Asymptotic inference for maximum likelihood estimators under the special exponential family with double-truncation

Biased sampling affects the inference for population parameters of interest if the sampling mechanism is not appropriately handled. This paper considers doubly-truncated data arising in lifetime data analysis in which samples are subject to both left- and right-truncations. To correct for the sampling bias with doubly-truncated data, maximum likelihood estimator (MLE) has been proposed under a parametric family called the special exponential family (Efron and Petrosian, in J Am Stat Assoc 94:824–834, 1999). However, there is still a lack of justifying the fundamental properties for the MLE, including consistency and asymptotic normality. In this paper, we point out that the classical asymptotic theory for the independent and identically distributed data is not suitable for studying the MLE under double-truncation due to the non-identical truncation intervals. Alternatively, we formalize the asymptotic results under independent but not identically distributed data that suitably takes into account for the between-sample heterogeneity of truncation intervals. We establish the consistency and asymptotic normality of the MLE under a reasonably simple set of regularity conditions. Then, we give asymptotically valid techniques to estimate standard errors and to construct confidence intervals. Simulations are conducted to verify the suggested techniques, and childhood cancer data are used for illustration.