ReIns (version 1.0.7)

crSurv: Non-parametric estimator of conditional survival function

Description

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

Usage

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".

Value

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

Details

We estimate the conditional survival function $$1-F_{Y|X}(y|x)$$ 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 estimator is given by $$1-\hat{F}_{Y|X}(y|x) = \prod_{\tilde{Y}_i \le y} (1-W_{n,i}(x;h_n)/(\sum_{j=1}^nW_{n,j}(x;h_n) I\{\tilde{Y}_j \ge \tilde{Y}_i\}))^{\Delta_i}$$ where \(\Delta_i=1\) when \((\tilde{X}_i, \tilde{Y}_i)\) is censored and 0 otherwise. The weights are given by $$W_{n,i}(x;h_n) = K((x-\tilde{X}_i)/h_n)/\sum_{\Delta_j=1}K((x-\tilde{X}_j)/h_n)$$ when \(\Delta_i=1\) and 0 otherwise.

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

crParetoQQ, crHill

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)

# 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")
# }