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Analyzing correlated data by goodness-of-fit type tests is a critical statistical problem in many applications. A unified framework is provided through a general family of goodness-of-fit tests (GGOF) to address this problem. The GGOF family covers many classic and newly developed tests, such as the minimal p-value test, Simes test, the GATES, one-sided Kolmogorov-Smirnov type tests, one-sided phi-divergence tests, the generalized Higher Criticism, the generalized Berk-Jones, etc. It is shown that the omnibus test that automatically adapts among GGOF statistics for given data, i.e., the GGOF-O, is still contained in the GGOF family and is computationally efficient. For analytically controlling the type I error rate of any GGOF tests, exact calculation is deduced under the Gaussian model with positive equal correlations. Based on that, the effective correlation coefficient (ECC) algorithm is proposed to address arbitrary correlations. Simulations are used to explore how signal and correlation patterns jointly influence typical GGOF tests' statistical power. The GGOF-O is shown robustly powerful across various signal and correlation patterns. As demonstrated by a study of bone mineral density, the GGOF framework has good potential for detecting novel disease genes in genetic summary data analysis. Computational tools are available in the R package SetTest on the CRAN.

New characterizations of the Poisson distribution based on an identity involving the Binomial thinning operator are presented. These characterizations allow the construction of statistics for testing the Poisson distribution against alternatives belonging to a large family called the LC-class, and against general alternatives. The usefulness and the power of the tests are illustrated with several examples of applications in Biodosimetry.

In this paper, using simulations, we compare specification procedures for testing the null hypothesis of a Gaussian distribution for the innovations of GARCH models. More precisely, Cramer-von Mises and Kolmogorov-Smirnov type statistics are computed for empirical processes based on the standardized residuals and their squares. For computing P-values, we use the parametric bootstrap method and the multipliers method. We also use the so-called Khmaladze's transform to obtain an approximated Brownian motion under the null hypothesis, for which we compute also Cramer-von Mises and Kolmogorov-Smirnov type statistics, using both the standardized residuals and their squares.

The dependence structure between WTI and Brent crude oil spot log-returns is analysed using modern copula techniques. In a first step, to account for autocorrelation and volatility clustering in the marginals, several single equation models are applied. Second, to select both copulas and tail copulas characterising the joint dynamics between the time series, newly introduced bootstrap-based goodness-of-fit tests are implemented and evaluated. Based on each approach, a comprehensive backtesting is performed by simulating and comparing the risk measures Value-at-Risk and Expected Shortfall with observed values.

Markovian binary trees form a class of continuous-time branching processes where the lifetime and reproduction epochs of individuals are controlled by an underlying Markov process. An Expectation-Maximization (EM) algorithm is developed to estimate the parameters of the Markov process from the continuous observation of some populations, first with information about which individuals reproduce or die (the distinguishable case), and second without this information (the indistinguishable case). The performance of the EM algorithm is illustrated with some numerical examples. Fits resulting from the distinguishable case are shown not to be significantly better than fits resulting from the indistinguishable case using some goodness of fit measures.

A goodness of fit test for the Pareto distribution, when the observations are subjected to Type II right censoring is proposed. The test statistic involves transformations of the original data and is based on the nonparametric Nelson-Aalen estimator of the cumulative hazard function. By Monte Carlo simulation, the empirical distribution of the test statistic is obtained and the power of the test is investigated for some alternative distributions. The power is compared with adaptations for Type II censored data of the Cramer-von Mises and Anderson-Darling tests, and a test based on Kullback-Leibler information. For some alternative distributions with monotone decreasing hazard function, the proposed test has higher power. The methodology is illustrated by reanalyzing two published data sets.

This paper proposes a bootstrap goodness of fit test for the Generalized Pareto distribution (GPd) with shape parameter @c. The proposed test is an intersection-union test which tests separately the cases of @c>=0 and @c

Generalizing the Cramer-von Mises and the Kolmogorov-Smirnov test, different integral statistics based on L"p-norms are compared with respect to local approximate Bahadur efficiency. Simulation results corroborate the theoretical findings. Several examples illustrate that goodness-of-fit testing based on L"p-norms should receive more attention. It is shown that, given a distribution function F"0 and a specific alternative, one can draw the plot of efficiency as a function of p and determine the value of p giving the maximum efficiency.

When continuous predictors are present, classical Pearson and deviance goodness-of-fit tests to assess logistic model fit break down. The Hosmer-Lemeshow test can be used in these situations. While simple to perform and widely used, it does not have desirable power in many cases and provides no further information on the source of any detectable lack of fit. Tsiatis proposed a score statistic to test for covariate regional effects. While conceptually elegant, its lack of a general rule for how to partition the covariate space has, to a certain degree, limited its popularity. We propose a new method for goodness-of-fit testing that uses a very general partitioning strategy (clustering) in the covariate space and either a Pearson statistic or a score statistic. Properties of the proposed statistics are discussed, and a simulation study demonstrates increased power to detect model misspecification in a variety of settings. An application of these different methods on data from a clinical trial illustrates their use. Discussions on further improvement of the proposed tests and extending this new method to other data situations, such as ordinal response regression models are also included.

The possibility of considering random projections to identify probability distributions belonging to parametric families is explored. The results are based on considerations involving invariance properties of the family of distributions as well as on the random way of choosing the projections. In particular, it is shown that if a one-dimensional (suitably) randomly chosen projection is Gaussian, then the distribution is Gaussian. In order to show the applicability of the methodology some goodness-of-fit tests based on these ideas are designed. These tests are computationally feasible through the bootstrap setup, even in the functional framework. Simulations providing power comparisons of these projections-based tests with other available tests of normality, as well as to test the Black-Scholes model for a stochastic process are presented.

A goodness of fit test for copulas based on Rosenblatt's transformation is investigated. This test performs well if the marginal distribution functions are known and are used in the test statistic. If the marginal distribution functions are unknown and are replaced by their empirical estimates, then the test's properties change significantly. This is shown in detail by simulation for special cases. A bootstrap version of the test is suggested and it is shown by simulation that it performs well. An empirical application of this test to daily returns of German assets reveals that a Gaussian copula is unsuitable to describe their dependence structure. A t"@n-copula with low degrees of freedom such as @n=4 or 5 fits the data in some cases.

Some of the most powerful techniques currently available to test the goodness of fit of a hypothesized continuous cumulative distribution function (CDF) use statistics based on the empirical distribution function (EDF), such as those of Kolmogorov, Cramer-von Mises and Anderson-Darling, among others. The use of EDF statistics was analyzed for estimation purposes. In this approach, maximum goodness-of-fit estimators (also called minimum distance estimators) of the parameters of the CDF can be obtained by minimizing any of the EDF statistics with respect to the unknown parameters. The results showed that there is no unique EDF statistic that can be considered most efficient for all situations. Consequently, the possibility of defining new EDF statistics is entertained; in particular, an Anderson-Darling statistic of degree two and one-sided Anderson-Darling statistics of degree one and two appear to be notable in some situations. The procedure is shown to be able to deal successfully with the estimation of the parameters of homogeneous and heterogeneous generalized Pareto distributions, even when maximum likelihood and other estimation methods fail.

Genes that are more closely spaced on the chromosome than expected by chance are said to be spatially clustered. Standard tests of clustering versus uniformity do not take into account two important features of genes-the high variability of gene length and the low probability that gene locations overlap (exclusion). We show by simulation that the standard null distributions which ignore length and exclusion do not appropriately approximate the true null distributions of standard tests such as the chi-squared test. We therefore recommend bootstrap sampling to estimate the null distributions. Simulations demonstrate that the chi-squared goodness-of-fit test is a more powerful test of clustering than two other commonly used tests-Kolmogorov and Cramer-von Mises-when the distribution of gene lengths and locations is modeled by a mixture of exponentials and there is a single cluster. The chi-squared test requires binning the gene locations-the number of genes in the bin can be compared to the expected maximum number under random distribution to determine the location of gene clusters and gene deserts. The bootstrap method to test clustering is illustrated using data from human chromosome 22.

Change point models are often used to model longitudinal data. To estimate the change point, Bayesian (Biometrika 62 (1975) 407; Appl. Statist. 41 (1992) 389; Biometrics 51 (1995) 236) or profile likelihood (Statist. Med. 19 (2000) 1555) methods may be used.We compare and contrast the two methods in analyzing longitudinal cognitive data from the Bronx Aging Study. The Bayesian method has advantages over the profile likelihood method in that it does not require all subjects to have the same change point. Caution must be taken regarding sensitivity to choice of prior distribution, identifiability, and goodness of fit. Analyses show that decline in memory precedes diagnosis of dementia by 7.5-8 years, and individual change points are not needed to model heterogeneity across subjects.

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.

Based on the notion of multivariate quantiles in terms of minimum volume ellipsoids as introduced in Einmahl and Mason (1992), a generalized chi-square quantile plot is constructed from which tests for multivariate normality are derived. The generalized quantile plot also offers some extra possibilities for data analysis, e.g. in identifying and describing non-normality. An algorithm is proposed to approximate the generalized quantiles, extending existing methods to approximate median quantiles with only a minor extra computational cost and preserving the effectivity of the method. A power analysis is included and the method is illustrated with some case-studies.

Because of the low power of the Kolmogorov-Smirnov goodness-of-fit test in case of multimodal alternatives this paper investigates a family of tests indexed by an integer-valued parameter K≥2. The sensitivity of these tests to multimodal alternatives increases as K increases. The Kolmogorov-Smirnov test is obtained for K=2. The other elements of this family measure local deviations from uniformity in the same way as the Kolmogorov-Smirnov test measures the global deviation. Hence they may be regarded as generalized Kolmogorov-Smirnov tests. Tables of critical values are given and applications to the problem of testing for white noise are discussed. In addition, a further family of related tests is proposed. A Monte Carlo power study compares all the tests proposed in this note with Neyman's smooth test and different versions of the length test introduced by Reschenhofer and Bomze (Biometrika, 78 (1991) 207–216; Biometrika, 79 (1992) 859). A secondary result of this simulation study is that the possibly most widely used version of the length test can break down when a peak is located near the boundary.

Modified Kolmogorov-Smirnov (KS), Kuiper (V), Cramer-von Mises (CV), Watson (W), Anderson-Darling (AD) and sequential goodness-of-fit tests are developed for the inverse Gaussian distribution with unknown parameters. A Monte Carlo procedure is employed to generate critical values for a wide range of sample sizes and shape parameters. Power studies indicate that the W test is most effective against alternate distributions that are very similar in shape to the null inverse Gaussian distribution. Otherwise, the modified AD test generally demonstrates the highest power among single tests. To eliminate the need for extensive critical value tables, functional relationships between critical values, sample sizes, and shape parameters are reported.

The Berry-Esseen theorem is used to study the error of the normal approximation to the distribution of several goodness of fit tests for discrete data. A practical bound is proposed and confirmed by simulations.

A graphical test statistic is proposed for testing the goodness of fit in the case of the multinomial distribution. The exact power of the test statistic is calculated and the comparison between the test statistic and the traditional test statistics is discussed with respect to the same example.

A method used to compensate for nonresponse is to impute missing values; that is, to replace each missing value with a respondent value selected from all observed values or from a subset of observed values. The imputation procedure used in this paper selects imputed values from the respondent data using simple random sampling with replacement within homogeneous subsets and replaces the missing values with these values to complete the data set. The empirical distribution of the goodness-of-fit chi-square statistic computed from the ‘completed’ data set is compared to its asymptotic distribution and to the distribution of the traditional chi-square test statistic applied to the completed data set by ignoring the imputation. At nominal levels of five and ten percent, the asymptotic distribution of the goodness-of-fit chi-square statistic computed from the completed data set is shown to have a good empirical behavior at moderate sample sizes. When the imputed values are treated as actual responses and the imputation is ignored, the empirical levels of significance are much larger than the nominal levels.