shen: NPMLE computation with Shen algorithm
In DTDA: Doubly Truncated Data Analysis

Description Usage Arguments Details Value Author(s) References See Also Examples

This function computes the NPMLE for the cumulative distribution function of X 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.

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)

`X`	Numeric vector with the values of the target variable.
`U`	Numeric vector with the values of the left truncation variable. If there are no truncation values from the left, put `U`=`NA`.
`V`	Numeric vector with the values of the right truncation variable. If there are no truncation values from the right, 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 pointwise error when estimating the density associated to X (f) in two consecutive steps. If this is missing, it is $1e-06$.
`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 represent the joint distribution of the truncation times. This must be one of "image" or "persp", with default `NULL`.

The NPMLE for the cumulative distribution function is computed by a single algorithm proposed in Shen (2010). This is an iterative 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.

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 probability of `X` falling into each truncation interval.
`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`.
`sd.boot`	The bootstrap standard deviation of F estimator.
`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 Uña-Álvarez and Rosa Crujeiras

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.

Shen P-S (2010) Nonparametric analysis of doubly truncated data. Annals of the Institute of Statistical Mathematics 62, 835-853.

Xiao J, Hudgens MG (2020) On nonparametric maximum likelihood estimation with double truncation. Biometrika 106, 989-996.

lynden

##  Generating data which are doubly truncated

set.seed(4321)
n<-100
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=FALSE, plot.joint=TRUE, plot.type="persp")