Continuous statistical models: With or without truncation parameters?

Lifetime data are usually assumed to stem from a continuous distribution supported on [0, b) for some b ≤ ∞. The continuity assumption implies that the support of the distribution does not have atom points, particularly not at 0. Accordingly, it seems reasonable that with an accurate measurement tool all data observations will be positive. This suggests that the true support may be truncated from the left. In this work we investigate the effects of adding a left truncation parameter to a continuous lifetime data statistical model. We consider two main settings: right truncation parametric models with possible left truncation, and exponential family models with possible left truncation. We analyze the performance of some optimal estimators constructed under the assumption of no left truncation when left truncation is present, and vice versa. We investigate both asymptotic and finite-sample behavior of the estimators. We show that when left truncation is not assumed but is, in fact present, the estimators have a constant bias term, and therefore will result in inaccurate and inefficient estimation. We also show that assuming left truncation where actually there is none, typically does not result in substantial inefficiency, and some estimators in this case are asymptotically unbiased and efficient.