{"title":"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","authors":"D. Ferger","doi":"10.1214/23-bjps564","DOIUrl":null,"url":null,"abstract":"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 $F \\neq F_0$.","PeriodicalId":51242,"journal":{"name":"Brazilian Journal of Probability and Statistics","volume":" ","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2022-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Exact and asymptotic goodness-of-fit tests based on the maximum and its location of the empirical process\",\"authors\":\"D. Ferger\",\"doi\":\"10.1214/23-bjps564\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"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 $F \\\\neq F_0$.\",\"PeriodicalId\":51242,\"journal\":{\"name\":\"Brazilian Journal of Probability and Statistics\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2022-07-16\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Brazilian Journal of Probability and Statistics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1214/23-bjps564\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"STATISTICS & PROBABILITY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Brazilian Journal of Probability and Statistics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1214/23-bjps564","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
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 $F \neq F_0$.
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.