# crParetoQQ: Conditional Pareto quantile plot for right censored data In ReIns: Functions from "Reinsurance: Actuarial and Statistical Aspects"

## Description

Conditional Pareto QQ-plot adapted for right censored data.

## Usage

 1 2 3 crParetoQQ(x, Xtilde, Ytilde, censored, h, kernel = c("biweight", "normal", "uniform", "triangular", "epanechnikov"), plot = TRUE, add = FALSE, main = "Pareto QQ-plot", type = "p", ...) 

## Arguments

 x Value of the conditioning variable X at which to make the conditional Pareto QQ-plot. Xtilde Vector of length n containing the censored sample of the conditioning variable X. Ytilde Vector of length n containing the censored sample of the variable Y. censored A logical vector of length n indicating if an observation is censored. h Bandwidth of the non-parametric estimator for the conditional survival function (crSurv). kernel Kernel of the non-parametric estimator for the conditional survival function (crSurv). One of "biweight" (default), "normal", "uniform", "triangular" and "epanechnikov". plot Logical indicating if the quantiles should be plotted in a Pareto QQ-plot, default is TRUE. add Logical indicating if the quantiles should be added to an existing plot, default is FALSE. main Title for the plot, default is "Pareto QQ-plot". type Type of the plot, default is "p" meaning points are plotted, see plot for more details. ... Additional arguments for the plot function, see plot for more details.

## Details

We construct a Pareto QQ-plot for Y conditional on X=x using the censored sample (\tilde{X}_i, \tilde{Y}_i), for i=1,…,n, where X and Y are censored at the same time. We assume that Y and the censoring variable are conditionally independent given X.

The conditional Pareto QQ-plot adapted for right censoring is given by

( -\log(1-\hat{F}_{Y|X}(\tilde{Y}_{j,n}|x)), \log \tilde{Y}_{j,n} )

for j=1,…,n-1, with \tilde{Y}_{i,n} the i-th order statistic of the censored data and \hat{F}_{Y|X}(y|x) the non-parametric estimator for the conditional CDF of Akritas and Van Keilegom (2003), see crSurv.

See Section 4.4.3 in Albrecher et al. (2017) for more details.

## Value

A list with following components:

 pqq.the Vector of the theoretical quantiles, see Details. pqq.emp Vector of the empirical quantiles from the log-transformed Y data.

Tom Reynkens

## References

Akritas, M.G. and Van Keilegom, I. (2003). "Estimation of Bivariate and Marginal Distributions With Censored Data." Journal of the Royal Statistical Society: Series B, 65, 457–471.

Albrecher, H., Beirlant, J. and Teugels, J. (2017). Reinsurance: Actuarial and Statistical Aspects, Wiley, Chichester.

crSurv, crHill, cParetoQQ
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 # Set seed set.seed(29072016) # Pareto random sample Y <- rpareto(200, shape=2) # Censoring variable C <- rpareto(200, shape=1) # Observed (censored) sample of variable Y Ytilde <- pmin(Y, C) # Censoring indicator censored <- (Y>C) # Conditioning variable X <- seq(1, 10, length.out=length(Y)) # Observed (censored) sample of conditioning variable Xtilde <- X Xtilde[censored] <- X[censored] - runif(sum(censored), 0, 1) # Conditional Pareto QQ-plot crParetoQQ(x=1, Xtilde=Xtilde, Ytilde=Ytilde, censored=censored, h=2) # Plot Hill-type estimates crHill(x=1, Xtilde, Ytilde, censored, h=2, plot=TRUE)