Density Estimation Under Random Censorship and Order Restrictions

Abstract

Do a random censorship and/or order restrictions (e.g., nonnegativity, monotonicity, convexity) affect estimation of a smooth density under mean integrated squared error (MISE)? Under mild assumptions, the known asymptotic results, which are concerned only with rates, answer “no.” This answer, especially for censored data, contradicts practical experience and statistical intuition. So what can be said about constants of MISE convergence? It is shown that asymptotically (a) censorship does affect the constant, and this allows one to find a relationship between sample sizes of directly observed and censored datasets that implies the same precision of estimation, and (b) an order restriction does not affect the constant, and thus no isotonic estimation is needed. Intensive Monte Carlo simulations show that the lessons of the sharp asymptotics are valuable for small sample sizes. Also, the estimator developed is illustrated both on simulated data and a dataset of lifetimes of conveyer blades used at wastewater treatment plants.

KEY WORDS:

• Coefficient of difficulty
• Data-driven estimation
• Isotonic estimation
• Nonparametric
• Survival analysis