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Applications of Sequentially Estimating the Mean in a Normal Distribution Having Equal Mean and Variance

机译：均值和方差相等的正态分布中顺序估计均值的应用

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We suppose that we have at our disposal a sequence of independent observations from a N(θ,θ) distribution where θ( > 0) is an unknown parameter. Such a model will make good sense to approximate a Poisson(θ) distribution, especially when θ( > 0) is moderately large. The Fisher-information contained in the sample mean, the sample variance, and the MLE are included (Appendix A). The derivation of the UMVUE of θ and some remarks regarding computational complexities in numerically evaluating its expression, even for small fixed-sample-size n, are also included (Appendix B). Assuming the availability of a lower bound θ_L( > 0) for θ, both purely sequential and two-stage bounded risk methodologies are developed for estimating θ. We have considered the analogs of fixed-sample-size MLE, sample mean, sample median, sample variance, and the UMVUE. First-order asymptotic properties of both purely sequential as well as two-stage sample sizes and the associated risk-bound for an analog of the MLE have been found. Extensive investigations based on computer-simulations have been carried out and the previously stated estimators of θ are compared with one another. The effect of the ratio θ_L/θ on the pilot sample size is critically examined. Overall, we have found that an analog of the fixed-sample-size MLE performs most satisfactorily. In the end, both proposed methodologies are successfully implemented to investigate how these work in two practical situations with the help of real datasets. They show that these methodologies stay fairly robust under mild departures from normality.
机译：我们假设我们可以从N（θ，θ）分布中获得一系列独立的观察结果，其中θ（> 0）是未知参数。这样的模型对于近似泊松（θ）分布非常有意义，尤其是当θ（> 0）适中时。样本均值，样本方差和MLE中包含的Fisher信息也包括在内（附录A）。还包括θ的UMVUE的推导以及有关在数值上评估其表达式的计算复杂性的一些说明，即使对于较小的固定样本大小的n也是如此（附录B）。假设θ的下界θ_L（> 0）可用，则开发了纯顺序风险法和两阶段有界风险方法来估计θ。我们考虑了固定样本大小MLE，样本均值，样本中位数，样本方差和UMVUE的类似物。已发现纯连续样本和两阶段样本量的一阶渐近性质以及与MLE类似物相关的风险约束。已经进行了基于计算机模拟的广泛研究，并且将先前所述的θ估计量相互比较。严格检验了比率θ_L/θ对导频样本大小的影响。总体而言，我们发现固定样本大小MLE的模拟效果最佳。最后，两种提出的方​​法均成功实施，以借助实际数据集研究它们在两种实际情况下的工作方式。他们表明，在轻微偏离常态的情况下，这些方法仍然相当健壮。

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