Nonparametric analysis of doubly truncated data |
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Authors: | Pao-sheng Shen |
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Institution: | (1) Department of Mathematics and Computer Science, Illinois Wesleyan University, Bloomington, IL 61702, USA;(2) Division of Epidemiology and Clinical Applications, National Heart, Lung and Blood Institute, Bethesda, MD 20892, USA |
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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), ^(F)]{\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 ^(F)]{\hat F} under certain condition. For testing the null hypothesis Hq0: T*(q0) = T*-q0·log(1 + Z*){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. |
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Keywords: | |
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