Density estimation under random censorship and order restrictions: From asymptotic to small samples

Do a random censorship and/or order restrictions (e.g., nonnegativity, mono tonicity, convexity) affect estimation of a smooth density under mean integ rated squared error (MISE)? Under mild assumptions, the known asymptotic re sults, which are concerned only with rates, answer "no." This answer, espec ially for censored data, contradicts practical experience and statistical i ntuition. So what can be said about constants of MISE convergence? It is sh own that asymptotically (a) censorship does affect the constant, and this a llows 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 e stimation is needed. Intensive Monte Carlo simulations show that the lesson s of the sharp asymptotics are valuable for small sample sizes. Also, the e stimator developed is illustrated both on simulated data and a dataset of l ifetimes of conveyer blades used at wastewater treatment plants.