Nonparametric analysis of doubly truncated data

  • Pao-sheng ShenEmail author


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 θ 0· log(1  +  Z*), and T*(θ 0) = T*  −  θ 0· log(1  +  Z*) is independent of Z*, where Z* is the redshift of a quasar and θ 0 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*(θ 0) is independent of (U*(θ 0), V*(θ 0)), where U*(θ 0) = U*  −  θ 0· log(1  +  Z*) and V*(θ 0) = V*  −  θ 0· log(1  +  Z*). Under this assumption, the nonparametric maximum likelihood estimate (NPMLE) of the hazard function of T*(θ 0) (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*(θ 0), \({\hat F}\) , 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 \({\hat F}\) under certain condition. For testing the null hypothesis \({H_{\theta_0}: T^{\ast}(\theta_0) = T^{\ast}-\theta_0\cdot \log(1 + Z^{\ast})}\) 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.


Double truncation Nonparametric MLE Kendall’s tau 


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Copyright information

© The Institute of Statistical Mathematics, Tokyo 2008

Authors and Affiliations

  1. 1.Department of StatisticsTunghai UniversityTaichungTaiwan

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