March 2023 Exact and asymptotic goodness-of-fit tests based on the maximum and its location of the empirical process
Dietmar Ferger
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Braz. J. Probab. Stat. 37(1): 140-154 (March 2023). DOI: 10.1214/23-BJPS564


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 F0 (hypothesis) or FF0 (one-sided alternative). Since (The Annals of Statistics 7 (1979) 108–115) 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/F0(x)(1F0(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 (The Annals of Statistics 11 (1983) 933–946). Our methodology also works for the two-sided alternative FF0.


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Dietmar Ferger. "Exact and asymptotic goodness-of-fit tests based on the maximum and its location of the empirical process." Braz. J. Probab. Stat. 37 (1) 140 - 154, March 2023.


Received: 1 June 2022; Accepted: 1 January 2023; Published: March 2023
First available in Project Euclid: 27 April 2023

MathSciNet: MR4580888
zbMATH: 07692853
Digital Object Identifier: 10.1214/23-BJPS564

Keywords: empirical process , goodness of fit , measurability and continuity of the argmax-functional

Rights: Copyright © 2023 Brazilian Statistical Association


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Vol.37 • No. 1 • March 2023
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