基于经验过程最大值及其位置的精确和渐近拟合优度检验

IF 0.6 4区 数学 Q4 STATISTICS & PROBABILITY
D. Ferger
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引用次数: 0

摘要

标准化经验过程的上确界是检验i.i.d.实随机变量的分布函数$F$是否等于给定的分布函数%F_0$(假设)或$F\geF_0$(单侧替代)的一个很有前途的统计量。由于\ cite{r5},众所周知,当样本大小趋于无穷大时,上确界的仿射线性变换在分布上收敛于Gumbel定律。这使得能够构建一个渐进水平-$\alpha$测试。然而,收敛速度极为缓慢。因此,即使样本量超过$10000$,I型错误的概率也远大于$\alpha$。现在,标准化由权重函数$1/\sqrt{F_0(x)(1-F_0(x))}$组成。用一个合适的随机常数代替权重函数会得到一个新的检验统计量,我们可以在假设下推导出它的精确分布(和极限分布)。通过蒙特卡罗模拟进行的比较表明,新的测试一致优于Smirnov测试和由于cite{r20}而适当修改的测试。我们的方法也适用于双边备选方案$F\neqF_0$。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Exact and asymptotic goodness-of-fit tests based on the maximum and its location of the empirical process
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-$\alpha$ test. However, the rate of convergence is extremely slow. As a consequence the probability of the type I error is much larger than $\alpha$ 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 $F \neq F_0$.
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来源期刊
CiteScore
1.60
自引率
10.00%
发文量
30
审稿时长
>12 weeks
期刊介绍: The Brazilian Journal of Probability and Statistics aims to publish high quality research papers in applied probability, applied statistics, computational statistics, mathematical statistics, probability theory and stochastic processes. More specifically, the following types of contributions will be considered: (i) Original articles dealing with methodological developments, comparison of competing techniques or their computational aspects. (ii) Original articles developing theoretical results. (iii) Articles that contain novel applications of existing methodologies to practical problems. For these papers the focus is in the importance and originality of the applied problem, as well as, applications of the best available methodologies to solve it. (iv) Survey articles containing a thorough coverage of topics of broad interest to probability and statistics. The journal will occasionally publish book reviews, invited papers and essays on the teaching of statistics.
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