Exploratory statistical theory after searching the underlying distribution

1.作为位置的统计估计过程，在单样本模型中引入了样本均值、Hodges和Lehman的R-估计和Huber的M-估计。三个估计量的渐近分布理论和模拟均方误差给出了各自估计量的特征，这取决于潜在的分布。基于这些特征，我们提出了一个估计过程，在搜索到一个接近底层分布的分布后，从三个估计中选择一个估计。结果表明，新的估计量的均方误差比三种估计量更稳定。接下来，作为无分布检验程序，介绍了条件t检验、Wilcoxon符号秩检验和M检验。渐近相对效率和模拟功率的相应的测试进行了研究。基于它们的特点，我们提出了一个稳定的测试过程，在搜索到一个接近底层分布的分布后，从三个测试中选择一个。 ...更多信息 .在单向方差分析模型中，Tukey（1953）、Dunnett（1955）和Scheffe（1953）讨论了基于尺度不变M-统计量的单步多重比较方法。虽然正态理论过程的分布是由二重积分给出的，但所提出的过程的渐近分布表示为单重积分。表的渐近临界值提供了拟议的M程序。此外，虽然在Huber的M-估计量的渐近理论中需要潜在分布的对称性，但所提出的方法并不要求对称性。研究发现，除了底层分布为正态分布的情况外，M-方法上级经典的正态理论方法.在单向布局中，假设底层分布是正态的，我们可以执行Tukey-Kramer多重比较程序来搜索位置的所有成对差异。对于样本量不等的情况，Tukey-Kramer（T-K）方法是保守的。T-K方法是利用学生化极差分布A（t）的上c点给出的。A（t）是Tukey-Kramer统计量分布的下限。我们导出了分布B（t），它给出了Tukey-Kramer统计量分布的一个上界。通过数值二重积分，我们证明了B（t）的值略大于A（t）的值。结果表明，T-K方法的保守性较小。少

1. As statistical estimation procedures for location, the sample mean, Hodges and Lehman's R-estimator, and Huber's M-estimator are introduced in a one-sample model. The asymptotic distributional theory for the three estimators and simulated mean squared errors give the features of the respective estimators depending on the underlying distribution. Based on the features, we propose an estimation procedure selecting one of the three estimators after searching a distribution near to the underlying distribution. It is shown that the mean squared error of the new estimator is more stable than the three estimators. Next, as distribution-free test procedures, the conditional t-test, Wilcoxon's signed rank test, and the M-test are introduced. Asymptotic relative efficiency and simulated power of the respective tests are investigated. Based on their features, we propose a stable test procedure selecting one of the three tests after searching a distribution near to the underlying distribution.2 … More . In a one-way analysis of variance model, robust versions based on scale-invariant M-statistics are proposed for single-step multiple comparisons procedures discussed by Tukey (1953), Dunnett (1955), and Scheffe (1953). Although the distributions for the normal theory pocedures are given by double integrals, the asymptotic distributions for the proposed procedures are expressed as single integrals. Tables of asymptotic critical values are provided for the proposed M procedures. Furthermore although the symmetry of the underlying distribution is needed in the asymptotic theory of Huber's M-estimators, the proposed procedures do not demand the symmetry. It is found that the M-procedures are superior to the classical normal theory procedures except the case that an underlying distribution is normal.3. In the one-way layout assuming that the underlying distribution is normal, we may execute Tukey-Kramer multiple comparisons procedure for searching all pairwise differences of locations. For the unequal sample sizes, the Tukey-Kramer (T-K) method is conservative. The T-K method is given by using the upper c point of the studentized range distribution A(t). A(t) is a lower bound for the distribution of the Tukey-Kramer statistic. We derive the distribution B(t) which gives an upper bound for the distribution of Tukey-Kramer statistic. By using numerical double integration, we show that the value of B(t) is a little larger than that of A(t). As the result, we may verify that the conservativeness of the T-K method is small. Less