zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
A Berry-Esseen type bound in kernel density estimation for strong mixing censored samples. (English) Zbl 1159.62019
Summary: We discuss the estimation of a density function based on censored data by the kernel smoothing method when the survival and the censoring times form a stationary α-mixing sequence. A Berry-Esseen type bound is derived for the kernel density estimator at a fixed point x. For practical purposes, a randomly weighted estimator of the density function is also constructed and investigated.
MSC:
62G07Density estimation
62G20Nonparametric asymptotic efficiency
62N01Censored data models
62N02Estimation (survival analysis)
References:
[1]Breslow, N.; Crowley, J.: A large sample study of the life table and product limit estimates under random censorship, Ann. statist. 2, 437-453 (1974) · Zbl 0283.62023 · doi:10.1214/aos/1176342705
[2]Földes, A.; Rejtö, L.: A LIL type result for the product limit estimator, Z. wahrsch. Verw. gebiete 56, 75-84 (1981) · Zbl 0443.62031 · doi:10.1007/BF00531975
[3]Gu, M. G.; Lai, T. L.: Functional laws of the iterated logarithm for the product-limit estimator of a distribution function under random censorship or truncation, Ann. probab. 18, 160-189 (1990) · Zbl 0705.62040 · doi:10.1214/aop/1176990943
[4]Gill, R.: Censoring and stochastic integrals, Mathematical centre tracts 124 (1980) · Zbl 0456.62003
[5]Kang, S. S.; Koehler, K. J.: Modification of the greenwood formula for correlated failure times, Biometrics 53, 885-899 (1997) · Zbl 0890.62081 · doi:10.2307/2533550
[6]Wei, L. J.; Lin, D. Y.; Weissfeld, L.: Regression analysis of multivariate incomplete failure times data by modelling marginal distributions, J. amer. Statist. assoc. 84, 1064-1073 (1989)
[7]Shumway, R. H.; Azari, A. S.; Johnson, P.: Estimating mean concentrations under transformation for environmental data with detection limits, Technometrics 31, 347-356 (1988)
[8]Koehler, K. J.; Symanowski, J.: Constructing multivariate distributions with specific marginal distributions, J. multivariate anal. 55, 261-282 (1995) · Zbl 0863.62048 · doi:10.1006/jmva.1995.1079
[9]Koehler, K. J.: An analysis of temperature effects on bean leaf beetle egg hatch times, Proceedings of the 1994 KSU conference on applied statistics in agriculture 6, 215-229 (1995)
[10]Ying, Z.; Wei, L. J.: The kaplan–meier estimate for dependent failure time observations, J. multivariate anal. 50, 17-29 (1994) · Zbl 0798.62048 · doi:10.1006/jmva.1994.1031
[11]Lecoutre, J. P.; Ould-Saïd, E.: Convergence of the conditional kaplan–meier estimate under strong mixing, J. statist. Plann. inference 44, 359-369 (1995) · Zbl 0813.62042 · doi:10.1016/0378-3758(94)00084-9
[12]Cai, Z. W.: Asymptotic properties of kaplan–meier estimator for censored dependent data, Statist. probab. Lett. 37, 381-389 (1998) · Zbl 0902.62040 · doi:10.1016/S0167-7152(97)00141-7
[13]Cai, Z. W.: Estimating a distribution function for censored time series data, J. multivariate anal. 78, 299-318 (2001) · Zbl 1057.62068 · doi:10.1006/jmva.2000.1953
[14]Mielniczuk, J.: Some asymptotic properties of kernel estimators of a density function in case of censored data, Ann. statist. 14, 766-773 (1986) · Zbl 0603.62047 · doi:10.1214/aos/1176349954
[15]Diehl, S.; Stute, W.: Kernel density and hazard function estimation in the presence of censoring, J. multivariate anal. 25, 299-310 (1988) · Zbl 0661.62028 · doi:10.1016/0047-259X(88)90053-X
[16]Xiang, X.: Law of the logarithm for density and hazard rate estimation for censored data, J. multivariate anal. 49, 278-286 (1994) · Zbl 0795.62044 · doi:10.1006/jmva.1994.1027
[17]Lo, S. H.; Mack, Y. P.; Wang, J. L.: Density and hazard rate estimation for censored data via strong representation of the kaplan–meier estimator, Probab. theory related fields 80, No. 4, 461-473 (1989) · Zbl 0637.62039 · doi:10.1007/BF01794434
[18]Sun, L. Q.; Zhu, L. X.: A Berry–Esseen type bound for kernel density estimators under random censorship, Acta. math. Sinica 42, No. 4, 627-636 (1999) · Zbl 1115.62319
[19]Xue, L. G.: Approximation rates of the error distribution of wavelet estimators of a density function under censorship, J. statist. Plann. inference 118, 167-183 (2004) · Zbl 1031.62031 · doi:10.1016/S0378-3758(02)00393-2
[20]Li, L.: Non-linear wavelet-based density estimators under random censorship, J. statist. Plann. inference 117, No. 1, 35-58 (2003) · Zbl 1022.62038 · doi:10.1016/S0378-3758(02)00366-X
[21]A. Rodríguez-Casal, J. de Uña-Álvarez, Nonlinear wavelet density estimation under the Koziol-Green model, in: The International Conference on Recent Trends and Directions in Nonparametric Statistics, J. Nonparametr. Statist. 16 (1–2) (2004) 91–109 · Zbl 1063.62049 · doi:10.1080/10485250310001622875
[22]Liebscher, E.: Kernel density and hazard rate estimation for censored data under α-mixing condition, Ann. inst. Statist. math. 54, No. 1, 19-28 (2002) · Zbl 0991.62029 · doi:10.1023/A:1016157519826
[23]Doukhan, P.: Mixing: properties and examples, Lecture notes in statistics 85 (1994) · Zbl 0801.60027
[24]Zhou, Y.; Liang, H.: Asymptotic normality for L1 norm kernel estimator of conditional median under α-mixing dependence, J. multivariate anal. 73, 136-154 (2000) · Zbl 0948.62026 · doi:10.1006/jmva.1999.1876
[25]Masry, E.: Asymptotic normality for deconvolution estimators of multivariate densities of stationary processes, J. multivariate anal. 44, 47-68 (1993) · Zbl 0783.62065 · doi:10.1006/jmva.1993.1003
[26]Masry, E.: Strong consistency and rates for deconvolution estimators of multivariate densities of stationary processes, Stochastic process. Appl. 47, 53-74 (1993) · Zbl 0797.62071 · doi:10.1016/0304-4149(93)90094-K
[27]Petrov, V. V.: Limit theorems of probability theory, (1995)
[28]Chang, M. N.; Rao, P. V.: Berry–Esseen bound for the kaplan–meier estimator, Comm. statist. Theory methods 18, No. 12, 4647-4664 (1989) · Zbl 0707.62079 · doi:10.1080/03610928908830180
[29]Hall, P.; Heyde, C. C.: Martingale limit theory and its application., (1980) · Zbl 0462.60045
[30]Yang, S. C.: Maximal moment inequality for partial sums of strong mixing sequences and application, Acta. math. Sinica, English series 23, 1013-1024 (2007) · Zbl 1121.60017 · doi:10.1007/s10114-005-0841-9
[31]Yang, S. C.; Li, Y. M.: Uniformly asymptotic normality of the regression weighted estimator for strong mixing samples, Acta. math. Sinica 49, No. 5, 1163-1170 (2006) · Zbl 1114.62019
[32]Cai, Z. W.; Roussas, G. G.: Uniform strong estimation under α-mixing, with rate, Statist. probab. Lett. 15, 47-55 (1992) · Zbl 0757.62024 · doi:10.1016/0167-7152(92)90284-C
[33]Liebscher, E.: Estimation of the density and the regression function under mixing conditions, Statist. decisions 19, 9-26 (2001) · Zbl 1179.62051
[34]Liebscher, E.: Strong convergence of sums of α-mixing random variables with applications to density estimation, Stochastic processes appl. 65, 69-80 (1996) · Zbl 0885.62045 · doi:10.1016/S0304-4149(96)00096-8