Performances of some goodness-of-fit tests for sampling designs in ranked set sampling

In this study, we consider different sampling designs of ranked set sampling (RSS) and give empirical distribution function (EDF) estimators for each sampling designs. We provide comparative graphs for the EDFs. Using these EDFs, power of five goodness-of-fit tests are obtained by Monte Carlo simulations for Tukey's g–h distributions under RSS and simple random sampling (SRS). Performances of these tests are compared with the tests based on the SRS. Also, critical values belong to these tests are obtained for different set and cycle sizes.     

A modification of watson's statistic for goodness-of-fit

A modification of Watson's statistic is suggested for dealing with stochastically ordered alternatives in the goodness-of-fit setting.     

Pearson-type goodness-of-fit tests for regression

A procedure for testing the goodness of fit of linear regression models is introduced. For a given partition of the real line into cells, the proposed test is a quadratic form based on the vector of observed minus expected frequencies of the residuals obtained by maximum-likelihood estimation of the regression parameters. The quadratic form is of the same computational difficulty as the traditional Pearson-type tests with uncensored data. A statistic based on only one cell is particularly easy to apply and is used for testing the normality assumption in a real data set from astronomy. A simulation study examines the finite-sample properties of the proposed tests.     

Percentage points for directional anderson-darling goodness-of-fit tests

This paper discusses the problem of assessing the asymptotic distribution when parameters of the hypothesized distribution are estimated from a sample, pointing out a common mistake included in the paper by Sinclair, Spurr, and Ahmad (1990) which introduced two modifications of the Anderson-Darling goodness-of-fit test statistic. Their two test statistics modify the popular Anderson-Darling test statistic to be sensitive to departures of the fitted distribution from the true distribution in one or the other of the tails. This paper uses these new test statistics to develop tests of fit for the normal and exponential distributions. Easy to use formulas are given so the reader can perform these tests at any sample size without consulting exhaustive tables of percentage points. Finally a power study is given to demonstrate the test statistics’ viability against a broad range of alternatives.     

A power study of goodness-of-fit tests for multivariate normality implemented in R

Multivariate statistical analysis procedures often require data to be multivariate normally distributed. Many tests have been developed to verify if a sample could indeed have come from a normally distributed population. These tests do not all share the same sensitivity for detecting departures from normality, and thus a choice of test is of central importance. This study investigates through simulated data the power of those tests for multivariate normality implemented in the statistic software R and pits them against the variant of testing each marginal distribution for normality. The results of testing two-dimensional data at a level of significance α=5% showed that almost one-third of those tests implemented in R do not have a type I error below this. Other tests outperformed the naive variant in terms of power even when the marginals were not normally distributed. Even though no test was consistently better than all alternatives with every alternative distribution, the energy-statistic test always showed relatively good power across all tested sample sizes.     

Negative exponential disparity based family of goodness-of-fit tests for multinomial models

This paper investigates a new family of goodness-of-fit tests based on the negative exponential disparities. This family includes the popular Pearson's chi-square as a member and is a subclass of the general class of disparity tests (Basu and Sarkar, 1994) which also contains the family of power divergence statistics. Pitman efficiency and finite sample power comparisons between different members of this new family are made. Three asymptotic approximations of the exact null distributions of the negative exponential disparity famiiy of tests are discussed. Some numerical results on the small sample perfomance of this family of tests are presented for the symmetric null hypothesis. It is shown that the negative exponential disparity famiiy, Like the power divergence family, produces a new goodness-of-fit test statistic that can be a very attractive alternative to the Pearson's chi-square. Some numerical results suggest that, application of this test statistic, as an alternative to Pearson's chi-square, could be preferable to the I ^2/3 statistic of Cressie and Read (1984) under the use of chi-square critical values.     

Zipf's law for randomly generated frequencies: explicit tests for the goodness-of-fit

The question, whether Zipf's law arises only in consequence of interactions in a complex system or if it is also valid for random structures, has been controversially discussed in the literature over several decades. We show by means of simulations that the frequency distributions of simple random sequences usually obey this regularity. For tens of thousands of cases the goodness-of-fit is explicitly demonstrated by estimating the parameter and calculating the corresponding chi-square value. From the study it becomes clear that some existing results in the literature should be revised, and some ideas concerning the explanation of Zipf's law are provided.     

Asymptotics of goodness-of-fit tests based on minimum p-value statistics

This paper provides some new results on the asymptotics of goodness-of-fit (GOF) tests based on minimum p-value statistics. In connection with detectability of sparse signals in high-dimensional data, various tests were proposed and investigated during the last decade, especially with respect to asymptotic properties. Minimum p-value GOF statistics were already investigated as minimum level attained statistics by Berk and Jones with respect to Bahadur efficiency. The distribution of minimum p-value GOF statistics is closely related to the distribution of higher criticism statistics, the distribution of the supremum of a normalized Brownian bridge, and the supremum of an Ornstein–Uhlenbeck process.     

On a data-dependent choice of the tuning parameter appearing in certain goodness-of-fit tests

We propose a data-dependent method for choosing the tuning parameter appearing in many recently developed goodness-of-fit test statistics. The new method, based on the bootstrap, is applicable to a class of distributions for which the null distribution of the test statistic is independent of unknown parameters. No data-dependent choice for this parameter exists in the literature; typically, a fixed value for the parameter is chosen which can perform well for some alternatives, but poorly for others. The performance of the new method is investigated by means of a Monte Carlo study, employing three tests for exponentiality. It is found that the Monte Carlo power of these tests, using the data-dependent choice, compares favourably to the maximum achievable power for the tests calculated over a grid of values of the tuning parameter.     

On goodness-of-fit tests for Aalen's additive model based on left-truncated right-censored or doubly censored data

ABSTRACT

Gandy and Jensen (2005 Gandy, A., Jensen, U. (2005). On goodness-of-fit tests for Aalen's additive risk model. Scan. J. Stat. 32:425–445.[Crossref], [Web of Science ®] , [Google Scholar]) proposed goodness-of-fit tests for Aalen's additive risk model. In this article, we demonstrate that the approach of Gandy and Jensen (2005 Gandy, A., Jensen, U. (2005). On goodness-of-fit tests for Aalen's additive risk model. Scan. J. Stat. 32:425–445.[Crossref], [Web of Science ®] , [Google Scholar]) can be applied to left-truncated right-censored (LTRC) data and doubly censored data. A simulation study is conducted to investigate the performance of the proposed tests. The proposed tests are illustrated using heart transplant data.     

New invariant and consistent chi-squared type goodness-of-fit tests for multivariate normality and a related comparative simulation study

ABSTRACT

New invariant and consistent goodness-of-fit tests for multivariate normality are introduced. Tests are based on the Karhunen–Loève transformation of a multidimensional sample from a population. A comparison of simulated powers of tests and other well-known tests with respect to some alternatives is given. The simulation study demonstrates that power of the proposed McCull test almost does not depend on the number of grouping cells. The test shows an advantage over other chi-squared type tests. However, averaged over all of the simulated conditions examined in this article, the Anderson–Darling type and the Cramer–von Mises type tests seem to be the best.     

Robust durbin's and portmanteau goodness-of-fit test statistics and their limit distributions

Robust analogue of Durbin's(1970)statistic is derived and its limiting distribution is obtained under both null and alternative hypotheses. Also, robust version of the portmanteau goodness-of-fit test statistic for AR(p)model is given and the asymptotic distribution is derived.     

An overview of tests for exponentiality

Developments since 1960 in goodness-of-fit tests for the one and two parameter exponential models using both complete and censored samples are reviewed. Special attention is given to both the omnibus or general alternative and to specialized alter-natives such as the class of distributions with increasing failure rates. The use of transformations in developing tests is also discussed.     

Entropy estimation and goodness-of-fit tests for the inverse Gaussian and Laplace distributions using paired ranked set sampling

ABSTRACT

In this paper, Vasicek [A test for normality based on sample entropy. J R Stat Soc Ser B. 1976;38:54–59] entropy estimator is modified using paired ranked set sampling (PRSS) method. Also, two goodness-of-fit tests using PRSS are suggested for the inverse Gaussian and Laplace distributions. The new suggested entropy estimator and goodness-of-fit tests using PRSS are compared with their counterparts using simple random sampling (SRS) via Monte Carlo simulations. The critical values of the suggested tests are obtained, and the powers of the tests based on several alternatives hypotheses using SRS and PRSS are calculated. It turns out that the proposed PRSS entropy estimator is more efficient than the SRS counterpart in terms of root mean square error. Also, the proposed PRSS goodness-of-fit tests have higher powers than their counterparts using SRS for all alternative considered in this study.     

Remember the curse of dimensionality: the case of goodness-of-fit testing in arbitrary dimension

Despite a substantial literature on nonparametric two-sample goodness-of-fit testing in arbitrary dimensions, there is no mention there of any curse of dimensionality. In fact, in some publications, a parametric rate is derived. As we discuss below, this is because a directional alternative is considered. Indeed, even in dimension one, Ingster, Y. I. [(1987). Minimax testing of nonparametric hypotheses on a distribution density in the l_p metrics. Theory of Probability & Its Applications, 31(2), 333–337] has shown that the minimax rate is not parametric. In this paper, we extend his results to arbitrary dimension and confirm that the minimax rate is not only nonparametric, exhibits but also a prototypical curse of dimensionality. We further extend Ingster's work to show that the chi-squared test achieves the minimax rate. Moreover, we show that the test adapts to the intrinsic dimensionality of the data. Finally, in the spirit of Ingster, Y. I. [(2000). Adaptive chi-square tests. Journal of Mathematical Sciences, 99(2), 1110–1119], we consider a multiscale version of the chi-square test, showing that one can adapt to unknown smoothness without much loss in power.     

A class of goodness-of-fit test for the additive hazards model with case-cohort data

The case-cohort design is commonly used in epidemiological studies due to its cost-effectiveness. The additive hazards model is widely used in survival analysis when the hazards difference is constant. In this article, we propose a class of goodness-of-fit test statistics for the assumption of the additive hazards model with case-cohort data through a class of asymptotically mean-zero multiparameter stochastic processes. We also establish the asymptotic theory of the proposed test statistics and a resampling scheme is adopted to approximate its asymptotic distribution. The performance of the proposed test statistics is evaluated through simulation studies and a real dataset is analyzed to illustrate the proposed method.     

Modified maximum likelihood estimator under the Jones and Faddy's skew t-error distribution for censored regression model

It is well-known that classical Tobit estimator of the parameters of the censored regression (CR) model is inefficient in case of non-normal error terms. In this paper, we propose to use the modified maximum likelihood (MML) estimator under the Jones and Faddy''s skew t-error distribution, which covers a wide range of skew and symmetric distributions, for the CR model. The MML estimators, providing an alternative to the Tobit estimator, are explicitly expressed and they are asymptotically equivalent to the maximum likelihood estimator. A simulation study is conducted to compare the efficiencies of the MML estimators with the classical estimators such as the ordinary least squares, Tobit, censored least absolute deviations and symmetrically trimmed least squares estimators. The results of the simulation study show that the MML estimators work well among the others with respect to the root mean square error criterion for the CR model. A real life example is also provided to show the suitability of the MML methodology.     

Asyhptotic efficiency of sequential and non-sequential procedures based on fisher's method of combining tests

In this paper we propose test statistics based on Fisher's method of combining tests for hypotheses involving two or more parameters simultaneously, It Is shown that these tests are asymptotically efficient In the sense of Bahadur, It is then shown how these tests can be modified to give sequential test procedures which are efficient in the sense of Berk and Brown (1978).

The results in section 3 generalize the work of Perng (1977) and Durairajan (1980).     

Improvement of goodness-of-fit test for normal distribution based on entropy and power comparison

Vasicek's entropy test for normality is based on sample entropy and a parametric entropy estimator. These estimators are known to have bias in small samples. The use of Vasicek's test could affect the capability of detecting non-normality to some extent. This paper presents an improved entropy test, which uses bias-corrected entropy estimators. A Monte Carlo simulation study is performed to compare the power of the proposed test under several alternative distributions with some other tests. The results report that as anticipated, the improved entropy test has consistently higher power than the ordinary entropy test in nearly all sample sizes and alternatives considered, and compares favorably with other tests.