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While time-dependent covariates naturally arise in time-to-event analysis, existing works in the literature discuss treatments for data collected either under an independent censoring mechanism or a longitudinal setting. Our formulation extends the current scope so that the conventional setting of time-dependent covariates can be properly handled. The new framework also generalizes the definition of quantiles and offers a new dynamic perspective for interpretation. Asymptotic properties of the recursive estimator are established. 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