CLT for integrated square error of density estimators with censoring indicators missing at random
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CLT for integrated square error of density estimators with censoring indicators missing at random Yu-Ye Zou1 · Han-Ying Liang2 Received: 4 October 2017 / Revised: 20 November 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract A popular stochastic measure of the distance between the density of the lifetimes and its estimator is the integrated square error (ISE) and Hellinger distance (HD). In this paper, we focus on the right-censored model when the censoring indicators are missing at random. Based on two density estimators defined by Wang et al. (J Multivar Anal 100:835–850, 2009), and another new kernel estimator of the density, we established the asymptotic normality of the ISE and HD for the proposed estimators. In addition, the uniformly strongly consistency of the new kernel estimator of the density is discussed. Also, a simulation study is conducted to compare finite-sample performance of the proposed estimators. Keywords Asymptotic normality · Hellinger distance · Integrated square error · Missing at random · Strong consistency Mathematics Subject Classification 62N01 · 62G07
1 Introduction When data are observed completely, the problem of density estimation has received considerable attention. For example, Hosseinioun et al. (2012), Wied and Weißbach (2012), Chatrabgoun et al. (2017) and Liu and Lu (2011). In many situations, as for example in medical follow-up or in engineering life testing studies, one may not be able to observe the variable of interest. The incomplete data are often encountered such
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Han-Ying Liang [email protected] Yu-Ye Zou [email protected]
1
College of Economics and Management, Shanghai Maritime University, Shanghai 201306, People’s Republic of China
2
School of Mathematical Science, Tongji University, Shanghai 200092, People’s Republic of China
123
Y.-Y. Zou, H.-Y. Liang
as right-censored or/and left-truncated as well as missing data for various reasons. The censored or truncated models have been receiving considerable attention for a long time, such as Liang and Baek (2016), Liang and Liu (2013), Zou and Liang (2014) and Zamini et al. (2015), whereas missing data are relatively new and only in recently years have become appealing interest, see Xu et al. (2017), Zou and Liang (2017a), Zou et al. (2015), Nguyen and Tsoy (2017), Yang and Liu (2016) and Luo and Zhang (2016). In this paper, we focus on the survival data which are subject to both right censoring and missing at random. Let T be a random variable representing lifetime with distribution function (df) F and density f , and let C denote a random variable describing right censoring time with a continuous df G. It is assumed that T is independent of C. For the right censoring model, one observes only (X , δ), where X = min{T , C} and δ = I (T ≤ C). Let H and h be the df and density of X , respectively, then 1 − H (x) = [1 − F(x)][1 − G(x)]. Without loss of generality, we assume that T and C are nonnegative random variables, as usual in survival analysis. It is well known that almost all of
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