ReIns (version 1.0.7)

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", ...)

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.

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.

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.

See Also

crSurv, crHill, cParetoQQ

Examples

# NOT RUN {
# 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)
# }