Geodesic
Uniform Limit Theorems under length-biased sampling and type I censoring
Teoriâ slučajnyh processov, Tome 25 (2020) no. 2, pp. 93-106.

Voir la notice de l'article provenant de la source Math-Net.Ru

In recent years, in view of theory of empirical processes, authors have become more interested in the uniform analogue of the three fundamental theorems: the uniform law of large numbers of Glivenko-Cantelli type, the uniform central limit theorem for Donsker type and the functional law of the iterated logarithm (LIL). In this paper, under the bracketing entropy conditions, the uniform law of large numbers, uniform central limit theorem and the uniform LIL of Strassen type have been investigated in the case of length-biased and type I censoring.
Keywords: Bracketing entropy, Length-biased sampling, Uniform limit theorems. 
@article{THSP_2020_25_2_a9,
     author = {Raheleh Zamini and Sarah Jomhoori},
     title = {Uniform {Limit} {Theorems} under length-biased sampling and type {I} censoring},
     journal = {Teori\^a slu\v{c}ajnyh processov},
     pages = {93--106},
     publisher = {mathdoc},
     volume = {25},
     number = {2},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/THSP_2020_25_2_a9/}
}
TY  - JOUR
AU  - Raheleh Zamini
AU  - Sarah Jomhoori
TI  - Uniform Limit Theorems under length-biased sampling and type I censoring
JO  - Teoriâ slučajnyh processov
PY  - 2020
SP  - 93
EP  - 106
VL  - 25
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/THSP_2020_25_2_a9/
LA  - en
ID  - THSP_2020_25_2_a9
ER  - 
%0 Journal Article
%A Raheleh Zamini
%A Sarah Jomhoori
%T Uniform Limit Theorems under length-biased sampling and type I censoring
%J Teoriâ slučajnyh processov
%D 2020
%P 93-106
%V 25
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/THSP_2020_25_2_a9/
%G en
%F THSP_2020_25_2_a9
Raheleh Zamini; Sarah Jomhoori. Uniform Limit Theorems under length-biased sampling and type I censoring. Teoriâ slučajnyh processov, Tome 25 (2020) no. 2, pp. 93-106. http://geodesic.mathdoc.fr/item/THSP_2020_25_2_a9/

[1] M. Asgharian, C. E. M'Lan, D. B. Wolfson, “Length-biased sampling with right-censoring: an unconditional approach”, J. Amer. Statist. Assoc., 97 (2002), 201–209

[2] J. Bae, S. Kim, “The empirical LIL for the Kaplan-Meier integral process”, Bull. Korean Math. Soc., 40:2 (2003a), 269–279

[3] J. Bae, S. Kim, “The uniform law of large numbers for the Kaplan-Meier integral process”, Bull. Austral. Math., 67 (2003b), 459–465

[4] J. Bae, S. Kim, “The uniform central limit theorem for the Kaplan-Meier integral process”, Bull. Austral. Math., 67 (2003c), 467–480

[5] J. DeHardt, “Generalizations of the Glivenko-Cantelli theorem”, Ann. Math. Stat., 42 (1971), 2050–2055

[6] J. De U$\tilde{n}$a-$\acute{A}$lvarez, “Nonparametric estimation under length-biased sampling and Type I censoring: a moment based approach”, Ann. Inst. Statist. Math., 56 (2004), 667–681 J. De U$\tilde{n}$a-$\acute{A}$lvarez, Nonparametric estimation under length-biased sampling and Type I censoring: a moment based approach, Ann. Inst. Statist. Math. 56 (2004), 667–681.

[7] R. M. Dudley, “Central limit theorems for empirical measures”, Ann. Probab., 6:6 (1978), 899–929

[8] R. M. Dudley, W. Philipp, “Invariance principles for sums of Banach space valued random elements and empirical processes”, Z. Wahrsch. Verw. Gebiete, 62 (1983), 509–552

[9] J. Hoffmann-Jörgensen, Stochastic processes on Polish spaces, Aarhus Universite, Matematisk Institut, Aarhus, 1991

[10] J. Kuelbs, R. M. Dudley, “Log log law for empirical measures”, Ann. Probab, 8 (1980), 405–418

[11] T. Lancaster, The econometric analysis of truncation data, Cambridge University Press, Cambridge, 1990

[12] M. Ossiander, “A central limit theorem under metric entropy with $L_2$ bracketing”, Ann. Probab., 15 (1987), 897–919

[13] G. Pisier, Le theorem de la limite centrale et la loi du logarithme itere dans les espaces de Banach, Seminaire Maurey-Schwarz 1975-1976 exposes Nos. 3 et 4, 1975

[14] D. Pollard, “Empirical processes: theory and applications”, Regional conference series in Probability and Statistics, 2, Inst. Math. Statist, Hayward CA, 1990

[15] S. Van de Geer, Applications of empirical process theory, Cambridge Series in Statistical and probabilistic mathematics, Cambridge University Press, Cambridge, U. K., 2000

[16] A. W. Van der Vaart, J. A. Wellner, Weak Convergence and Empirical Processes, Springer, New York, 1996

[17] B. Van Es, C. A. J. Klaassen, K. Oudshoorn, “Survival analysis under cross-sectional sampling: Length bias and multiplicative censoring”, J. Stat. Plann. Inf., 91 (2000), 295–312

[18] M.-C. Wang, “Nonparametric estimation from cross-sectional survival data”, J. American Stat. Assoc., 86 (1991), 130–143

[19] B. B. Winter, A. F$\ddot{o}$ldes, “A product-limit estimator for use with length-biased data”, Canadian J. Stat., 16 (1988), 337–355