Nonparametric rank based estimation of bivariate densities given censored data conditional on marginal probabilities

Alan D Hutson

doi:10.1186/s40488-016-0047-y

Abstract

In this note we develop a new Kaplan-Meier product-limit type estimator for the bivariate survival function given right censored data in one or both dimensions. Our derivation is based on extending the constrained maximum likelihood density based approach that is utilized in the univariate setting as an alternative strategy to the approach originally developed by Kaplan and Meier (1958). The key feature of our bivariate survival function is that the marginal survival functions correspond exactly to the Kaplan-Meier product limit estimators. This provides a level of consistency between the joint bivariate estimator and the marginal quantities as compared to other approaches. The approach we outline in this note may be extended to higher dimensions and different censoring mechanisms using the same techniques.

Highlights

1 Introduction In this note we develop a new Kaplan-Meier product-limit type estimator for the bivariate survival function given right censored data in one or both dimensions
Our derivation is based on extending the constrained maximum likelihood density based approach (Satten and Datta 2001; Zhou 2005) that is utilized in the univariate setting as an alternative strategy to the classical discrete nonparametric hazard function approach (Kaplan and Meier 1958)
To the best of our knowledge one of the key limitations of all of the completely nonparametric bivariate survival function estimators developed to-date is that they yield marginal estimators that may not be equivalent to the product-limit estimator corresponding to each dimension

Summary

Introduction

In this note we develop a new Kaplan-Meier product-limit type estimator for the bivariate survival function given right censored data in one or both dimensions. We start by outlining the nonparametric maximum likelihood based density estimator in the univariate setting given right censored data We can utilize this estimator to define a survival function estimator, which is equivalent to the productlimit estimator. In terms of determining the relevant parameters for use in the nonparametric likelihood model we need to define an indicator function as a type of bookkeeping feature given censoring information from both marginal distributions In the specific case where there is no censoring for both X and Y the number of parameters is of size n and the corresponding maximum likelihood estimator for πrsi ,rtj is 1/n as per the standard empirical density estimator, i.e. there is a point mass of 1/n per each set of paired observations. For moderate to large samples and from a programming point of view, this becomes a rather complex computational problem such that we would recommend either bootstrap or jackknife methodologies for the purpose of variance estimation

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