crParetoQQ: Conditional Pareto quantile plot for right censored data

Description

Conditional Pareto QQ-plot adapted for right censored data.

Usage

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

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.

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.

Author

Tom Reynkens

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

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.

Examples

Run this code

# 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)

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