Diagnostic accuracy studies have progressed in the past decade to consider survival outcomes beyond the traditional dichotomous outcome. Another recent advance is the appearance of novel measures for diagnostic accuracy improvement by adding new markers. In this paper we attempt to integrate these two evolving areas and contribute a discussion on assessing diagnostic accuracy improvement for censored survival outcomes. More importantly, we consider competing-risk censoring in addition to independent censoring, and provide inferential procedures. Particularly, we consider fitting regression models based on cumulative incidence functions for the primary event, and propose parallel estimators for the adapted accuracy improvement measures based on inverse probability weighting and bivariate cumulative incidence function estimation. Both estimators perform very well in simulations and in an application to a breast cancer study.