Nonparametric analysis of doubly truncated data

One of the principal goals of the quasar investigations is to study luminosity evolution. A convenient one-parameter model for luminosity says that the expected log luminosity, T ∗, increases linearly as θ0 · log(1+ Z∗), and T ∗(θ0) = T ∗ − θ0 · log(1 + Z∗) is independent of Z∗, where Z∗ is the redshift of a quasar and θ0 is the true value of evolution parameter. Due to experimental constraints, the distribution of T ∗ is doubly truncated to an interval (U∗, V ∗) depending on Z∗, i.e., a quadruple (T ∗, Z∗,U∗, V ∗) is observable only when U∗ ≤ T ∗ ≤ V ∗. Under the one-parameter model, T ∗(θ0) is independent of (U∗(θ0), V ∗(θ0)), where U∗(θ0) = U∗ − θ0 · log(1 + Z∗) and V ∗(θ0) = V ∗ − θ0 · log(1 + Z∗). Under this assumption, the nonparametric maximum likelihood estimate (NPMLE) of the hazard function of T ∗(θ0) (denoted by ĥ) was developed by Efron and Petrosian (J Am Stat Assoc 94:824–834, 1999). In this note, we present an alternative derivation of ĥ. Besides, the NPMLE of distribution function of T ∗(θ0), F̂ , will be derived through an inverse-probability-weighted (IPW) approach. Based on Theorem 3.1 of Van der Laan (1996), we prove the consistency and asymptotic normality of the NPMLE F̂ under certain condition. For testing the null hypothesis Hθ0 : T ∗(θ0) = T ∗−θ0 · log(1+ Z∗) is independent of Z∗, (Efron and Petrosian in J Am Stat Assoc 94:824–834, 1999). proposed a truncated version of the Kendall’s tau statistic. However, when T ∗ is exponential distributed, the testing procedure is futile. To circumvent this difficulty, a modified testing procedure is proposed. Simulations show that the proposed test works adequately for moderate sample size.