Author: José Carlos Soage González
Date: 2022-04-12

# Package DTDA.ni

#### José Carlos Soage González

#### 2022-04-12

### 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). 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). 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

    # 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)
    #> Call:
    #> DTDAni(x = x, u = u, tau = tau)
    plot(res, ecdf = FALSE) # Plot without ecdf (Default)
    abline(a = 0, b = 1, col = "green")  # The true cumulative distribution

    # Calculating the reverse estimator:
    plot(res)

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

    # 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