shen {DTDA}R Documentation

Doubly truncated data analysis with the Shen algorithm

Description

This function computes the NPMLE of a lifetime distribution function observed under one-sided (right or left) and two-sided (double) truncation. The NPMLE of the joint distribution of the truncation times along with its marginal distributions are also computed. It provides bootstrap pointwise confidence limits too.

Usage

shen(X, U = NA, V = NA, wt = NA, error = NA,
         nmaxit = NA, boot = TRUE, boot.type = "simple",
                 B = NA, alpha = NA, display.FS = FALSE, 
                        display.UV = FALSE, plot.joint = FALSE, plot.type = NULL)

Arguments

X Numeric vector with the times of ultimate interest.
U Numeric vector with the left truncation times. If there are no truncation times from the left, put U=NA.
V Numeric vector with the right truncation times. If there are no truncation times from the left, put V=NA.
wt Numeric vector of non-negative initial solution, with the same length as X. Default value is set to 1/n, being n the length of X.
error Numeric value. Maximum distance among the densities computed in two successive steps. If this is missing, it is set to 10% of 1/n.
nmaxit Numeric value. Maximum number of iterations. If this is missing, it is set to nmaxit =100 .
boot Logical. If TRUE (default), the simple bootstrap method is applied to lifetime and truncation times distributions estimation. Pointwise confidence bands are provided.
boot.type A character string giving the bootstrap type to be used. This must be one of "simple" or "obvious", with default "simple".
B Numeric value. Number of bootstrap resamples . The default NA is equivalent to B =500 .
alpha Numeric value. (1-alpha) is the nominal coverage for the pointwise confidence intervals.
display.FS Logical. Default is FALSE. If TRUE, the estimated cumulative distribution function and the estimated survival function associated to X, (F) and (S) respectively, are plotted.
display.UV Logical. Default is FALSE. If TRUE, the marginal distributions of U (fU) and V (fV), are plotted.
plot.joint Logical. Default is FALSE. If TRUE, the joint distribution of the truncation times is plotted.
plot.type A character string giving the plot type to be used to representing the joint distribution of the truncation times. This must be one of "image" or "persp", with default NULL.

Details

The NPMLE of the lifetime is computed by a single algorithm proposed in Shen (2008). This is an alternative algorithm which converges to the NMPLE after a number of iterations. Initial solutions are given by the ordinary empirical distribution functions. If the second (respectively third) argument is missing, computation of the Lynden-Bell estimator for right-truncated (respectively left-truncated) data is obtained. Note that individuals with NAs in the three first arguments will be automatically excluded.

Value

A list containing the following values:

time The timepoint on the curve.
n.event The number of events that ocurred at time t.
events The total number of events.
density The estimated density values associated to X.
cumulative.df The estimated cumulative distribution values of X.
truncation.probs The probabilities of truncation values, in each region.
S0 error reached in the algorithm.
Survival The estimated survival values.
density.joint The estimated joint densities values associated to (U,V).
marginal.U The estimated cumulative univariate marginal values of the U.
marginal.V The estimated cumulative univariate marginal values of the V.
cumulative.joint The estimated joint cumulative distribution values.
n.iterations The number of iterations used by this algorithm.
biasf The estimated probabilities of observing the lifetimes.
Boot The type of bootstrap method applied.
B Number of bootstrap resamples computed.
alpha The nominal level used to construct the confidence intervals.
upper.df The estimated upper limits of the confidence intervals for F.
lower.df The estimated lower limits of the confidence intervals for F.
upper.Sob The estimated upper limits of the confidence intervals for S.
lower.Sob The estimated lower limits of the confidence intervals for S.
upper.fU The estimated upper limits of the confidence intervals for fU.
lower.fU The estimated lower limits of the confidence intervals for fU.
upper.fV The estimated upper limits of the confidence intervals for fV.
lower.fV The estimated lower limits of the confidence intervals for fV.

Author(s)

Carla Moreira, Jacobo de Uña-Álvarez and Rosa Crujeiras

References

Lynden-Bell, D. (1971) A method of allowing for known observational selection in small samples applied to 3CR quasars. Monograph National Royal Astronomical Society 155, 95-118. Astronomical Society, 155, 95-118.

Shen, P-S. (2008) Nonparametric analysis of doubly truncated data. Annals of the Institute of Statistical Mathematics DOI 10.1007/s10463-008-0192-2.

See Also

lynden

Examples


##  Generating data which are doubly truncated

n<-25
X<-runif(n,0,1)
U<-runif(n,0,0.67)
V<-runif(n,0.33,1)
for (i in 1:n){
        while (X[i]<U[i]|X[i]>V[i]){
        U[i]<-runif(1,0,0.67)
        X[i]<-runif(1,0,1)
        V[i]<-runif(1,0.33,1)
                                        }
                }

res<-shen(X,U,V,boot.type="obvious",plot.joint=TRUE, plot.type="persp")

##  Generating data which are left truncated

n<-25
X<-runif(n,0,1)
U<-runif(n,0,0.67)
for (i in 1:n){
        while (X[i]<U[i]){
        U[i]<-runif(1,0,0.67)
        X[i]<-runif(1,0,1)
                                }
                  }

res<-shen(X,U,V=NA,boot=FALSE)


[Package DTDA version 1.2-1 Index]