Ren, Jian-Jian; Zhou, Mai L-estimators and M-estimators for doubly censored data. (English) Zbl 0881.62054 J. Nonparametric Stat. 8, No. 1, 1-20 (1997). Summary: Motivated by estimation and testing with doubly censored data, we study (robust) \(L\)-estimators and \(M\)-estimators based on such data. These estimators are given through functionals of the self-consistent estimator \(S^{(n)}_X\) for the survival function with doubly censored data. We show that a Hadamard differentiable functional of \(S^{(n)}_X\) is asymptotically normal. Thereby, the asymptotic normality of the proposed \(L\)-estimators and \(M\)-estimators for doubly censored data are derived via their Hadamard differentiability properties. The estimates for the asymptotic variances of the proposed estimators are also given and shown to be strongly consistent. The proposed estimators are applied to a doubly censored data set encountered in breast cancer research. Cited in 4 Documents MSC: 62G20 Asymptotic properties of nonparametric inference 62G35 Nonparametric robustness 62G05 Nonparametric estimation 62E20 Asymptotic distribution theory in statistics Keywords:\(M\)-estimators; \(L\)-estimators; self-consistent estimator; strong consistency; asymptotic normality; Fredholm integral equation; doubly censored data; Hadamard differentiability; asymptotic variances PDFBibTeX XMLCite \textit{J.-J. Ren} and \textit{M. Zhou}, J. Nonparametric Stat. 8, No. 1, 1--20 (1997; Zbl 0881.62054) Full Text: DOI References: [1] Andersen P. K., Statistical Models Based on Counting Processes (1993) · Zbl 0769.62061 · doi:10.1007/978-1-4612-4348-9 [2] DOI: 10.1214/aos/1176350608 · Zbl 0629.62040 · doi:10.1214/aos/1176350608 [3] DOI: 10.1214/aos/1176347506 · Zbl 0706.62044 · doi:10.1214/aos/1176347506 [4] Delves L. M., Numerical Solution of Integral Equations (1974) · Zbl 0294.65068 [5] DOI: 10.1017/CBO9780511569609 · Zbl 0592.65093 · doi:10.1017/CBO9780511569609 [6] Fernholz L. T., Von Mises Calculus for Statistical Functional (1983) · Zbl 0525.62031 · doi:10.1007/978-1-4612-5604-5 [7] Gill R. D., Scand. J. Statis. 16 pp 97– (1989) [8] DOI: 10.1214/aos/1176349140 · Zbl 0788.62029 · doi:10.1214/aos/1176349140 [9] DOI: 10.1002/0471725250 · Zbl 0536.62025 · doi:10.1002/0471725250 [10] Iranpour R., Basic Stochastic Processes-The Mark Kac Lectures (1988) · Zbl 0681.60035 [11] DOI: 10.1214/aos/1032298293 · Zbl 0867.62019 · doi:10.1214/aos/1032298293 [12] DOI: 10.1002/1097-0142(19930601)71:11<3547::AID-CNCR2820711114>3.0.CO;2-C · doi:10.1002/1097-0142(19930601)71:11<3547::AID-CNCR2820711114>3.0.CO;2-C [13] Reeds J. A. On the definition of Von Mises functional Ph. D. dissertation Harvard University Cambridge, MA 1976 [14] Ren J., Ann. Inst. Statist. Math. 47 pp 525– (1995) [15] Ren J., J. Statist. Plan. Infer. (1994) [16] DOI: 10.1016/0047-259X(91)90003-K · Zbl 0739.62037 · doi:10.1016/0047-259X(91)90003-K [17] DOI: 10.1006/jmva.1995.1064 · Zbl 0898.62019 · doi:10.1006/jmva.1995.1064 [18] DOI: 10.1002/9780470316481 · Zbl 0538.62002 · doi:10.1002/9780470316481 [19] DOI: 10.2307/2285518 · Zbl 0281.62044 · doi:10.2307/2285518 [20] Von Mises R., AMS 18 pp 309– (1947) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.