lynden: NPMLE computation with the second Efron-Petrosian 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. It provides simple bootstrap pointwise confidence limits too. This function allows for ties in the samples of X, U and V.

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lynden(X, U = NA, V = NA, error = NA, nmaxit = NA,
		 boot = TRUE, B = NA, alpha = NA,
			display.F = FALSE, display.S = FALSE)

`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`.
`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 distribution estimation. Pointwise confidence bands are provided.
`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.F`	Logical. Default is FALSE. If TRUE, the estimated cumulative distribution function associated to `X`, (F) is plotted.
`display.S`	Logical. Default is FALSE. If TRUE, the estimated survival function associated to `X`, (S) is plotted.

The NPMLE for the cumulative distribution function is computed by the second algorithm proposed in Efron and Petrosian (1999). This is an iterative algorithm which converges to the NMPLE after a number of iterations. If the second (respectively third) argument is missing, 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.
`NJ`	The number of individuals in risk considering the left truncation times.
`density`	The estimated density values.
`cumulative.df`	The estimated cumulative distribution values.
`truncation.probs`	The probability of `X` falling into each truncation interval.
`hazard`	The estimated hazard values.
`S0`	`error` reached in the algorithm.
`Survival`	The estimated survival values.
`n.iterations`	The number of iterations used by this algorithm.
`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.
`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

Efron B and Petrosian V (1999) Nonparametric methods for doubly truncated data. Journal of the American Statistical Association 94, 824-834.

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.

efron.petrosian

#  Generating data which are doubly truncated

set.seed(4321)

n<-25
X<-runif(n,0,1)
U<-runif(n,0,0.25)
V<-runif(n,0.75,1)

for (i in 1:n){
	while (X[i]<U[i]|X[i]>V[i]){
	U[i]<-runif(1,0,0.25)
	X[i]<-runif(1,0,1)
	V[i]<-runif(1,0.75,1)
					   }
		 }
res<-lynden(X=X, U=U, V=V, boot=FALSE, display.F=TRUE, display.S=TRUE)

# Generating data which are right truncated

set.seed(4321)

n<-25
X<-runif(n,0,1)
V<-runif(n,0.75,1)

for (i in 1:n){
	while (X[i]>V[i]){
	X[i]<-runif(1,0,1)
	V[i]<-runif(1,0.75,1)
			     }
		 }
res<-lynden(X=X,U=NA, V=V, boot=FALSE)