Empirical Distribution Function Tests Research Articles

Tests of fit for the Gumbel distribution: EDF-based tests against entropy-based tests

In this article, we propose some tests of fit based on sample entropy for the composite Gumbel (Extreme Value) hypothesis. The proposed test statistics are constructed using different entropy estimates. Through a Monte Carlo simulation, critical values of the test statistics for various sample sizes are obtained. Since the tests based on the empirical distribution function (EDF) are commonly used in practice, the power values of the entropy-based tests with those of the EDF tests are compared against various alternatives and different sample sizes. Finally, two real data sets are modeled by the Gumbel distribution.

Michael Arthur Stephens, 1927–2019

Michael Arthur Stephens, 1927–2019

Tests based on EDF statistics for randomly censored normal distributions when parameters are unknown

Goodness-of-fit techniques are an important topic in statistical analysis. Censored data occur frequently in survival experiments; therefore, many studies are conducted when data are censored. In this paper we mainly consider test statistics based on the empirical distribution function (EDF) to test normal distributions with unknown location and scale parameters when data are randomly censored. The most famous EDF test statistic is the Kolmogorov-Smirnov; in addition, the quadratic statistics such as the Cramer-von Mises and the Anderson- Darling statistic are well known. The Cramer-von Mises statistic is generalized to randomly censored cases by Koziol and Green (Biometrika, 63, 465-474, 1976). In this paper, we generalize the Anderson-Darling statistic to randomly censored data using the Kaplan-Meier estimator as it was done by Koziol and Green. A simulation study is conducted under a particular censorship model proposed by Koziol and Green. Through a simulation study, the generalized Anderson-Darling statistic shows the best power against almost all alternatives considered among the three EDF statistics we take into account.

EDF goodness-of-fit tests based on centre-outward ordering

ABSTRACTThe Kolmogorov-Smirnov (KS) test, the Cramér-von Mises (CvM) test, and the Anderson-Darling (AD) test are widely used goodness-of-fit tests based on the empirical distribution function (EDF). When defining the EDF in those tests, the data are ordered from the smallest to the largest. This left-to-right ordering partly explains why the three EDF tests are only sensitive to location differences but not to scale differences. Motivated by this observation, we propose three EDF tests based on the centre-outward ordering and they are shown to be more powerful than the KS, CvM and AD tests for detecting scale differences. To detect any arbitrary distributional differences, we further propose to combine our newly developed centre-outward ordering based EDF tests with the original EDF tests. The combined tests are easy to implement and are shown to have good performance detecting a variety of distributional differences. The extension of the results to the two-sample comparison problem is also discussed.

Most Frequent Value Statistics and the Hubble Constant

The measurement of Hubble constant (H0) is clearly a very important task in astrophysics and cosmology. Based on the principle of minimization of the information loss, we propose a robust most frequent value (MFV) procedure to determine H0, regardless of the Gaussian or non-Gaussian distributions. The updated data set of H0 contains the 591 measurements including the extensive compilations of Huchra and other researchers. The calculated result of the MFV is H0 = 67.498 km s−1 Mpc−1, which is very close to the average value of recent Planck H0 value (67.81 ± 0.92 km s−1 Mpc−1 and 66.93 ± 0.62 km s−1 Mpc−1) and Dark Energy Survey Year 1 Results. Furthermore, we apply the bootstrap method to estimate the uncertainty of the MFV of H0 under different conditions, and find that the 95% confidence interval for the MFV of H0 measurements is [66.319, 68.690] associated with statistical bootstrap errors, while a systematically larger estimate is (systematic uncertainty). Especially, the non-Normality of error distribution is again verified via the empirical distribution function test including Shapiro–Wilk test and Anderson–Darling test. These results illustrate that the MFV algorithm has many advantages in the analysis of such statistical problems, no matter what the distributions of the original measurements are.

Goodness-of-Fit Tests for Generalized Normal Distribution for Use in Hydrological Frequency Analysis

The use of three-parameter generalized normal (GNO) as a hydrological frequency distribution is well recognized, but its application is limited due to unavailability of popular goodness-of-fit (GOF) test statistics. This study develops popular empirical distribution function (EDF)-based test statistics to investigate the goodness-of-fit of the GNO distribution. The focus is on the case most relevant to the hydrologist, namely, that in which the parameter values are unidentified and estimated from a sample using the method of L-moments. The widely used EDF tests such as Kolmogorov–Smirnov, Cramer von Mises, and Anderson–Darling (AD) are considered in this study. A modified version of AD, namely, the Modified Anderson–Darling (MAD) test, is also considered and its performance is assessed against other EDF tests using a power study that incorporates six specific Wakeby distributions (WA-1, WA-2, WA-3, WA-4, WA-5, and WA-6) as the alternative distributions. The critical values of the proposed test statistics are approximated using Monte Carlo techniques and are summarized in chart and regression equation form to show the dependence of shape parameter and sample size. The performance results obtained from the power study suggest that the AD and a variant of the MAD (MAD-L) are the most powerful tests. Finally, the study performs case studies involving annual maximum flow data of selected gauged sites from Irish and US catchments to show the application of the derived critical values and recommends further assessments to be carried out on flow data sets of rivers with various hydrological regimes.

Random Weighting Empirical Distribution Function and its Applications to Goodness-of-Fit Testing

In this article, a randomized estimator of the empirical distribution function (EDF) called random weighting empirical distribution function (RWEDF) is introduced, one special case of which is just equivalent to the Bayesian bootstrap. The consistency of the RWEDF is established under certain conditions. By substituting this new EDF for the classical EDF, we obtain new versions of some EDF test statistics for goodness-of-fit. The simulation results show that the new tests are more powerful than the corresponding tests based on the classical EDF under some cases.

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.

Comparison of Structural Change Tests in Linear Regression Models

The actual power performance of historical structural change tests are compared under various alternatives. The tests of interest are F, CUSUM, MOSUM, Moving Estimates and empirical distribution function tests with both recursive and ordinary least-squares residuals. Our comparison of the structural tests involves limiting distributions under the hypothesis, the ability to detect the alternative hypotheses under one or double structural change, and smooth change in parameters. Even though no version is uniformly superior to the other, the knowledge about the properties of those tests and connections between these tests can be used in practical structural change tests and in further research on other change tests.

Identifying All Distinct Sample P-P Plots, with an Application to the Exact Finite Sample Distribution of the L<sub>1</sub>-FCvM Test Statistic

P-p plots contain all the information that is needed for scale-invariant comparisons. Indeed, Empirical Distribution Function (EDF) tests translate sample p-p plots into a single number. In this paper we characterize the set of all distinct p-p plots for two balanced sample of size n absent ties. Distributions of EDF test statistics are embedded in this set. It is thus used to derive the exact finite sample distribution of the L1-version of the Fisz-Cramer-von Mises test. Comparing this distribution with the (known) limiting distribution shows that the latter can always be used for hypothesis testing: although for finite samples the critical percentiles of the limiting distribution differ from the exact values, this will not lead to differences in the rejection of the underlying hypothesis.

Evaluation of the influence of black carbon on the distribution of PAHs in sediments from along the entire Swedish continental shelf

A growing literature proposes that combusted-derived black carbon (BC) dominates the sorption and, by inference, the environmental distribution and bioavailability of many hydrophobic pollutant classes. There is a paucity of studies simultaneously evaluating the distribution of both BC geosorbents and pollutant sorbates in the actual field. Here, 120 surface sediments collected by the Geological Survey of Sweden along the 2000 km continental shelf along the Swedish coast facilitated evaluation of the relative influences of BC and non-BC organic carbon (OC) on the spatial distribution of polycyclic aromatic hydrocarbons (PAHs). The sum of 15 out of the 16 EPA PAHs ranged from 0.12 to 9.6 μg/g dry weight (dw), with the highest levels being found in the southern half of the Swedish Shelf (SS) area and in the vicinity of larger cities (Stockholm, Göteborg, Malmö and Umeå). Source-diagnostic PAH ratios such as ANT/(PHE + ANT), FLT/(FLT + PYR), BaA/(BaA + BPE), IPY/(IPY + BPE), CombPAH/ΣPAH and LMW/HMW suggested that pyrogenic sources are dominating the load of PAHs in Swedish Baltic and North Sea sediments. The sediment TOC was 4.8–168 mg/gdw (median 43 mg/gdw), while a BC concentration of 0.6–18 mg/gdw (median 1.8 mg/gdw) yielded BC:TOC ratios spanning a wide range of 1.7–47% (median 4.6%). Empirical distribution function tests indicated that the use of linear regression statistics was inappropriate. Instead, evaluation with the non-parametric Spearman function yielded higher correlation coefficient ( r S) for total PAHs versus BC (0.54, p < 0.01) than versus either TOC (0.28, p < 0.01) or OC (TOC-BC; 0.26, p < 0.01). The results from this field study, encompassing an order of magnitude more observations than any previous sediment study, constitute a broad field manifestation of the importance of BC in affecting the distribution of planar aromatic pollutants in aquatic environments.

A K-Sample Homogeneity Test Based on the Quantification of the p-p Plot: The Harmonic Weighted Mass Index

We propose a quantification of the p-p plot that assigns equal weight to all distances between the respective distributions: the surface between the p-p plot and the diagonal. This surface is labelled the Harmonic Weighted Mass (HWM) index. We introduce the diagonal-deviation (d-d) plot that allows the index to be computed exactly under all circumstances. For two balanced samples absent ties the finite sample distribution of the HWM index is derived. Simulations show that in most cases unbalanced samples and ties have little effect on this distribution. The d-d plot allows for a straightforward extension to the K-sample HWM index. As we have not been able to derive the distribution of the index for K>2, we simulate significance tables for K=3,...,15. An example involving economic growth rates of the G7 countries illustrates that the HWM test can have better power than alternative Empirical Distribution Function tests.

Assessing the Goodness-of-Fit of Statistical Distributions When Data Are Grouped

Modeling statistical distributions of phenomena can be compromised by the choice of goodness-of-fit statistics. The Pearson chi-square test is the most commonly used test in the geosciences, but the lesser known empirical distribution function (EDF) statistics should be preferred in many test situations. Using a data set from geomorphology, the Anderson–Darling test for grouped exponential distributions is employed to illustrate ease of use and statistical advantages of this EDF test. Attention to the issues discussed will result in more informed statistic selection and increased rigor in the identification of distribution functions that describe random variables.

Moment-Based Goodness-of-Fit Tests for the Inverse Gaussian Distribution

Data in statistical practice often consist of nonnegative measurements that exhibit positive skewness. The inverse Gaussian (IG) family of distributions provides a versatile and flexible model for analyzing such data. The most appealing feature of an IG model relates to the many similarities that it shares with the ubiquitous Gaussian (G) model; for example, the optimal procedure for testing homogeneity of several means involves an F-distribution analogous to the F-statistic associated with ANOVA. Mudholkar and Natarajan, in their investigation of G–IG analogies, presented two functions (δ1, δ2) of the moments, termed the coefficients of IG skewness and IG kurtosis. They also discussed the parallelism of these functions with the classical coefficients of skewness and kurtosis, which have extensive goodness-of-fit applications. In this article we consider the sample versions of these functions for constructing goodnessof-fit tests for the IG models. The test statistics are proposed, and their asymptotic normal and chi-squared distributions are derived. Monte Carlo simulations are used to refine the asymptotic distributions and to study the operating characteristics of the tests. We demonstrate that the approach proposed in the article offers an advantage over the empirical distribution function tests, enabling one to address 10 distinct goodness-of-fit problems involving IG models against a variety of restricted IG-skewness and IG-kurtosis alternatives.

Goodness-of-Fit Test for Exponentiality Based on Kullback–Leibler Information

In this paper, we discuss goodness-of-fit tests of the exponential distribution based on Kullback–Leibler information. To construct test statistics, Van Es' and Correa's entropy estimator are used as an estimator of Shannon's entropy. Critical values for various sample sizes determined by means of Monte Carlo simulations are presented for each of test statistics. Also, formulas for calculating approximate critical values for given sample sizes are provided for the respective test statistics. A Monte Carlo power analysis is performed for various alternatives and sample sizes in order to compare the proposed tests with several existing goodness-of-fit tests based on the empirical distribution function (EDF). Simulation results show that Kullback–Leibler information tests have higher powers than the EDF tests.

TESTS FOR THE GOODNESS-OF-FIT OF THE LAPLACE DISTRIBUTION

ABSTRACT Tests based on EDF (the empirical distribution function) are given for the goodness-of-fit of the two-parameter Laplace distribution when one parameter or both parameters are unknown. Coefficients of polynomial functions for computing critical values of these tests are tabled. Power studies are reported to compare among these tests. The Anderson-Darling statistic A 2 gives the overall most powerful EDF tests followed by the Cramer-Von Mises statistic W 2.

Goodness-of-fit tests for generalized logarithmic series distribution

Goodness-of-fit test statistics based on the empirical distribution function (EDF) are considered for the generalized logarithmic series distribution. The α levels of the tests for small or moderate sample sizes are close to the chosen nominal 5% and 10% significance levels. For small or moderate sample sizes, the tests are compared with respect to their simulated power of detecting some alternative hypotheses against a null hypothesis of generalized logarithmic series distribution. The discrete version of the Cramer–von Mises and Anderson–Darling tests are found to be the most powerful pair among the EDF tests.

EDF Tests for the Generalized Poisson Distribution

Empirical distribution function (EDF) goodness-of-fit tests are considered for the generalized Poisson distribution. The tests are compared with respect to their simulated small sample power of detecting some alternative distributions. The small sample levels of the tests are close to the nominal significance levels. The discrete version of the Anderson-Darling test is found to be the most powerful among the EDF tests. A numerical example is used to illustrate the application of the goodness-of-fit tests.

Bootstrap based tests for generalized negative binomial distribution

Goodness of fit test statistics based on the empirical distribution function (EDF) are considered for the generalized negative binomial distribution. The small sample levels of the tests are found to be very close to the nominal significance levels. For small sample sizes, the tests are compared with respect to their simulated power of detecting some alternative hypotheses against a null hypothesis of generalized negative binomial distribution. The discrete Anderson—Darling test is the most powerful among the EDF tests. Two numerical examples are used to illustrate the application of the goodness of fit tests.

GOODNESS OF FIT FOR THE BINOMIAL DISTRIBUTION

Goodness of fit testing for the binomial distribution can be carried out using Pearson's X2p statistic and its components. Applications of this technique are considered and compared with recently suggested empirical distribution function tests. Diagnostic use of components is discussed.