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Hypothesis testing for the exponential distribution family consistently gains significant attentions from the statistical community, especially in situations characterized by small sample sizes. In this paper, we propose a novel approach for constructing a Kolmogorov-Smirnov type test based on representative points. This testing statistic is derived through a comparison between two empirical functions: one from the observed data and another from chosen representative points. A distribution-free quantile estimator is used for sample standardization. We also discuss how to select an optimal number of representative points based on the loss function, which is recommended for constructing the empirical function from the exponential distribution family. To evaluate the efficiency of this new test, a comprehensive Monte Carlo simulation is conducted to compare this new test with tailor-made tests. Simulation results show that our test statistic will be more powerful especially for small sample sizes, and it is competitive in the comparison with tailor-made tests. Some datasets are analyzed to further illustrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

AbstractHypothesis testing for the Weibull distribution always raises attention in the literature. It is challenging especially in small-sample scenarios. In this paper, we propose a new Kolmogorov-Smirnov type test based on the distance between two empirical functions, one from the data and another from the representative points of the underlying distribution. A bias correction technique is used to estimate unknown parameters in the Weibull distribution. We also discuss how to choose a suitable number of representative points based on the proposed loss function, which is recommended for constructing the empirical function from the Weibull distribution. To compare with tailor-made tests in the literature, a simulation study of the empirical type I error and testing power is conducted. It shows that our proposed test statistic is more powerful in most alternative scenarios, and it significantly improves the power of testing in small-sample scenarios. Finally, two real-world datasets are conducted to further demonstrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]