Description Usage Arguments Details Value Author(s) References See Also Examples
This function computes the NPMLE of a lifetime distribution function observed under onesided (right or left) and twosided (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.
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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 
V 
Numeric vector with the right truncation times. If there are no truncation times from the left, put 
wt 
Numeric vector of nonnegative initial solution, with the same length as 
error 
Numeric value. Maximum pointwise error when estimating the density associated to X (f) in two consecutive steps. If this is missing, it is $1e06$. 
nmaxit 
Numeric value. Maximum number of iterations. If this is missing, it is set to 
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 
B 
Numeric value. Number of bootstrap resamples . The default 
alpha 
Numeric value. (1 
display.FS 
Logical. Default is FALSE. If TRUE, the estimated cumulative distribution function and the estimated survival function associated to 
display.UV 
Logical. Default is FALSE. If TRUE, the marginal distributions of 
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 represent the joint distribution of the truncation times.
This must be one of "image" or "persp", with default 
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 LyndenBell estimator for righttruncated (respectively lefttruncated) data is obtained. Note that individuals with NAs in the three first arguments will be automatically excluded.
A list containing the following values:
time 
The timepoint on the curve. 
n.event 
The number of events that ocurred at time 
events 
The total number of events. 
density 
The estimated density values associated to 
cumulative.df 
The estimated cumulative distribution values of 
truncation.probs 
The probabilities of truncation values, in each region. 
S0 

Survival 
The estimated survival values. 
density.joint 
The estimated joint densities values associated to 
marginal.U 
The estimated cumulative univariate marginal values of the 
marginal.V 
The estimated cumulative univariate marginal values of the 
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 
lower.fU 
The estimated lower limits of the confidence intervals for 
upper.fV 
The estimated upper limits of the confidence intervals for 
lower.fV 
The estimated lower limits of the confidence intervals for 
sd.boot 
The bootstrap standard deviation. 
Boot.Repeat 
The number of resamples done in each bootstrap call to ensure the existence and uniqueness of the bootstrap NPMLE. 
Carla Moreira, Jacobo de UñaÁlvarez and Rosa Crujeiras
LyndenBell D (1971) A method of allowing for known observational selection in small samples applied to 3CR quasars. Monograph National Royal Astronomical Society 155, 95118.
Shen PS (2010) Nonparametric analysis of doubly truncated data. Annals of the Institute of Statistical Mathematics 62, 835853.
Xiao J, Hudgens MG (2020) On nonparametric maximum likelihood estimation with double truncation. Biometrika 106, 989996.
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