The central limit theorem under random truncation
Under left truncation, data (X(i), Y(i)) are observed only when Y(i) ≤ X(i). Usually, the distribution function F of the X(i) is the target of interest. In this paper, we study linear functionals ∫ φ dF(n) of the nonparametric maximum likelihood estimator (MLE) of F, the Lynden-Bell estimator F(n)....
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| Published in | Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability Vol. 14; no. 3; p. 604 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
England
01.08.2008
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| Online Access | Get more information |
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| Summary: | Under left truncation, data (X(i), Y(i)) are observed only when Y(i) ≤ X(i). Usually, the distribution function F of the X(i) is the target of interest. In this paper, we study linear functionals ∫ φ dF(n) of the nonparametric maximum likelihood estimator (MLE) of F, the Lynden-Bell estimator F(n). A useful representation of ∫ φ dF(n) is derived which yields asymptotic normality under optimal moment conditions on the score function φ. No continuity assumption on F is required. As a by-product, we obtain the distributional convergence of the Lynden-Bell empirical process on the whole real line. |
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| ISSN: | 1350-7265 |
| DOI: | 10.3150/07-BEJ116 |