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$cumprobAcknowledgements
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