论文标题
根据经验过程的最大及其位置的精确和渐近型测试
Exact and asymptotic goodness-of-fit tests based on the maximum and its location of the empirical process
论文作者
论文摘要
标准化经验过程的最高措施是测试分配功能$ f $ i.i.d.的有希望的统计数据。真实的随机变量要么等于给定的分布函数$ f_0 $（假设）或$ f \ ge f_0 $（单面替代）。由于\ cite {r5}众所周知，由于样本量趋向于无穷大，因此，超层的仿射线性转化会融合到牙胶定律中。这样可以构建渐近级别$α$测试。但是，收敛速度非常慢。结果，即使对于$ 10.000 $ $ 10.000的样本量，I型错误的概率也大得多。现在，标准化由$ 1/\ sqrt {f_0（x）（1-f_0（x））} $组成。通过合适的随机常数代替重量功能会导致一种新的测试统计，为此，我们可以在假设下得出确切的分布（和极限分布）。通过蒙特卡洛模拟的比较表明，由于\ cite {r20}，新测试均匀均匀地比Smirnov检验和适当修改的测试。我们的方法还适用于双面替代$ f \ neq f_0 $。
The supremum of the standardized empirical process is a promising statistic for testing whether the distribution function $F$ of i.i.d. real random variables is either equal to a given distribution function $F_0$ (hypothesis) or $F \ge F_0$ (one-sided alternative). Since \cite{r5} it is well-known that an affine-linear transformation of the suprema converge in distribution to the Gumbel law as the sample size tends to infinity. This enables the construction of an asymptotic level-$α$ test. However, the rate of convergence is extremely slow. As a consequence the probability of the type I error is much larger than $α$ even for sample sizes beyond $10.000$. Now, the standardization consists of the weight-function $1/\sqrt{F_0(x)(1-F_0(x))}$. Substituting the weight-function by a suitable random constant leads to a new test-statistic, for which we can derive the exact distribution (and the limit distribution) under the hypothesis. A comparison via a Monte-Carlo simulation shows that the new test is uniformly better than the Smirnov-test and an appropriately modified test due to \cite{r20}. Our methodology also works for the two-sided alternative $F \neq F_0$.
