crParetoQQ function

Conditional Pareto quantile plot for right censored data

Conditional Pareto QQ-plot adapted for right censored data.

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 $i=1,\ldots,n$, 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

for $j=1,\ldots,n-1,$

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.

Returns

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.

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.

Author(s)

Tom Reynkens

See Also

crSurv, crHill, cParetoQQ

Examples

# 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)