Nonparametric priors for vectors of survival functions

The paper proposes a new nonparametric prior for two-dimensional vectors of survival functions (S1,S2). The definition we introduce is based on the notion of Levy copula and it will be used to model, in a nonparametric Bayesian framework, two-sample survival data. Such an application will yield a natural extension of the more familiar neutral to the right process of Doksum (1974) adopted for drawing inferences on single survival functions. We, then, obtain a description of the posterior distribution of (S1,S2), conditionally on possibly right-censored data. As a by-product of our analysis, we find out that the marginal distribution of a pair of observations from the two samples coincides with the Marshall-Olkin or the Weibull distribution according to specific choices of the marginal Levy measures.

[1]  N. Hjort.  Nonparametric Bayes Estimators Based on Beta Processes in Models for Life History Data , 1990 .

[2]  S. Walker,et al.  Beta-Stacy processes and a generalization of the Polya urn scheme , 1997 .

[3]  Albert Y. Lo,et al.  On a class of Bayesian nonparametric estimates: II. Hazard rate estimates , 1989 .

[4]  J. E. Griffin,et al.  Order-Based Dependent Dirichlet Processes , 2006 .

[5]  S. Walker,et al.  BAYESIAN NONPARAMETRIC ESTIMATION OF A BIVARIATE SURVIVAL FUNCTION , 2007 .

[6]  S. Walker,et al.  Posterior analysis for some classes of nonparametric models , 2008 .

[7]  T. Ferguson,et al.  Bayesian Nonparametric Estimation Based on Censored Data , 1979 .

[8]  Yongdai Kim,et al.  On posterior consistency of survival models , 2001 .

[9]  Lancelot F. James.  Bayesian Poisson process partition calculus with an application to Bayesian Lévy moving averages , 2005, math/0508283.

[10]  R. Schilling.  Financial Modelling with Jump Processes , 2005 .

[11]  R. Ramamoorthi,et al.  Consistency of Dykstra-Laud priors , 2003 .

[12]  P. Tankov,et al.  Characterization of dependence of multidimensional Lévy processes using Lévy copulas , 2006 .

[13]  A. P. Basu,et al.  BIVARIATE FAILURE RATE , 1971 .

[14]  Hemant Ishwaran,et al.  SERIES REPRESENTATIONS FOR MULTIVARIATE GENERALIZED GAMMA PROCESSES VIA A SCALE INVARIANCE PRINCIPLE , 2009 .

[15]  R. Gill,et al.  A Survey of Product-Integration with a View Toward Application in Survival Analysis , 1990 .

[16]  Robert F. Woolson,et al.  Rank tests for censored matched pairs , 1980 .

[17]  S. MacEachern,et al.  An ANOVA Model for Dependent Random Measures , 2004 .

[18]  Igor Prunster,et al.  ASYMPTOTICS FOR POSTERIOR HAZARDS , 2009, 0908.1882.

[19]  Purushottam W. Laud,et al.  A Bayesian Nonparametric Approach to Reliability , 1981 .

[20]  T. Ferguson.  Prior Distributions on Spaces of Probability Measures , 1974 .

[21]  A. Gelfand,et al.  The Nested Dirichlet Process , 2008 .

[22]  J. Ghosh,et al.  Bayesian bivariate survival estimation , 2006 .

[23]  M. R. Leadbetter.  Poisson Processes , 2011, International Encyclopedia of Statistical Science.

[24]  Stephen G. Walker,et al.  A Bayesian semi-parametric bivariate failure time model , 2007, Comput. Stat. Data Anal..

[25]  K. Doksum.  Tailfree and Neutral Random Probabilities and Their Posterior Distributions , 1974 .

[26]  Michael,et al.  On a Class of Bayesian Nonparametric Estimates : I . Density Estimates , 2008 .

[27]  Zhiliang Ying,et al.  A simple nonparametric estimator of the bivariate survival function under univariate censoring , 1993 .

[28]  S. MacEachern.  Decision Theoretic Aspects of Dependent Nonparametric Processes , 2000 .

[29]  D. Dunson,et al.  Kernel stick-breaking processes. , 2008, Biometrika.

[30]  R. Ramamoorthi,et al.  Some Aspects Of Neutral To Right Priors , 2003 .

[31]  Stephen G. Walker,et al.  A bivariate Dirichlet process , 2003 .