Estimators of parameters of truncated distributions, namely the truncated normal distribution, have been widely studied for a known truncation region. There is also literature for estimating the unknown bounds for known parent distributions. In this work, we develop a novel algorithm under the expectation-solution (ES) framework, which is an iterative method of solving nonlinear estimating equations, to estimate both the bounds and the location and scale parameters of the parent normal distribution utilizing the theory of best linear unbiased estimates from location-scale families of distribution and unbiased minimum variance estimation of truncation regions. The conditions for the algorithm to converge to the solution of the estimating equations for a fixed sample size are discussed, and the asymptotic properties of the estimators are characterized using results on M- and Z-estimation from empirical process theory. The proposed method is then compared to methods utilizing the known truncation bounds via Monte Carlo simulation.

翻译：对于已知截断区域的截断分布（特别是截断正态分布），其参数估计量已有广泛研究。同时，现有文献也探讨了在已知母分布情况下对未知边界的估计问题。本研究基于期望-解（ES）框架——一种求解非线性估计方程的迭代方法，提出了一种新颖算法。该算法结合位置-尺度分布族的最佳线性无偏估计理论以及截断区域的无偏最小方差估计理论，能够同时估计母正态分布的边界、位置参数与尺度参数。我们讨论了固定样本量下算法收敛至估计方程解的条件，并利用经验过程理论中M估计与Z估计的相关结果，刻画了估计量的渐近性质。最后，通过蒙特卡洛模拟将所提方法与已知截断边界下的估计方法进行了对比分析。