Exploratory statistical theory after searching the underlying distribution

1. 作为位置的统计估计程序，在单样本模型中引入了样本均值、Hodges和Lehman的r估计量和Huber的m估计量。三种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随底层分布的特征。基于这些特征，我们提出了一种估计过程，即在搜索接近底层分布的分布后，从三个估计量中选择一个估计量。结果表明，新估计量的均方误差比三种估计量更稳定。接下来，作为无分布的检验程序，介绍了条件t检验、Wilcoxon有符号秩检验和m检验。研究了各测试的渐近相对效率和模拟功率。根据它们的特征，我们提出了一个稳定的测试程序，在搜索接近底层分布的分布后，从三个测试中选择一个。2 .更多。在单向方差分析模型中，针对Tukey（1953）、Dunnett（1955）和Scheffe（1953）讨论的单步多重比较过程，提出了基于尺度不变m统计的稳健版本。虽然正常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是用单积分表示的。给出了M过程的渐近临界值表。此外，虽然在Huber的m估计的渐近理论中需要底层分布的对称性，但所提出的过程并不要求对称性。除了底层分布为正态分布的情况外，m -过程优于经典正态理论过程。在单向布局中，假设底层分布为正态分布，我们可以执行Tukey-Kramer多重比较程序来搜索所有位置的两两差异。对于不相等的样本量，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统计量分布的下界。我们导出了分布B(t)，它给出了Tukey-Kramer统计量分布的上界。通过数值二重积分，我们证明了B(t)的值略大于a (t)的值。因此，我们可以验证T-K方法的保守性很小。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