Kaplan-Meier Estimate on the Plane

Estimation of the bivariate survival function from censored data is considered. The product integral representation of univariate survival functions is generalized to the bivariate case and used to determine identifiability of the survival function of the partially observed data. A bivariate analogue of the Kaplan-Meier estimate is introduced and its almost sure consistency is studied. Extensions to the general multivariate case are sketched.

[1]  M. Shaked,et al.  On the Identifiability of Multivariate Life Distribution Functions. , 1982 .

[2]  Gregory Campbell.  Asymptotic properties of several nonparametric multivariate distribution function estimators under random censoring , 1982 .

[3]  J. Wellner,et al.  Empirical Processes with Applications to Statistics , 2009 .

[4]  F. Ruymgaart.  Some properties of bivariate empirical hazard processes under random censoring , 1989 .

[5]  O. Pons,et al.  A test of independence between two censored survival times , 1986 .

[6]  Richard D. Gill,et al.  Product-integrals and counting processes , 1987 .

[7]  Sue Leurgans,et al.  Nonparametric Estimation of a Bivariate Survival Function in the Presence of Censoring , 1986 .

[8]  J. Wellner.  A Heavy Censoring Limit Theorem for the Product Limit Estimator , 1985 .

[9]  D. Clayton,et al.  Multivariate generalizations of the proportional hazards model , 1985 .

[10]  Jane-ling Wang,et al.  Representations for the bivariate product limit estimators and the bootstrap versions , 1989 .

[11]  Lajos Horváth.  The rate of strong uniform consistency for the multivariate product-limit estimator , 1983 .

[12]  T. H. Hildebrandt.  Introduction to the theory of integration , 1963 .

[13]  M. D. Burke.  AN ALMOST-SURE APPROXIMATION OF A MULTIVARIATE PRODUCT-LIMIT ESTIMATOR UNDER RANDOM CENSORSHIP , 1988 .

[14]  J A Hanley,et al.  Nonparametric estimation of a multivariate distribution in the presence of censoring. , 1983, Biometrics.

[15]  A. V. Peterson.  Expressing the Kaplan-Meier estimator as a function of empirical subsurvival functions , 1977 .

[16]  D. Clayton.  A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence , 1978 .

[17]  B. Yandell,et al.  BOOTSTRAPPED MULTI‐DIMENSIONAL PRODUCT LIMIT PROCESS , 1988 .

[18]  Ren Johansen.  An Empirical Transition Matrix for Non-homogeneous Markov Chains Based on Censored Observations , 1978 .

[19]  B. Efron.  The two sample problem with censored data , 1967 .

[20]  Gregory Campbell,et al.  Nonparametric bivariate estimation with randomly censored data , 1981 .

[21]  R. Gill.  Censoring and stochastic integrals , 1980 .

[22]  J. A. Clarkson,et al.  On definitions of bounded variation for functions of two variables , 1933 .

[23]  E. Kaplan,et al.  Nonparametric Estimation from Incomplete Observations , 1958 .

[24]  A. Földes,et al.  STRONG UNIFORM CONSISTENCY FOR NONPARAMETRIC SURVIVAL CURVE ESTIMATORS FROM RANDOMLY CENSORED DATA , 1981 .

[25]  L. Horváth,et al.  The rate of strong uniform consistency for the product-limit estimator , 1983 .