Cramer-von Mises Test Research Articles

Tests of fit for the power function lognormal distribution.

In this study, tests of fit for the power function lognormal distribution is considered. The probability plot, probability plot correlation coefficient, and goodness-of-fit tests-the Kolmogorov-Smirnov (KS), Cramér-von Mises (CvM), and Anderson-Darling (AD) tests are provided. Tables of critical values are presented by using simulation techniques, and the AD test outperforms KS and CvM tests based on power comparisons. Finally, to illustrate these test procedures, we fit this distribution to the data which represent the survival times of 121 breast cancer patients from one hospital.

Enhanced symplectic geometry mode decomposition and its application to rotating machinery fault diagnosis under variable speed conditions

Tacholess order tracking (TLOT) has been one of the most powerful and applicable rotating machinery fault diagnosis methods. However, it is still a challenge to accurately estimate the instantaneous phase of the reference shaft under variable speed conditions. Although current available signal decomposition methods can deal with non-stationary vibration signal, they are probably interfered by some tough problems, such as prior knowledge requirement or heavy computational burden. In this paper, the aforementioned problems are addressed and an adaptive harmonic components extraction framework is proposed. Firstly, the vibration signal is low-pass filtered and down-sampled for computation speed acceleration. Harmonic components are subsequently extracted by enhanced symplectic geometry mode decomposition (ESGMD), which is based on Cramer-von Mises criterion. Secondly, surrogate data test is applied for pseudo components identification, thereby interferences induced by background noise can be filtered out adaptively. Finally, Hilbert transform is applied to obtain the instantaneous phase of the decomposed fundamental harmonic component, and thereby TLOT is conducted to realize fault diagnosis. The effectiveness of the proposed method has been validated by both simulated and experimental tests. The results exhibit that the proposed method outperforms conventional approaches in harmonic components extraction for rotating machinery fault diagnosis under variable speed conditions.

Techniques of quality of adjustment of statistical models with evaluation of probability distributions using production data of laying quails

The goal of our study was to evaluate the quality of fit from different types of probability distributions for continuous data. For this, performance traits and quality of quail egg in the production of nutraceutical eggs were used as a continuous data source. The data were collected over 42 days, the experimental design was completely randomized with 7 treatments, 6 repetitions, with 252 animals allocated in 36 cages. The distributions for continuous data used were the exponential, gamma, gaussian, and lognormal. The R Open Source and SAS® University Edition software was used to perform the analysis. The graphical analysis of the traits was performed from the predicted versus observed values, Cumulative Distribution Function (CDF), and skewness-kurtosis. The fits were also evaluated by the Akaike information criterion (AIC), Bayesian information criterion (BIC), Conditional model of adjusted R-Square (), Conditional model of adjusted concordance correlation (), Kolmogorov-Smirnov test (KS), Cramer-von Mises test (CvM), Anderson-Darling test (AD), Watanabe-Akaike Information Criterion (WAIC) and Leave-one-out cross-validation (LOO). All the tests indicated the Gaussian distribution as the most suitable and they excluded the exponential distribution for all the evaluated characteristics.

A simulation study of a class of nonparametric test statistics: a close look of empirical distribution function-based tests

Kolmogorov–Smirnov (KS) statistic is a non-parametric statistic based on the empirical distribution function. For the one-sample case, it uses the supremum distance between an empirical distribution function (EDF) and a pre-specified cumulative distribution function (CDF). For two-sample case, it measures the maximum of the distance between two EDFs. KS test, as well as other EDF-based tests such as the Anderson-Darling (AD) test and Cramer-von Mises (CvM) test, has been widely used in statistical analysis. To address and compare the performance of these test statistics, we have conducted a simulation study comparing the type I error and power of the KS test, the CvM test, the AD test, and the Chi-squared test. Our study includes both one sample and two sample tests and for both independent and correlated samples. Our study showed that if we do not have prior information about the tested distributions, EDF-based tests are better. However, so long as we have prior information about the tested distribution and the density of two distributions is bell-shaped and we are expecting differences in variance/sparseness, then the Chi-squared test may be more preferable. When correlation exists between tested samples, adjustment on the informative sample size is important and required.

On the Choice of Test Statistic for Conditional Moment Inequalities

This paper derives asymptotic power functions for Cramer-von Mises (CvM) style tests for conditional moment inequality models in the set identified case. Combined with power results for Kolmogorov-Smirnov (KS) tests, these results can be used to choose the optimal test statistic, weighting function and, for tests based on kernel estimates, kernel bandwidth. The results show that KS tests are preferred to CvM tests, and that a truncated variance weighting is preferred to bounded weightings under a minimax criterion, and for a class of alternatives that arises naturally in these models. The results also provide insight into how moment selection and the choice of instruments affect power. Such considerations have a large effect on power for instrument based approaches when a CvM statistic or an unweighted KS statistic is used and relatively little effect on power with optimally weighted KS tests.

Properties and approximate p-value calculation of the Cramer test

Two-sample tests are probably the most commonly used tests in statistics. These tests generally address one aspect of the samples' distribution, such as mean or variance. When the null hypothesis is that two distributions are equal, the Anderson–Darling (AD) test, which is developed from the Cramer–von Mises (CvM) test, is generally employed. Unfortunately, we find that the AD test often fails to identify true differences when the differences are complex: they are not only in terms of mean, variance and/or skewness but also in terms of multi-modality. In such cases, we find that Cramer test, a modification of the CvM test, performs well. However, the adaptation of the Cramer test in routine analysis is hindered by the fact that the mean, variance and skewness of the test statistic are not available, which resulted in the problem of calculating the associated p-value. For this purpose, we propose a new method for obtaining a p-value by approximating the distribution of the test statistic by a generalized Pareto distribution. By approximating the distribution in this way, the calculation of the p-value is much faster than e.g. bootstrap method, especially for large n. We have observed that this approximation enables the Cramer test to have proper control of type-I error. A simulation study indicates that the Cramer test is as powerful as other tests in simple cases and more powerful in more complicated cases.

Application of Generalized Pareto in Non-Life Insurance

This paper focuses on the modeling and estimation of tail loss distribution parameters from Egyptian’s commercial fire loss severities. Using theoretical extreme value, we use the generalized distribution of Pareto (GPD) and compare it to standard parametric modeling based on exp, Weibull, gumbel, frechet, lognormal and gamma distributions. The goodness-of-fit tests included Kolmogorov-Smirnov, Anderson and Cramer-von Mises test is carried out, and the calculation of the value-at-risk and expected shortfall are performed. We use the bootstrap approach to create confidence intervals for the estimates.

Goodness-of-fit tests for the Compound Rayleigh distribution with application to real data

An important problem in statistics is to obtain information about the form of the population from which the sample is drawn. Goodness of fit (GOF) tests is employed to determine how well the observed sample data “fits” some proposed model. The well known standard goodness of fit tests; Kolomogorov-Smirnov (KS), Cramer von Mises (CVM) and Anderson-(AD) tests are used for continuous distributions. When the parameters are unknown, the standard tables for these tests are not valid. The complete sample procedures of goodness of fit tests are inappropriate for use with censored samples. The critical values obtained from published tables of the complete sample test statistic are necessarily conservative.In this paper, we obtain the tables of critical values of modified Kolmogorov-Smirnov (KS) test, Cramer-Von Mises (CVM) test and Anderson-Darling (AD) test for the Compound Rayleigh (CR) distribution with unknown parameters in the case of complete and type II censored samples. Furthermore, we present power comparison between KS test, CVM test and AD test for a number of alternative distributions. Applications of the considered distribution to real medical data sets given by Stablein et al. (1981) are presented.

Modulation Classification Using a Goodness of Fit Test

In this paper, we propose an innovative approach using Anderson-Darling (A-D) test for signal modulation classification with less signal samples in AWGN channels. The A-D test is a non-parametric approach to measure the goodness of fit test. It is based on Cramer-Von Mises (CVM) test, using the mean square integral of the difference between the empirical cumulative distribution functions (ECDFs) from received signals and cumulative distribution functions (CDFs) of the signal under different candidate modulation format. In order to avoid the default of CVM, such as it is less sensitive at heads and tails of the distributions, more weights are given to heads and tails in A-D test. Massive simulation results show that compared with the Kolmogorov-Smirnov (K-S) classifiers and the traditional high-order cumulant-based classifiers, the A-D classifiers show better classification performance at different SNRs with less signal samples for M-QAM and M-PSK modulations in AWGN channels.

Failure intensity of offshore power plants under varying maintenance policies

Abstract System reliability of an offshore power plant with several gas turbine engines is analyzed in this study to understand the failure intensity of a selected gas turbine engines under varying maintenance activities. A set of event data of a selected gas turbine engine is considered to identify system failure intensity levels, where unknown maintenance actions were implemented (i.e. various repairs disturb the failure rate). A non-homogeneous Poisson process (NHPP) is used to model the age dependent failure intensity of the same gas turbine engine and the maximum likelihood estimation (MLE) approach for calculating the respective model parameters is proposed. Several failure intensity rates (i.e. varying failure trends) in these models (i.e. during the system age of the gas turbine engine) are observed. Furthermore, these varying failure trend models are evaluated with actual failure events of the same gas turbine engine by considering two goodness-of-fit tests: Cramer-von Mises and Chi-square tests. Finally, system reliability of the gas turbine engine under the failure transition, failure intensity, mission reliability and mean time between failures (MTBF) is also discussed in this study.

Stochastic demand side management in smart grid system

Two-way communication of smart grid allows customer to see their energy consumption in demand side, and makes the energy supplier informed to manage economic dispatch based on load demand variability. Statistical analysis of demand-side management gives us the fitness of hourly load demand with proposed probability density functions. We proposed Lognormal, Gamma and Weibull probability density function for estimating the load demand variability at Banaras Hindu University (BHU) campus. We have collected the hourly load demand data of BHU campus for a year from the Electricity & Water Supply Service Center, BHU. The fitness of data with theoretical probability density function has been analysed with three goodness-of-fit tests named Kolmogorov-Smirnov, Anderson-Darling, and Cramer-von Mises tests. Modern R programming language software has been used for the computational work and graphical analysis. With this, we can manage optimal power flow between demand and supply to increase the system reliability.

Stochastic demand side management in smart grid system

Two-way communication of smart grid allows customer to see their energy consumption in demand side, and makes the energy supplier informed to manage economic dispatch based on load demand variability. Statistical analysis of demand-side management gives us the fitness of hourly load demand with proposed probability density functions. We proposed Lognormal, Gamma and Weibull probability density function for estimating the load demand variability at Banaras Hindu University (BHU) campus. We have collected the hourly load demand data of BHU campus for a year from the Electricity & Water Supply Service Center, BHU. The fitness of data with theoretical probability density function has been analysed with three goodness-of-fit tests named Kolmogorov-Smirnov, Anderson-Darling, and Cramer-von Mises tests. Modern R programming language software has been used for the computational work and graphical analysis. With this, we can manage optimal power flow between demand and supply to increase the system reliability.

Testing nonstationary and absolutely regular nonlinear time series models

We study some general methods for testing the goodness-of-fit of a general nonstationary and absolutely regular nonlinear time series model. These testing methods are based on some marked empirical processes that we show to converge in distribution to a zero-mean Gaussian process with respect to the Skorohod topology. We investigate the behavior of this process under fixed alternatives and under a sequence of local alternatives. Our results are applied to testing a general class of nonlinear semiparametric time series models. A simulation experiment shows that the Cramer–von Mises test studied behaves well on the examples considered.

Methods of Estimating the Parameters of the Quasi Lindley Distribution

In this paper, we review the quasi Lindley distribution and established its quantile function. A simulation study is conducted to examine the bias and mean square error of the parameter estimates of the distribution through the method of moment estimation and the maximum likelihood estimation. Result obtained shows that the method of maximum likelihood is a better choice of estimation method for the parameters of the quasi Lindley distribution. Finally, an applicability of the quasi Lindley disttribution to a waiting time data set suggests that the distribution demonstrates superiority over the power Lindley distribution, Sushila distribution and the classical oneparameter Lindley distribution in terms of the maximized loglikelihood, the Akaike information criterion, the Kolmogorov-Smirnov and Cramer von Mises test statistic.

Goodness of t tests for logistic distribution based on Phi-divergence

Some goodness of fit tests for logistic distribution based on Phi-divergenceare developed. The powers of the introduced tests are compared with sometraditional goodness of t tests including Kolmogorov-Smirnov, Anderson-Darling and Cramer-von Mises tests for logistic distribution using MonteCarlo simulation. It is shown the proposed tests have good performance ascompared with their competitors in the literature. A real data set is used forillustration.

On the choice of test statistic for conditional moment inequalities

This paper derives asymptotic approximations to the power of Cramer–von Mises (CvM) style tests for inference on a finite dimensional parameter defined by conditional moment inequalities in the case where the parameter is set identified. Combined with power results for Kolmogorov–Smirnov (KS) tests, these results can be used to choose the optimal test statistic, weighting function and, for tests based on kernel estimates, kernel bandwidth. The results show that, in the setting considered here, KS tests are preferred to CvM tests, and that a truncated variance weighting is preferred to bounded weightings.

Power Comparison of Various Normality Tests

The assumption of normality is very important because it is used in many statistical procedures such as Analysis of variance, linear regression analysis, discriminant analysis and t-tests. The three common procedures are used for assessing the assumption of normality that is graphical methods, numerical methods and formal normality tests. In the literature, significant amount of normality tests are available. In this paper, only eight different tests of normality are discussed. The tests consider in the present study are Shapiro Wilk, Shapiro Francia, Kolmogrov Smirnov, Anderson Darling, Cramer von Mises, Jarque Bera, Geary and Lilliefors test. Power comparisons of each test are obtained by using Monte Carlo computation of sample data generated from different alternate distributions by using 5% level of significance. The results show that power of each test is affected by sample size and alternate distribution. Shapiro Francia and Kolmogrov Smirnov test perform well for Cauchy exponential distribution respectively. For t-distribution Geary, Shapiro Francia and Jarque Bera test perform well for degrees of freedom 5, 10 and 15 respectively.

A constructive hypothesis test for the single-index models with two groups

Comparison of two-sample heteroscedastic single-index models, where both the scale and location functions are modeled as single-index models, is studied in this paper. We propose a test for checking the equality of single-index parameters when dimensions of covariates of the two samples are equal. Further, we propose two test statistics based on Kolmogorov–Smirnov and Cramer–von Mises type functionals. These statistics evaluate the difference of the empirical residual processes to test the equality of mean functions of two single-index models. Asymptotic distributions of estimators and test statistics are derived. The Kolmogorov–Smirnov and Cramer–von Mises test statistics can detect local alternatives that converge to the null hypothesis at a parametric convergence rate. To calculate the critical values of Kolmogorov–Smirnov and Cramer–von Mises test statistics, a bootstrap procedure is proposed. Simulation studies and an empirical study demonstrate the performance of the proposed procedures.

Goodness of fit tests for Rayleigh distribution based on Phi-divergence

In this paper, we develop some goodness of fit tests for Rayleigh distribution based on Phi-divergence. Using Monte Carlo simulation, we compare the power of the proposed tests with some traditional goodness of fit tests including Kolmogorov-Smirnov, Anderson-Darling and Cramer von-Mises tests. The results indicate that the proposed tests perform well as compared with their competing tests in the literature. Finally, the proposed procedures are illustrated via a real data set.

Sample size determination of a nonparametric test based on weighted [formula omitted]-Wasserstein distance

In this paper we derive a formula for the sample size of a nonparametric test based on weighted L2-Wasserstein distance with a known alternative. As an example we apply the formula to determine the sample size for a hypothesis testing from the Tukey’s contaminated distribution model. The reliability of the sample size formula is evaluated in terms of type I errors and the powers of the test. The Monte Carlo simulation results show that the test based on the weighted L2-Wasserstein distance performs equally well as the Anderson–Darling test and is superior to the Kolmogorov–Smirnov and the Cramer–von Mises tests.