Modified Anderson-Darling Test Research Articles

Modified Goodness of Fit Tests for Rayleigh Distribution

The modified goodness of fit tests for the Rayleigh distribution are studied. The critical values of modified Kolmogorov-Smirnov, Cramer-von-Mises and Anderson-Darling tests are obtained by Monte Carlo simulation for different sample sizes and significant levels. The type I error rate and power of these tests are studied and compared. The results show that all of the three tests have type I error rate close to the significant levels. Under several alternative distributions, it is founded that when the sample size is large, modified Anderson-Darling has the largest power in all cases. However, when the sample size is small, skewness of the distribution plays an important role. For the more skewed distribution, the modified Anderson-Darling test has more power than the others, while the modified Cramer-von-Mises has the largest power when the distribution is less skewed.

Modified likelihood ratio tests for extreme value distributions

The modified Anderson-Darling test statistics have been suggested in testing the highly skewed distributions to detect more possible departures in the right (or left) tail. In this paper, we propose the corresponding modified likelihood ratio test statistics, and compare their performances for the Gumbel distribution with the modified Anderson-Darling test statistics. We also provide the approximate critical values for the generalized extreme value distribution and generalized Pareto distribution where the unknown shape parameter is present.

3변수 Weibull 분포형의 형상매개변수 및 극치값 가중치를 고려한 EDF 검정에 대한 연구

적절한 확률분포형을 결정하고 그에 따른 확률수문량을 산정하는 것은 빈도해석에서 가장 중요한 절차이며, 이를 수행하기 위해서는 경험적 확률분포에서 얻어지는 자료와 가정한 확률분포에서 얻어지는 자료의 일치 정도를 판별하는 적합도 검정을 거쳐야 한다. 지금까지 일반적으로 적용된 적합도 검정 방법은 분포형의 전체적인 적합정도를 판별하여 최근의 기상이변으로 인한 극치 사상에 대하여는 충분히 고려하지 못하고 있다. 따라서 본 연구에서는 분포형의 극치 사상에 가중치를 주는 modified Anderson-Darling(AD) 검정 방법을 3변수 Weibull 분포형에 적용하여 검정통계량 한계값과 기각력을 살펴보았으며 이를 실제자료에 적용한 결과, modified AD 검정 방법이 다른 기존의 적합도 검정보다 더 우수한 기각력을 가지고 있음을 확인하였다. 이는 앞으로 3변수 Weibull 분포형을 이용한 극치 수문량 선정에 있어 modified AD 방법이 하나의 기준으로 작용할 수 있을 것이라 판단된다. The most important procedure in frequency analysis is to determine the appropriate probability distribution and to estimate quantiles for a given return period. To perform the frequency analysis, the goodness-of-fit tests should be carried out for judging fitness between obtained data from empirical probability distribution and assumed probability distribution. The previous goodness-of-fit could not consider enough extreme events from the recent climate change. In this study, the critical values of the modified Anderson-Darling test statistics were derived for 3-parameter Weibull distribution and power test was performed to evaluate the performance of the suggested test. Finally, this method was applied to 50 sites in South Korea. The result shows that the power of modified Anderson-Darling test has better than other existing goodness-of-fit tests. Thus, modified Anderson-Darling test will be able to act as a reference of goodness-of-fit test for 3-parameter Weibull model.

Approximation of modified Anderson–Darling test statistics for extreme value distributions with unknown shape parameter

Studies of the goodness-of-fit test, which describes how well a model fits a set of observations with an assumed distribution, have long been the subject of statistical research. The selection of an appropriate probability distribution is generally based on goodness-of-fit tests. This test is an effective means of examining how well a sample data set agrees with an assumed probability distribution that represents its population. However, the empirical distribution function test gives equal weight to the differences between the empirical and theoretical distribution functions corresponding to all observations. The modified Anderson–Darling test, suggested by Ahmad et al. (1988), uses a weight function that emphasizes the tail deviations at the upper or lower tails. In this study, we derive new regression equation forms of the critical values for the modified Anderson–Darling test statistics considering the effect of unknown shape parameters. The regression equations are derived using simulation experiments for extreme value distributions such as the log-Gumbel, generalized Pareto, GEV, and generalized logistic models. In addition, power test and at-site frequency analyses are performed to evaluate the performance and to explain the applicability of the modified Anderson–Darling test.

Assessment of modified Anderson–Darling test statistics for the generalized extreme value and generalized logistic distributions

An important problem in frequency analysis is the selection of an appropriate probability distribution for a given sample data. This selection is generally based on goodness-of-fit tests. The goodness-of-fit method is an effective means of examining how well a sample data agrees with an assumed probability distribution as its population. However, the goodness of fit test based on empirical distribution functions gives equal weight to differences between empirical and theoretical distribution functions corresponding to all observations. To overcome this drawback, the modified Anderson–Darling test was suggested by Ahmad et al. (1988b). In this study, the critical values of the modified Anderson–Darling test statistics are revised using simulation experiments with extensions of the shape parameters for the GEV and GLO distributions, and a power study is performed to test the performance of the modified Anderson–Darling test. The results of the power study show that the modified Anderson–Darling test is more powerful than traditional tests such as the χ2, Kolmogorov–Smirnov, and Cramer von Mises tests. In addition, to compare the results of these goodness-of-fit tests, the modified Anderson–Darling test is applied to the annual maximum rainfall data in Korea.

Analysis of horizontal machining center field failure data based on generalized linear mixed model—a case study

AbstractIn order to assess the operational reliability of the horizontal machining center (HMC), field failure data for 14 HMCs over one year are collected from an engine plant of a large automobile manufacturing company in China. In contrast with the usual approach, which just pools the data from all copies under the assumption that each copy is modeled by the same power law processes (PLP), a new model based on the generalized linear mixed model (GLMM) is proposed for analyzing the failure data from all copies of HMC. A basic idea of this method is to assume heterogeneity among all copies of HMC; it is also found that the underlying model for each individual copy is a PLP model with different shape parameters and scale parameters in the GLMM model. This method can make inferences about both the population and each individual copy. Meanwhile, the modified Anderson–Darling test is adapted to the goodness‐of‐fit test of the model. The results of the analysis suggest that this method is effective to analyze reliability of HMC. Copyright © 2010 John Wiley & Sons, Ltd.

Gumbel 분포형의 수정 Anderson-Darling 검정통계량 유도 및 기각력 검토

빈도해석에 있어서 중요한 문제는 특정 재현기간에 대한 수문량의 크기를 산정하는 것으로, 빈도해석에서는 일반적으로 관측기간보다 긴 재현기간에 해당하는 수문량의 크기를 산정하기 위해 가정된 확률분포형을 표본 자료에 적합시키게 된다. 따라서 적절한 확률분포형의 선정이 무엇보다 중요하며 이는 일반적으로 대상 자료로부터 얻어지는 경험적 빈도분포와 가정한 확률분포의 일치 정도를 판단하는 적합도 검정 방법을 이용하게 된다. 일반적으로 많이 사용되는 적합도검정 방법들은 모든 표본 자료들의 적합 정도를 동일하게 고려하기 때문에 극치사상의 크기 증가에 따른 영향은 반영하기 힘든 방법들이다. 따라서 본 연구에서는 모의실험을 통해 극치사상에 대하여 가중치를 주는 modified Anderson-Darling (AD) 검정 방법의 Gumbel 분포형에 대한 검정 통계량 한계값을 제시하였으며, 기존의 여러 적합도 검정 방법과의 기각력을 비교해 보고, 이를 실제 자료에 적용하여 그 결과를 살펴보았다. 그 결과 modified AD 검정 방법이 기존의 여러 가지 적합도 검정 방법보다도 기각력이 더 우수한 것으로 나타났으며, 기존의 적합도 검정 방법으로는 부족한 분포형 선정 기준의 부족한 부분을 어느 정도 보완해 줄 수 있을 것으로 판단되었다. An important problem in frequency analysis is the estimation of the quantile for a certain return period. In frequency analysis an assumed probability distribution is fitted to the observed sample data to estimate the quantile at the upper tail corresponding to return periods which are usually much larger than the record length. In most cases, the selection of an appropriate probability distribution is based on goodness of fit tests. The goodness of fit test method can be described as a method for examining how well sample data agrees with an assumed probability distribution as its population. However it gives generally equal weight to differences between empirical and theoretical distribution functions corresponding to all the observations. In this study, the modified Anderson-Darling (AD) test statistics are provided using simulation and the power study are performed to compare the efficiency of other goodness of fit tests. The power test results indicate that the modified AD test has better rejection performances than the traditional tests. In addition, the applications to real world data are discussed and shows that the modified AD test may be a powerful test for selecting an appropriate distribution for frequency analysis when extreme cases are considered.

Saddlepoint Approximations to the Limiting Distribution of the Modified Anderson–Darling Test Statistic

The approximation for the distribution function of test statistic is extremely important in statistics. The standard and higher-order saddlepoint approximations are considered in tails of the limiting distribution for the modified Anderson–Darling test. The saddlepoint approximations are compared with the approximation of Sinclair et al. (1990) for upper tail area. An empirical function is derived to estimate the critical values of a saddlepoint approximation.

Generalized Linear Mixed Models for Reliability Analysis of Multi-Copy Repairable Systems

The power law process (PLP) is usually applied to failure data from a single repairable system. When a system has a number of copies for analysis, the usual approach is to assume homogeneity among all system copies, and then to pool data from these copies. In the real world, however, it may be more reasonable to assume heterogeneity among the system copies. Therefore, this paper proposes a new generalized linear mixed model (GLMM), called PLP-GLMM, to analyse failure data from multi-copy repairable systems. In the PLP-GLMM, the underlying model for each system copy is assumed to be a PLP at Stage 1, and parameters vary among copies at Stage 2. The PLP-GLMM can make inferences about both the population, and each system copy when accounting for copy-to-copy variance. A modified Anderson-Darling test is adapted to the goodness-of-fit test of the PLP-GLMM. A numerical application is given to show the effectiveness of the model