Estimation of Monotone Density Function in Random Left Truncation Model
There is a considerably vast literature on curve estimation in complete data and many censorship models under shape constraint. In contrast, little attention has been paid to such estimations from random left truncation model. To the best of our knowledge, the current literature contains no result on non-parametric estimation of a probability density function from random left truncation model under monotone constraint using unconditional likelihood. A naive application of Grenander estimator, ignoring truncation model and using this estimator is an unsuitable method, which leads to inconsistency in estimating decreasing density function. Here, we derive an estimator which maximizes the likelihood function over the set of all non-increasing left-continuous step-wise density functions. Additionally, the weak convergence and the weak uniform consistency of the proposed estimator are established. Moreover, we study the small sample behavior of the estimator, empirically. Finally, an application of our introduced method to real data consists the last section of this paper