On the association between variables with lower detection limits

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In this paper, we define a modified version τ(b) of Kendall's tau to measure the association in a pair (X,Y) of random variables subject to fixed left censoring due to known lower detection limits. We provide a nonparametric estimator of τ(b) and investigate its asymptotic properties. We then assume an Archimedean copula for (X,Y) and express τ(b) in terms of the copula parameter α and the censoring fractions. We deduce estimators for α and for the global Kendall's tau. We develop a goodness-of-fit test for the assumed copula. We evaluate the finite-sample performance of the proposed methods by simulations and illustrate their use with a real data set on plasma and saliva viral loads.