---
title: "Package DTDA.ni"
author: "José Carlos Soage González"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Package DTDA.ni}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

###  Doubly Truncated Data Analysis, Non Iterative

### Description [package]

This package implements a non-iterative estimator for the cumulative distribution of a doubly truncated variable, see [de Uña-Álvarez (2018)](https://link.springer.com/chapter/10.1007/978-3-319-73848-2_37). The package is restricted to interval sampling. 

de Uña-Álvarez J. (2018) A Non-iterative Estimator for Interval Sampling and Doubly Truncated Data. In: Gil E., Gil E., Gil J., Gil M. (eds) The Mathematics of the Uncertain. Studies in Systems, Decision and Control, vol 142. Springer, Cham, pp. 387-400. 

## DTDAni(x, u , tau)

### Description [function DTDAni] 

This function calculates the non-iterative estimator for the cumulative distribution of a doubly truncated variable proposed by [de Uña-Álvarez (2018)](https://link.springer.com/chapter/10.1007/978-3-319-73848-2_37). The package works for interval sampling.

### Details [function DTDAni] 

The function DTDAni is adapted to the presence of ties. It can be used to compute the direct $(Fd)$ and the reverse $(Fr)$ estimators; see the example below. Both curves are valid estimators for the cumulative distribution $(F)$ of the doubly truncated variable. Weighted estimators $Fw = w*Fd + (1-w)*Fr$ with $0<w<1$ are valid too, the choice $w=1/2$ being recommended in practice (de Uña-Álvarez, 2018).

### Usage

In order to use this package, install it and load the library with:  `library(DTDA.ni)`

### Parameters

+ **x**: Numeric vector corresponding the variable of ultimate interest.
+ **u**: Numeric vector corresponding to the left truncation variable.
+ **tau**: Sampling interval width. The right truncation values will be internally calculated as v = u + tau.

### Return

+ **x** : The distinct values of the variable of interest.
+ **nx** : The absloute frequency of each x value.
+ **cumprob** : The estimated cumulative probability for each x value.
+ **P** : The auxiliary Pi used in the calculation of the estimator.
+ **L** : The auxiliary Li used in the calculation of the estimator.


### Usage

```{r, eval=T, fig.align='center',message = FALSE, fig.height = 4, fig.width = 6, warning = FALSE}

# Loading the package:
library(DTDA.ni)

# Generating data which are doubly truncated:
N <- 250
x0 <- runif(N)           # Original data
u0 <- runif(N, -0.25, 0.5) # Left-truncation times
tau <- 0.75               # Interval width
v0 <- u0 + tau

x <- x0[u0 <= x0 & x0 <= v0]
u <- u0[u0 <= x0 & x0 <= v0]
v <- v0[u0 <= x0 & x0 <= v0]
n <- length(x)  # Final sample size after the interval sampling

# Create an object wit DTDAni function
res <- DTDAni(x, u, tau)
plot(res, ecdf = FALSE) # Plot without ecdf (Default)
abline(a = 0, b = 1, col = "green")  # The true cumulative distribution

```
```{r, eval=T, fig.align='center',message = FALSE, fig.height = 4, fig.width = 6, warning = FALSE}
# Calculating the reverse estimator:
plot(res)

res2 <- DTDAni(-x, -u - 0.75, 0.75)
abline(a = 0, b = 1, col = "green") 
lines(-res2$x, 1 - res2$cumprob, type = "s", col = "blue", lty = 2)

```

```{r, eval=T, fig.align='center',message = FALSE, fig.height = 4, fig.width = 6, warning = FALSE}
# Weigthed estimator (recommended):

w <- 1/2

k <- length(res$x)

Fw <- w * res$cumprob + (1 - w) * (1 - res2$cumprob[k:1])

plot(res)
abline(a = 0, b = 1, col = "green") 
lines(-res2$x, 1 - res2$cumprob, type = "s", col = "blue", lty = 2)
lines(res$x, Fw, type = "s", col = 2)


# Using res$P and res$L to compute the estimator:

k <- length(res$x)
F <- rep(1, k)
for (i in 2:k) {
F[i] <- (F[i - 1] - res$P[i - 1]) / res$L[i - 1] + res$P[i - 1]
}

F0 <- F / max(F)  # This is equal to res$cumprob
```




### Author(s)

+ de Uña Álavarez, Jacobo.
+ Soage González, José Carlos.

Maintainer: José Carlos Soage González (jsoage@uvigo.es)

### Acknowledgements

+ Jacobo de Uña-Álvarez was supported by Grant MTM2014-55966-P, Spanish Ministry of Economy and Competitiveness .

+ José Carlos Soage was supported by Red Tecnológica de Matemática Industrial (Red TMATI), Cons. de Cultura, Educación e OU, Xunta de Galicia (ED341D R2016/051) and by Grupos de Referencia Competitiva, Consolidación y Estructuración de Unidades de Investigación Competitivas del SUG, Cons. de Cultura, Educación e OU, Xunta de Galicia (GRC ED431C 2016/040).


### References

de Uña-Álvarez J. (2018) A Non-iterative Estimator for Interval Sampling and Doubly Truncated Data. In: Gil E., Gil E., Gil J., Gil M. (eds)
The Mathematics of the Uncertain. Studies in Systems, Decision and Control,
vol 142. Springer, Cham

