crSurv function

Non-parametric estimator of conditional survival function

Non-parametric estimator of the conditional survival function of $Y$ given $X$ for censored data, see Akritas and Van Keilegom (2003).

crSurv(x, y, Xtilde, Ytilde, censored, h, 
       kernel = c("biweight", "normal", "uniform", "triangular", "epanechnikov"))

Arguments

• x: The value of the conditioning variable $X$ to evaluate the survival function at. x needs to be a single number or a vector with the same length as y.
• y: The value(s) of the variable $Y$ to evaluate the survival function at.
• 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.
• kernel: Kernel of the non-parametric estimator. One of "biweight" (default), "normal", "uniform", "triangular" and "epanechnikov".

Details

We estimate the conditional survival function

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 estimator is given by

where $\Delta_i=1$ when $(\tilde{X}_i, \tilde{Y}_i)$ is censored and 0 otherwise. The weights are given by

when $\Delta_i=1$ and 0 otherwise.

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

Returns

Estimates for $1-F_{Y|X}(y|x)$ as described above.

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

crParetoQQ, crHill

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)

# Plot estimates of the conditional survival function
x <- 5
y <- seq(0, 5, 1/100)
plot(y, crSurv(x, y, Xtilde=Xtilde, Ytilde=Ytilde, censored=censored, h=5), type="l", 
     xlab="y", ylab="Conditional survival function")