Efron–Petrosian integrals for doubly truncated data with covariates: An asymptotic analysis

Abstract

In survival analysis, epidemiology and related fields there exists an increasing interest in statistical methods for doubly truncated data. Double truncation appears with interval sampling and other sampling schemes, and refers to situations in which the target variable is subject to two (left and right) random observation limits. Doubly truncated data require specific corrections for the observational bias, and this affects a variety of settings including the estimation of marginal and multivariate distributions, regression problems, and multi-state models. In this work multivariate Efron-Petrosian integrals for doubly truncated data are introduced. These integrals naturally arise when the goal is the estimation of the mean of a general transformation which involves the doubly truncated variable and covariates. An asymptotic representation of the Efron-Petrosian integrals as a sum of i.i.d. terms is derived and, from this, consistency and distributional convergence are established. As a by-product, uniform i.i.d. representations for the marginal nonparametric maximum likelihood estimator and its corresponding weighting process are provided. Applications to correlation analysis, regression, and competing risks models are presented. A simulation study is reported too.

Nonparametric estimation of the cumulative incidences of competing risks under double truncation

“…The formal investigation of the process has been somehow hidden or unsolved up to now. Shen (), and similarly Shen (), provided asymptotic results without complete proofs, and the existing gaps have been pointed out in the recent literature (de Uña‐Álvarez & Van Keilegom, ; Mandel et al., ). Interestingly, a deep investigation of the weighting process has recently been provided by de Uña‐Álvarez and Van Keilegom (); this has been used in the current paper to formally derive the uniform in‐probability consistency of the proposed empirical CIFs and the weak convergence of the corresponding processes.…”

“…Shen (), and similarly Shen (), provided asymptotic results without complete proofs, and the existing gaps have been pointed out in the recent literature (de Uña‐Álvarez & Van Keilegom, ; Mandel et al., ). Interestingly, a deep investigation of the weighting process has recently been provided by de Uña‐Álvarez and Van Keilegom (); this has been used in the current paper to formally derive the uniform in‐probability consistency of the proposed empirical CIFs and the weak convergence of the corresponding processes. These theoretical advances are relevant for the regression framework too, where the process has been used to weight the multivariate data in order to introduce consistent estimation procedures (Mandel et al., ; Moreira, de Uña‐Álvarez, & Meira‐Machado, ; Rennert & Xie, ; Shen, ).…”

Testing for an ignorable sampling bias under random double truncation

“…Theorem 1 follows from de Uña-Álvarez and Van Keilegom, 8 Theorem 2.1, where an asymptotic representation for the centered Efron-Petrosian NPMLE F n (x) − F(x) as a sum of zero-mean iid random variables is given. Under  0 that representation still holds for F n (x) − F * n (x), although the extra term F * (x) − F * n (x) naturally appears.…”

“…) are only slightly stronger versions of the aforementioned identifiability conditions for F and K. Finally, (C2) guarantees that the operator  appearing the asymptotic representation of F n is bounded; this is needed for obtaining the asymptotic properties of F n . See de Uña-Álvarez and Van Keilegom 8 for further details and discussion.…”

Nonparametric analysis of doubly truncated and interval-censored data

“…They also proposed a new condition for checking whether such an estimate exists. de Uña-Álvarez and Van Keilegom 19 formally established the consistency and distributional convergence of Efron-Petrosian integrals.…”