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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.

International audience; Many goodness-of-fit tests have been developed to assess the different assumptions of a (possibly heteroscedastic) regression model. Most of them are “directional” in that they detect departures from a given assumption of the model. Other tests are “global” (or “omnibus”) in that they assess whether a model fits a dataset on all its assumptions. We focus on the task of choosing the structural part of the regression function because it contains easily interpretable information about the studied relationship. We consider 2 nonparametric “directional” tests and one nonparametric “global” test, all based on generalizations of the Cramér-von Mises statistic. To perform these goodness-of-fit tests, we develop the R package cvmgof providing an easy-to-use tool for practitioners, available from the Comprehensive R Archive Network (CRAN). The use of the library is illustrated through a tutorial on real data. A simulation study is carried out in order to show how the package can be exploited to compare the 3 implemented tests.

The problem of parameters estimation plays a significant role in various areas of academic researches. In this article, we propose three different methods of estimation for the parameters of location-scale family under ranked set sampling in the view of missing data mechanism. Through a series of Monte Carlo simulations, it is well investigated that the proposed methods are relatively robust from violating the perfect ranking condition and provide better performance over their competitors using bias and MSE (mean square error) criteria. An empirical data set is also used for illustrative purposes.

Several commonly-used nonparametric change-point detection methods are analysed in terms of power, ability and accuracy of the estimated change-point location. The analysis is performed with synthetic data for different sample sizes, two types of change and different magnitudes of change. The methods studied are the Pettitt method, a method based on the Cramér von Mises (CvM) two-sample test statistic and a variant of the CUSUM method. The methods differ considerably in behaviour. For all methods the spread of estimated change-point location increases significantly for points near one of the ends of the sample. Series of annual maximum runoff for four stations on the Yangtze River in China are used to examine the performance of the methods on real data. It was found that the CvM-based test gave the best results, but all three methods suffer from bias and low detection rates for change points near the ends of the series.

This paper proposes some tests for lower tail dependence in extreme value models. Let (X,Y) be a random vector which follows in its lower tail a bivariate extreme value distribution with unit Frechet margins. We show that the conditional distribution function (df) of X+Y, given that X+Y

ENTHIS LINK GOES TO A ENGLISH SECTIONFRTHIS LINK GOES TO A FRENCH SECTION We propose nonparametric procedures for comparing the empirical distribution function of data from a complex survey with a hypothesized parametric reference distribution. The hypothesized distribution may be fully specified, or it may be a family with the parameters to be estimated from the data. Of the procedures studied, a modification of the Cramer–von Mises test proposed by Lockhart, Spinelli & Stephens [Lockhart, Spinelli and Stephens, The Canadian Journal of Statistics 2007; 35, 125–133] is supported theoretically and performs well in two simulation studies. The methods are applied to examine the distribution of body mass index in the U.S. National Health and Nutrition Examination Survey. The Canadian Journal of Statistics 47: 409–425; 2019 © 2019 Statistical Society of Canada

A new test of normality with unknown parameters is proposed in this article. We introduce a Cramer–von Mises type statistic with weight function equal to the inverse of the standard normal ...

In this article, the problem of estimating the model error distribution in a heteroscedastic varying coefficient regression model is considered. A residual-based estimator of the model error distri...

We propose a new framework for evaluating predictive densities in an environment where the estimation error of the parameters used to construct the densities is preserved asymptotically under the null hypothesis. The tests offer a simple way to evaluate the correct specification of predictive densities, where both the model specification and its estimation technique are evaluated jointly. Monte Carlo simulation results indicate that our tests are well sized and have good power in detecting misspecification. An empirical application to density forecasts of the Survey of Professional Forecasters shows the usefulness of our methodology.

Goodness of fit tests with two different methods are carried out in the present paper in order to recognize normal and exponential distributions within Type-I censored data. Here, we use the Lin-Wo...

Testing uniformity of a sample supported on the hypersphere is one of the first steps when analysing multivariate data for which only the directions (and not the magnitudes) are of interest. In this work, a projection-based Cramer–von Mises test of uniformity on the hypersphere is introduced. This test can be regarded as an extension of the well-known Watson test of circular uniformity to the hypersphere. The null asymptotic distribution of the test statistic is obtained and, via numerical experiments, shown to be tractable and practical. A novel study on the uniformity of the distribution of craters on Venus illustrates the usage of the test.

Goodness-of-fit tests of the Cramer–von Mises type are considered for testing hypotheses about the belonging of the observed random variable distribution to a given parametric family of distributions. The limiting distribution of the considered criteria statistic does not depend on the sample distribution in cases where such a distribution is specified exactly. However, for parametric families, the distribution of classical statistics can depend on the family under the hypothesis and the specific distribution function within this family. The limiting distribution of statistics using the Khmaladze transformation to an empirical process becomes independent of both the hypothetical family of distributions and the values of the unknown parameters of the sample distribution. The application of this criterion, however, remains very difficult and requires special approaches to each family of distributions. The paper recalls that for all the most commonly used families, the distribution of classical statistics does not depend on distribution parameters. For the remaining families, a new simple hypothesis testing procedure is proposed. This procedure is applied in work for a family of gamma distributions. The question of the possibility of using it for hyper-normal distribution families is also discussed. The obtained results are partially applicable also to the Kolmogorov–Smirnov test.

In this paper, we first study a new two parameter lifetime distribution. This distribution includes “monotone” and “non-monotone” hazard rate functions which are useful in lifetime data analysis and reliability. Some of its mathematical properties including explicit expressions for the ordinary and incomplete moments, generating function, Renyi entropy, δ-entropy, order statistics and probability weighted moments are derived. Non-Bayesian estimation methods such as the maximum likelihood, Cramer-Von-Mises, percentile estimation, and L-moments are used for estimating the model parameters. The importance and flexibility of the new distribution are illustrated by means of two applications to real data sets. Using the approach of the Bagdonavicius–Nikulin goodness-of-fit test for the right censored validation, we then propose and apply a modified chi-square goodness-of-fit test for the Burr X Weibull model. The modified goodness-of-fit statistics test is applied for the right censored real data set. Based on the censored maximum likelihood estimators on initial data, the modified goodness-of-fit test recovers the loss in information while the grouped data follows the chi-square distribution. The elements of the modified criteria tests are derived. A real data application is for validation under the uncensored scheme.

We consider the survival function for univariate right-censored event time data, when a cure fraction is present. This means that the population consists of two parts: the cured or non-susceptible group, who will never experience the event of interest versus the non-cured or susceptible group, who will undergo the event of interest when followed up sufficiently long. When modeling the data, a parametric form is often imposed on the survival function of the susceptible group. In this paper, we construct a simple novel test to verify the aptness of the assumed parametric form. To this end, we contrast the parametric fit with the nonparametric fit based on a rescaled Kaplan–Meier estimator. The asymptotic distribution of the two estimators and of the test statistic are established. The latter depends on unknown parameters, hence a bootstrap procedure is applied to approximate the critical values of the test. An extensive simulation study reveals the good finite sample performance of the developed test. To illustrate the practical use, the test is also applied on two real-life data sets.

We propose kernel-type smoothed Kolmogorov-Smirnov and Cram\'{e}r-von Mises tests for data on general interval, using bijective transformations. Though not as severe as in the kernel density estimation, utilizing naive kernel method directly to those particular tests will result in boundary problem as well. This happens mostly because the value of the naive kernel distribution function estimator is still larger than $0$ (or less than $1$) when it is evaluated at the boundary points. This situation can increase the errors of the tests especially the second-type error. In this article, we use bijective transformations to eliminate the boundary problem. Some simulation results illustrating the estimator and the tests' performances will be presented in the last part of this article.

The main purpose of this paper is to test the conditional independence of two random vectors Y and Z given another random vector X. A new test statistic is developed relying on the Cramer–von Mises...

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.

In this paper a test for model selection is proposed which extends the usual goodness-of-fit test in several ways. It is assumed that the underlying distribution H depends on a covariate value in a fixed design setting. Secondly, instead of one parametric class we consider two competing classes one of which may contain the underlying distribution. The test allows to select one of two equally treated model classes which fits the underlying distribution better. To define the distance of distributions various measures are available. Here the Cramer-von Mises has been chosen. The null hypothesis that both parametric classes have the same distance to the underlying distribution H can be checked by means of a test statistic, the asymptotic properties of which are shown under a set of suitable conditions. The performance of the test is demonstrated by Monte Carlo simulations. Finally, the procedure is applied to a data set from an endurance test on electric motors.

In this article, a new reciprocal Rayleigh extension called the Xgamma reciprocal Rayleigh model is defined and studied. The relevant statistical properties are derived, and the useful results related to the convexity and concavity are addressed. We discussed the estimation of the parameters using different estimation methods such as the maximum likelihood estimation method, the ordinary least squares estimation method, the weighted least squares estimation method, the Cramer-Von-Mises estimation method, and the bootstrapping method. A simulation study was conducted to assess the performances of the proposed estimation methods are investigated through a simulation study. Many bivariate and multivariate type model have also been derived based on Farlie-Gumbel-Morgenstern copula, the Clayton copula, Renyi’s entropy copula and the Ali-Mikhail-Haq copula. A modified Nikulin-Rao-Robson test for right-censored validation is applied to a censored real data set.

This paper proposes a new distribution-free control chart by integrating the powerful nonparametric two-sample Cramer-von Mises test and the exponentially weighted moving average control scheme to on-line monitoring. The proposed control chart can be used to monitor the location and the scale parameters of a univariate continuous distribution, simultaneously. The control limits based on Monte-Carlo simulation are provided in a table. The sensitivity analysis of effect of the number of reference samples on the control chart is studied in detail. Comparison results based on Monte-Carlo simulation show that the proposed chart is quite robust to non-normally distributed data, and moreover, it shows satisfactory performance in detecting various process shifts in terms of the average run length and standard deviation of run length. The application of our proposed chart is illustrated by a real data example for automobile engine piston rings.

We give a probabilistic interpretation of the Cramer–von Mises distance between continuous distribution functions F and . If F is unknown, we construct an asymptotic confidence interval for based on a random sample from F. Moreover, for given and some value , we propose an asymptotic equivalence test of the hypothesis that against the alternative . If such a ‘neighbourhood-of- validation test’, carried out at a small asymptotic level, rejects the hypothesis, there is evidence that F is within a distance of . As a neighbourhood-of-exponentiality test shows, the method may be extended to the case that is composite.

In this paper, we construct confidence sets for models defined by many conditional moment inequalities/equalities. The conditional moment restrictions in the models can be finite, countably infinite, or uncountably infinite. To deal with the complication brought about by the vast number of moment restrictions, we exploit the manageability (Pollard (1990)) of the class of moment functions. We verify the manageability condition in five examples from the recent partial identification literature. The proposed confidence sets are shown to have correct asymptotic size in a uniform sense and to exclude parameter values outside the identified set with probability approaching one. Monte Carlo experiments for a conditional stochastic dominance example and a random-coefficients binary-outcome example support the theoretical results.

Minimum Cramer-Von Mises distance estimation is extended to a simulated version. The simulated version consists of replacing the model distribution function with a sample distribution constructed using a simulated sample drawn from it. The method does not require an explicit form of the model density functions and can be applied to fitting many useful infinitely divisible distributions or mixture distributions without closed form density functions often encountered in actuarial science and finance. For these models likelihood estimation is difficult to implement and simulated Minimum Cramer-Von Mises (SMCVM) distance estimation can be used. Asymptotic properties of the SCVM estimators are established. The new method appears to be more robust and efficient than methods of moments (MM) for the models being considered which have more than two parameters. The method can be used as an alternative to simulated Hellinger distance (SMHD) estimation with a special feature: it can handle models with a discontinuity point at the origin with probability mass assigned to it such as in the case of the compound Poisson distribution where SMHD method might not be suitable. As the method is based on sample distributions instead of density estimates it is also easier to implement than SMHD method but it might not be as efficient as SMHD methods for continuous models.

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.

A sensitivity analysis represents crucial part of uncertainty quantification. The paper is focused on global sensitivity analysis, specifically moment-independent importance measure based on Cramer-von Mises distance. This type of sensitivity analysis takes whole probability distribution of random variables into account in contrast to commonly used Sobol’ indices. It leads to more precise sensitivity analysis. Nevertheless, such a method is highly computationally demanding. Therefore, novel idea of utilization the polynomial chaos expansion for the estimation of conditional distributions is presented herein. The paper represents a pilot study of performance of such method using simple example.

We study a rank based univariate two-sample distribution-free test. The test statistic is the difference between the average of between-group rank distances and the average of within-group rank distances. This test statistic is closely related to the two-sample Cramer–von Mises criterion. They are different empirical versions of a same quantity for testing the equality of two population distributions. Although they may be different for finite samples, they share the same expected value, variance and asymptotic properties. The advantage of the new rank based test over the classical one is its ease to generalize to the multivariate case. Rather than using the empirical process approach, we provide a different easier proof, bringing in a different perspective and insight. In particular, we apply the Hajek projection and orthogonal decomposition technique in deriving the asymptotics of the proposed rank based statistic. A numerical study compares power performance of the rank formulation test with other commonly-used nonparametric tests and recommendations on those tests are provided. Lastly, we propose a multivariate extension of the test based on the spatial rank.

This article presents different parameter estimation methods for flexible Weibull distribution introduced by Bebbington et al. (Reliability Engineering and System Safety 92:719-726, 2007), which is a modified version of the Weibull distribution and is suitable to model different shapes of the hazard rate. We consider both frequentist and Bayesian estimation methods and present a comprehensive comparison of them. For frequentist estimation, we consider the maximum likelihood estimators, least squares estimators, weighted least squares estimators, percentile estimators, the maximum product spacing estimators, the minimum spacing absolute distance estimators, the minimum spacing absolute log-distance estimators, Cramer von Mises estimators, Anderson Darling estimators, and right tailed Anderson Darling estimators, and compare them using a comprehensive simulation study. We also consider Bayesian estimation by assuming gamma priors for both shape and scale parameters. We use a Markov Chain Monte Carlo algorithm to compute the posterior summaries. A real data example is also a part of this work.

We propose the goodness of fit test for inhomogeneous Poisson processes with unknown scale and shift parameters. A test statistic of Cramer-von Mises type is proposed and its asymptotic behavior is studied. We show that under null hypothesis the limit distribution of this statistic does not depend on unknown parameters., Comment: 15 pages

In this paper, we propose a new class of distributions by compounding Lindley distributed random variates with the number of variates being zero-truncated Poisson distribution. This model is called a compound zero-truncated Poisson–Lindley distribution with two parameters. Different statistical properties of the proposed model are discussed. We describe different methods of estimation for the unknown parameters involved in the model. These methods include maximum likelihood, least squares, weighted least squares, Cramer–von Mises, maximum product of spacings, Anderson–Darling and right-tail Anderson–Darling methods. Numerical simulation experiments are conducted to assess the performance of the so obtained estimators developed from these methods. Finally, the potentiality of the model is studied using one real data set representing the monthly highest snowfall during February 2018, for a subset of stations in the Global Historical Climatological Network of USA.

The most popular isotropic yield conditions, verified for many ductile metals, were proposed by Tresca in 1864 (maximum shearing stresses theory) and von Mises in 1913. The von Mises yield theory (von Mises in Mathematisch-physikalische Klasse 582–592, 1913), also known as maximum distortion energy criterion, finds considerable experimental support, especially for very ductile materials and plane stress (Banabic et al. in Int. J. Mater. Form. 3:165–189, 2010). For this reason, and for its simplicity, it is common in design. During 100 years, this theory has been developed and improved systematically by Hosford, Christensen, Tsai-Hill, etc. The modified von Mises hypothesis combines the theories of maximum strain energy and maximum distortion energy, and it involves the Poisson ratio. It ensures a smooth transition from the von Mises to the Beltrami criterion. The results obtained by this new yield hypothesis are compared with those obtained both by the classic von Mises criterion and by experiments on different metallic materials. A quite good concordance is observed between these results.

This article gives the basics of EDF Statistics, and gives important tests with these statistics. Topics where the statistics are useful, or have been used, are summarized. Keywords: goodness-of-fit; distributional assumptions; model assessment

The von Mises distribution is widely used for modeling angular data. When such data are seen in a quality control setting, there may be interest in checking whether the values are in statistical control or have gone out of control. A cumulative sum (cusum) control chart has desirable properties for checking whether the distribution has changed from an in-control to an out-of-control setting. This paper develops cusums for a change in the mean direction and concentration of angular data and illustrates some of their properties.

For more than 60 years, the Anderson–Darling test is most frequently used among all Cramer–von Mises (omega-square) tests. This statistic modifies a classical empirical process defined within the [0, 1] interval by multiplying it by weighting function ψ(t) = (t(1–t))–1/2. The weighting function redistributes the test sensitivity to deviations of the distribution function of the observed stochastic quantity from a hypothetical distribution function in different its segments. However, the tests with other weighting functions may also be of interest in practice. New formulas for the eigenvalues of the Anderson–Darling statistic are proposed. The statistic “inverse” to the Anderson–Darling statistic with weighting function ψ(t) = (t(1–t))1/2 is considered. Tests with other weighting functions may also be of interest when weighted Cramer–von Mises statistics are used. The table of quantiles of statistics with weighting functions ψ(t) = tα(1–t)β, α >–1, β >–1 is presented. The quantiles are given for 36 different combinations of parameters α >–1 and β >–1. The table was calculated using accurate numerical methods and without application of modeling techniques.

We obtaine exact moment-type estimates for the real part and for the absolute value of a characteristic function with the first three moments being fixed. As a corollary, the Mises inequality for lattice distributions is improved and generalized to the case of fractional-order moments.

In this paper, a jackknife empirical likelihood based approach is developed to test whether the underlying distribution is equal to a specified one. The limiting distribution of the proposed testing statistic is derived under some mild conditions. It turns out that the proposed test is consistent and easy to be implemented. Some simulation studies are conducted to evaluate the finite sample behaviors by comparing the proposed method with the existing one. A real data example is also analyzed to illustrate the proposed test approach.

The performance of the existing spectrum sensing algorithms based on goodness of fit (GoF) tests are excellent, however, they are sensitive to the noise uncertainty. In this paper, the first component of Cramer-von Mises, insensitive to variances shift, is employed as test statistic in GoF test and a fast spectrum sensing based on the component of Cramervon Mises is proposed. The detection probability and false alarm probability of the proposed method are analyzed, and its theoretical threshold is derived. Simulation results demonstrate that the proposed method can overcome the noise uncertainty and has lower complexity compared with Cramer-Von Mises (CM) criterion, and the false alarm probability can satisfy the requirements of the standard.

International audience; In this paper, we first study a new sensitivity index that is based on higher moments and generalizes the so-called Sobol one. Further, following an idea of Borgonovo ([3]), we define and study a new sensitivity index based on the Cramér von Mises distance. This new index appears to be more general than the Sobol one as it takes into account, not only the variance, but the whole distribution of the random variable. Furthermore, we study the statistical properties of a Monte Carlo estimate of this new index.

Data generated by high-throughput screening (HTS) technologies are prone to spatial bias. Traditionally, bias correction methods used in HTS assume either a simple additive or, more recently, a simple multiplicative spatial bias model. These models do not, however, always provide an accurate correction of measurements in wells located at the intersection of rows and columns affected by spatial bias. The measurements in these wells depend on the nature of interaction between the involved biases. Here, we propose two novel additive and two novel multiplicative spatial bias models accounting for different types of bias interactions. We describe a statistical procedure that allows for detecting and removing different types of additive and multiplicative spatial biases from multiwell plates. We show how this procedure can be applied by analyzing data generated by the four HTS technologies (homogeneous, microorganism, cell-based, and gene expression HTS), the three high-content screening (HCS) technologies (area, intensity, and cell-count HCS), and the only small-molecule microarray technology available in the ChemBank small-molecule screening database. The proposed methods are included in the AssayCorrector program, implemented in R, and available on CRAN.

In order to overcome the nonlinearity of Mises criterion, a new linear yield criterion with a dodecagon shape of the same perimeter as Mises criterion was derived by means of geometrical analysis. Its specific plastic work rate expressed as a linear function of the yield stress, the maximum and minimum strains was also deduced and compared with that of Mises criterion. The physical meaning of the proposed yield criterion is that yielding of materials begins when the shear yield stress τs reaches the magnitude of 0.594σs. By introducing the Lode parameter, validation of evolution expressions of the proposed yield criterion with those based on Tresca, Mises and TSS criteria as well as available classical yield experimental results of various metals shows that the present results intersect with Mises results and coincide well with experimental data. Moreover, further application to the limit analysis of circle plate as an example is performed to demonstrate the effectiveness of the proposed yield criterion, and the subsequent comparison of limit loads with the Tresca analytical solutions and Mises numerical results shows that the present results are higher than the Tresca analytical results, and are in good agreement with the Mises numerical results.

A regression model for correlated circular data is proposed by assuming that samples of angular measurements are drawn from a multivariate von Mises distribution with mean and concentration parameters that depend on covariates through link functions. The model can flexibly accommodate heteroscedasticity, unstructured correlation, and specific autoregressive correlation structures. Because the computation of the normalizing constant of the multivariate von Mises distribution is unfeasible, inference is based on a computationally tractable Monte Carlo approximation of the log-likelihood. These methods are illustrated by fitting a number of regression models in two case studies: a longitudinal study of animal orientation, involving multiple time series of directional observations, and a study of marine currents, involving a spatial series of sea current directions.

We provide a consistent specification test for generalized autoregressive conditional heteroscedastic (GARCH (1,1)) models based on a test statistic of Cramer-von Mises type. Because the limit distribution of the test statistic under the null hypothesis depends on unknown quantities in a complicated manner, we propose a model-based (semiparametric) bootstrap method to approximate critical values of the test and to verify its asymptotic validity. Finally, we illuminate the finite sample behaviour of the test by some simulations.

One of the key problems in statistics is to get information about the form of the population from which a sample is drawn. To check compatibility of a set of observed values with a presumed distribution one can apply various, so called, goodness-of-fit tests. It seems that the goodness-of-fit testing problem becomes much more complicated in the presence of imprecise observations. Actually, although many statistical procedure dedicated for specified types of distributions were generalized to fuzzy environment, still there are not too many tools that help under fuzzy data from the unknown distribution. Therefore, in the paper we suggest how to generalize the well-known one-sample goodness-of-fit tests based on the empirical distribution function, like the Kolmogorov test, the Cramer-von Mises test or the Anderson-Darling test, for fuzzy data.