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--- 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-Ãlvarez (2018)](https://link.springer.com/chapter/10.1007/978-3-319-73848-2_37). The package is restricted to interval sampling. de Uña-Ãlvarez 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-Ãlvarez (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 $0 <1$ are valid too, the choice $w=1/2$ being recommended in practice (de Uña-Ãlvarez, 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 Ãlavarez, Jacobo. + Soage González, José Carlos. Maintainer: José Carlos Soage González (jsoage@uvigo.es) ### Acknowledgements + Jacobo de Uña-Ãlvarez 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-Ãlvarez 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
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