $n$ with the lower boundaries of the intervals for interval censored data or the observed data for right censored data.
U: Vector of length $n$ with the upper boundaries of the intervals. By default, they are equal to L.
censored: A logical vector of length $n$ indicating if an observation is censored.
splicefit: A SpliceFit object, e.g. output from SpliceFiticPareto.
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.

Author

Tom Reynkens

Details

The LL-plot consists of the points $$(\log(L_{i,n}), \log(1-\hat{F}^{TB}(L_{i,n})))$$ for $i=1,\ldots,n$ with $n$ the length of the data, $x_{i,n}$ the $i$-th smallest observation and $\hat{F}^{TB}$ the Turnbull estimator for the distribution function. Then, the line $$(\log(x), \log(1-\hat{F}_{spliced}(x))),$$ with $\hat{F}_{spliced}$ the fitted spliced distribution function, is added.

Right censored data should be entered as L=l and U=truncupper, and left censored data should be entered as L=trunclower and U=u. The limits trunclower and truncupper are obtained from the SpliceFit object.

If the interval package is installed, the icfit function is used to compute the Turnbull estimator. Otherwise, survfit.formula from survival is used.

Use SpliceLL for non-censored data.

See Reynkens et al. (2017) and Section 4.3.2 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

Examples

Run this code

if (FALSE) {

# Pareto random sample
X <- rpareto(500, shape=2)

# Censoring variable
Y <- rpareto(500, shape=1)

# Observed sample
Z <- pmin(X,Y)

# Censoring indicator
censored <- (X>Y)

# Right boundary
U <- Z
U[censored] <- Inf

# Splice ME and Pareto
splicefit <- SpliceFiticPareto(L=Z, U=U, censored=censored, tsplice=quantile(Z,0.9))



x <- seq(0,20,0.1)

# 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 Turnbull survival function 
SpliceTB(x, L=Z, U=U, censored=censored, splicefit=splicefit)


# Log-log plot with Turnbull survival function and fitted survival function
SpliceLL_TB(x, L=Z, U=U, censored=censored, splicefit=splicefit)


# PP-plot of Turnbull survival function and fitted survival function
SplicePP_TB(L=Z, U=U, censored=censored, splicefit=splicefit)

# PP-plot of Turnbull survival function and 
# fitted survival function with log-scales
SplicePP_TB(L=Z, U=U, censored=censored, splicefit=splicefit, log=TRUE)

# QQ-plot using Turnbull survival function and fitted survival function
SpliceQQ_TB(L=Z, U=U, censored=censored, splicefit=splicefit)
}

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