M-Estimation of an Intensity Function and an Underlying Population Size Under Random Right Truncation

We consider a random right truncation observation scheme for estimation of a lifetime distribution where we observe only selected pairs of failure times and truncation times ( τ , T ) $$(\tau , T)$$ such that τ ≤ T $$\tau \le T$$ , and τ $$\tau $$ and T are independent. The number of units for which τ > T $$\tau > T$$ is not available and hence, information about the size of the original population is partially lost. Our interest is in estimating the marginal distribution of τ $$\tau $$ , and also in the population size. Similar problem was discussed already in 1980s. Since then several articles focusing on the maximum likelihood estimation (parametric as well as nonparametric) using forward as well as reverse hazard rates have been published. Here, we consider survival regression models when ( τ i , T i , x i ) , i ∈ ℰ $$(\tau _i, T_i, x_i), i \in \mathcal {E}$$ , are the observed data, where ℰ ∈ P $$\mathcal {E} \in \mathcal {P}$$ , and the size of P $$\mathcal {P}$$ unknown. Assuming parametric and semiparametric models, we discuss identifiability of the parameters and derive consistent M-estimators. The maximum likelihood estimator (MLE) is a special case of the proposed M-estimators and the M-estimators may be easier to obtain using an iterative procedure compared to the MLE. We also provide two estimators of the population size, which allow individual-level covariates. The method is illustrated using simulation studies.