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Large sample properties for a class of copulas in bivariate survival analysis. (English) Zbl 1365.62375

Summary: This work is concerned with asymptotic properties of the bivariate survival function estimator using the functional relationship between marginal survival functions and a class of copulas for the dependence structure. Specifically, we study consistency and weak convergence of the bivariate survival function estimator obtained considering a two-step procedure of estimation. The obtained results are found from a key decomposition of the bivariate survival function in quantities that can be studied separately. In particular, we use relating results to almost sure and weak convergence of estimators, almost sure convergence of uniformly equicontinuous functions, and the delta method for functionals.

MSC:

62N02 Estimation in survival analysis and censored data
62N01 Censored data models
62F10 Point estimation
62F12 Asymptotic properties of parametric estimators
62G05 Nonparametric estimation
62H20 Measures of association (correlation, canonical correlation, etc.)
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References:

[1] Clayton DG (1978) A model for association in bivariate life-tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika 65:141-151 · Zbl 0394.92021 · doi:10.1093/biomet/65.1.141
[2] Dabrowska DM (1989) Kaplan-Meier estimates on the plane: weak convergence, LIL and the Bootstrap. J Multivar Anal 29:308-325 · Zbl 0667.62025
[3] Delgado MA, Escanciano JC (2012) Distribution-free tests of stochastic monotonicity. J Econom 170:68-75 · Zbl 1443.62117 · doi:10.1016/j.jeconom.2012.02.005
[4] Duchateau L, Janssen P (2008) The frailty model. Springer, New York · Zbl 1210.62153
[5] Fernholz LT (1983) Von Mises calculus for statistical functionals. In: Lecture notes in statistics, vol 19. Springer, New York · Zbl 0525.62031
[6] Fermanian JD, Radulovic D, Wegkamp M (2004) Weak convergence of empirical copula processes. Bernoulli 10:847-860 · Zbl 1068.62059 · doi:10.3150/bj/1099579158
[7] Gaenssler P, Stute W (1987) Seminar on empirical processes. DMV, Sem 9, Birkhäuser, Basel · Zbl 0637.62047
[8] Genest C, MacKay J (1986) The joy of copulas: bivariate distribution with uniform marginals. Am Stat 40:280-283
[9] Gill RD (1989) Non- and semi-parametric maximum likelihood estimators and the von Mises method (part 1). Scand J Stat 16:97-128 · Zbl 0688.62026
[10] Goethals K, Janssen P, Duchateau L (2008) Frailty models and copulas: similarities and differences. J Appl Stat 35(9):1071-1079 · Zbl 1253.62074 · doi:10.1080/02664760802271389
[11] Hougaard P (2000) Analysis of multivariate survival data. Springer, New York · Zbl 0962.62096 · doi:10.1007/978-1-4612-1304-8
[12] Joe H (1997) Multivariate models and dependence concepts. Chapman & Hall, London · Zbl 0990.62517 · doi:10.1201/b13150
[13] Kim G, Silvapulle MJ, Silvapulle P (2007) Comparison of semiparametric and parametric methods for estimating copulas. Comput Stat Data Anal 51:2836-2850 · Zbl 1161.62364 · doi:10.1016/j.csda.2006.10.009
[14] Lee S, Linton OB, Whang Y-J (2009) Testing for stochastic monotonicity. Econometrica 77:585-602 · Zbl 1161.62080 · doi:10.3982/ECTA7145
[15] Li Y, Prentice RL, Lin X (2008) Semiparametric maximum likelihood estimation in normal transformation models for bivariate survival data. Biometrika 95:947-960 · Zbl 1437.62541 · doi:10.1093/biomet/asn049
[16] Massonneta G, Janssen P, Duchateau L (2009) Modelling udder infection data using copula models for quadruples. J Stat Plan Inference 139(11):3865-3877 · Zbl 1169.62348 · doi:10.1016/j.jspi.2009.05.025
[17] Nelsen R (2006) An introduction to copulas, 2nd edn. Springer, New York · Zbl 1152.62030
[18] Oakes D (1989) Bivariate survival models induced by frailties. J Am Stat Assoc 84:487-493 · Zbl 0677.62094 · doi:10.1080/01621459.1989.10478795
[19] Owzar K, Sen PK (2003) Copulas: concepts and novel applications. Metron LXI(3):323-353 · Zbl 1416.62613
[20] Romeo JS, Tanaka NI, Pedroso-de-Lima AC (2006) Bivariate survival modeling: a Bayesian approach based on copulas. Lifetime Data Anal 12:205-222 · Zbl 1134.62321 · doi:10.1007/s10985-006-9001-5
[21] Sen PK, Singer JM, Pedroso-de-Lima AC (2010) From finite sample to asymptotic methods in statistics. Cambridge University Press, Cambridge
[22] Shih JH, Louis TA (1995) Inferences on the association parameter in copula models for bivariate survival data. Biometrics 51:1384-1399 · Zbl 0869.62083 · doi:10.2307/2533269
[23] Silvapulle MJ, Sen PK (2004) Constrained statistical inference. Inequalities, order and shape restrictions. Wiley, Hoboken, NJ · Zbl 1077.62019
[24] Sklar A (1959) Fonctions de répartition à \[n\] dimensions et leurs marges. Publication Institute Statistique. Univ. Paris 8:229-231 · Zbl 0100.14202
[25] Therneau TM, Grambsch P (2000) Modeling survival data: extending the Cox model. Springer, New York · Zbl 0958.62094 · doi:10.1007/978-1-4757-3294-8
[26] van der Vaart AW, Wellner JA (1996) Weak convergence and empirical processes. Springer, New York · doi:10.1007/978-1-4757-2545-2
[27] Wang W, Wells MT (2000) Model selection and semi-parametric inference for bivariate failure-time data. J Am Stat Assoc 95:62-72 · doi:10.1080/01621459.2000.10473899
[28] Wienke A (2010) Frailty models in survival analysis. Chapman & Hall/CRC, Boca Raton · doi:10.1201/9781420073911
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