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Large-sample study of the kernel density estimators under multiplicative censoring. (English) Zbl 1246.62094

Summary: The multiplicative censoring model, introduced by Y. Vardi [Biometrika 76, No. 4, 751–761 (1989; Zbl 0678.62051)], is an incomplete data problem where two independent samples from the life time distribution \(G\), \(\mathcal{X}_{m}=(X_{1},\dots,X_{m})\) and \(\mathcal{Z}_{n}=(Z_{1},\dots,Z_{n})\), are observed subject to a form of coarsening. Specifically, the sample \(\mathcal X_{m}\) is fully observed while \(\mathcal Y_{n}=(Y_{1},\dots,Y_{n})\) is observed instead of \(\mathcal Z_{n}\), where \(Y_{i} = U_{i}Z_{i}\) and \((U_{1}, \dots , U_{n})\) is an independent sample from the standard uniform distribution. Vardi showed that this model unifies several important statistical problems, such as the deconvolution of an exponential random variable, estimation under a decreasing density constraint and an estimation problem in renewal processes.
We establish the large-sample properties of kernel density estimators under the multiplicative censoring model. We first construct a strong approximation for the process \(\sqrt{k}(\hat{G}-G)\), where \(\hat{G}\) is a solution of the nonparametric score equation based on \((\mathcal{X}_{m},\mathcal{Y}_{n})\), and \(k = m + n\) is the total sample size. Using this strong approximation and a result on the global modulus of continuity, we establish conditions for the strong uniform consistency of kernel density estimators. We also make use of this strong approximation to study the weak convergence and integrated squared error properties of these estimators. We conclude by extending our results to the setting of length-biased sampling.

MSC:

62G07 Density estimation
62N01 Censored data models
62G20 Asymptotic properties of nonparametric inference
60F15 Strong limit theorems

Citations:

Zbl 0678.62051
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References:

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