# Nonparametric analysis of doubly truncated data

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## Abstract

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.

## Keywords

Double truncation Nonparametric MLE Kendall’s tau## Preview

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