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Bivariate Density Estimation with Randomly Truncated Data

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  • Gürler, Ülkü
  • Prewitt, Kathryn

Abstract

In this study bivariate kernel density estimators are considered when a component is subject to random truncation. In bivariate truncation models one observes the i.i.d. samples from the triplets (T, Y, X) only if T[less-than-or-equals, slant]Y. In this set-up, Y is said to be left truncated by T and T is right truncated by Y. We consider the estimation of the bivariate density function of (Y, X) via nonparametric kernel methods where Y is the variable of interest and X a covariate. We establish an i.i.d. representation of the bivariate distribution function estimator and show that the remainder term achieves an improved order of O(n-1 ln n), which is desirable for density estimation purposes. Expressions are then provided for the bias and the variance of the estimators. Finally some simulation results are presented.

Suggested Citation

  • Gürler, Ülkü & Prewitt, Kathryn, 2000. "Bivariate Density Estimation with Randomly Truncated Data," Journal of Multivariate Analysis, Elsevier, vol. 74(1), pages 88-115, July.
    • Handle: RePEc:eee:jmvana:v:74:y:2000:i:1:p:88-115
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    References listed on IDEAS

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    1. van der Laan, Mark J., 1996. "Nonparametric Estimation of the Bivariate Survival Function with Truncated Data," Journal of Multivariate Analysis, Elsevier, vol. 58(1), pages 107-131, July.
    2. Gijbels, I. & Wang, J. L., 1993. "Strong Representations of the Survival Function Estimator for Truncated and Censored Data with Applications," Journal of Multivariate Analysis, Elsevier, vol. 47(2), pages 210-229, November.
    3. Gurler, Y & Gijbels, I, 1997. "A Bivariate Distribution Function Estimator and Its Variance under Left Truncation and Right Censoring," Papers 9702, Catholique de Louvain - Institut de statistique.
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