Family Of Test Statistics Research Articles

A family of consistent normally distributed tests for Poissonity

A family of consistent tests, derived from a characterization of the probability generating function, is proposed for assessing Poissonity against a wide class of count distributions, which includes some of the most frequently adopted alternatives to the Poisson distribution. Actually, the family of test statistics is based on the difference between the plug-in estimator of the Poisson cumulative distribution function and the empirical cumulative distribution function. The test statistics have an intuitive and simple form and are asymptotically normally distributed, allowing a straightforward implementation of the test. The finite sample properties of the test are investigated by means of an extensive simulation study. The test shows satisfactory behaviour compared to other tests with known limit distribution.

Extensions of empirical likelihood and chi-squared-based tests for ordered alternatives

Several methods for comparing k populations have been proposed in the literature. These methods assess the same null hypothesis of equal distributions but differ in the alternative hypothesis they consider. We focus on two important alternative hypotheses: monotone and umbrella ordering. Two new families of test statistics are proposed, including two known tests, as well as two new powerful tests under monotone ordering. Furthermore, these families are adapted for testing umbrella ordering. We compare some members of the families with respect to power and Type I errors under different simulation scenarios. Finally, the methods are illustrated in several applications to real data.

Robust Wald-type test statistics based on minimum C-divergence estimators

ABSTRACT Maji et al. [Robust statistical inference based on the C-divergence family. Ann Inst Stat Math. 2019;71:1289–1322] introduced the minimum C-divergence estimators and plugging them in the C-divergence measures give test statistics for testing simple null and composite null hypotheses. One inconvenience of these test statistics is that their asymptotic distribution is not, in general, a chi-square distribution but a linear combination of chi-square distributions. To overcome this inconvenience, in this paper we consider Wald-type test statistics based on minimum C-divergence estimators. We establish that this family of test statistics is a chi-square distribution and we get an approximation of the power function under simple null hypothesis and composite null hypothesis. We have calculated both first order and second order influence function of the Wald-type test statistics and based on it we can see the robustness of the family of test statistics considered in this paper. Both simulated and real data examples have been shown as part of numerical results.

Testing against umbrella or tree orderings for binomial proportions with an adaptation of an insect resistance case.

Alternative hypotheses for order restrictions, such as umbrella or inverse umbrella (a.k.a tree) orderings, have been studied extensively in the literature, although less so when the studied response for each individual is the presence or absence of the event of interest. Two families of test statistics for solving the problem of testing against an umbrella or a tree ordering when the responses are binomial proportions are studied in this work and their asymptotic distributions are derived. A simulation study is conducted to compare the empirical power of some members of the derived families of test statistics with competing approaches. The methodology developed here was driven by an applied problem arising in stored products research where despite universal mortality in the case of doses of 1000 ppm of the insecticide phosphine, unexpected survival was noted at higher doses.

Statistical inference for multinomial populations based on a double index family of test statistics

ABSTRACTIn the present work, we consider parameter estimation and hypothesis testing based on a general class of measures, namely the -power divergence family for multinomial populations. In particular, we propose a general family of double index -test statistics that involves two indices, the values of which play a key role in the effectiveness and accuracy of the proposed methodology. The asymptotic properties of the associated estimators are examined together with the power of the test and the asymptotic distribution under the contiguous alternatives. Finally, we explore through extensive simulations, the effect of the shape of the multinomial distribution on the performance of the proposed test.

Robust Estimators and Test Statistics for One-Shot Device Testing Under the Exponential Distribution

This paper develops a new family of estimators, the minimum density power divergence estimators (MDPDEs), for the parameters of the one-shot device model as well as a new family of test statistics, Z-type test statistics based on MDPDEs, for testing the corresponding model parameters. The family of MDPDEs contains as a particular case the maximum likelihood estimator (MLE) considered in Balakrishnan and Ling (2012). Through a simulation study, it is shown that some MDPDEs have a better behavior than the MLE in relation to robustness. At the same time, it can be seen that some Z-type tests based on MDPDEs have a better behavior than the classical Z-test statistic also in terms of robustness.

A tail adaptive approach for change point detection

For change point problems with Gaussian distributions, the CUSUM method is most efficient for detecting mean shifts. In contrast, it is not so efficient for heavy-tailed or contaminated data because of its sensitivity to outliers. To address this issue, Csörgő and Horváth (1988) introduced the Wilcoxon–Mann–Whitney test based on two-sample U-statistics. In practice, however, the tail structure of distributions is typically unknown. For example, Barndorff-Nielsen and Shephard (2001) showed that with higher frequency, stock returns’ tails become heavier. To our knowledge, there are no uniformly most powerful testing methods for both heavy and light-tailed distributions. To deal with this issue, we construct a new family of test statistics and combine them to adapt to different tails. As the final test statistic is complex, we design a low-cost bootstrap procedure to approximate its limiting distribution. To capture temporal data dependence, we assume that the data follow a near epoch dependent process (Borovkova et al., 2001), which includes ARMA and GARCH processes, among others. We explore the validity of our method both theoretically and through simulation. We also illustrate its use with data on the S&P 500 index.

Composite Likelihood Methods Based on Minimum Density Power Divergence Estimator.

In this paper, a robust version of the Wald test statistic for composite likelihood is considered by using the composite minimum density power divergence estimator instead of the composite maximum likelihood estimator. This new family of test statistics will be called Wald-type test statistics. The problem of testing a simple and a composite null hypothesis is considered, and the robustness is studied on the basis of a simulation study. The composite minimum density power divergence estimator is also introduced, and its asymptotic properties are studied.

Statistical inference in constrained latent class models for multinomial data based on $$\phi $$ ϕ -divergence measures

In this paper we explore the possibilities of applying $$\phi $$ -divergence measures in inferential problems in the field of latent class models (LCMs) for multinomial data. We first treat the problem of estimating the model parameters. As explained below, minimum $$\phi $$ -divergence estimators (M $$\phi $$ Es) considered in this paper are a natural extension of the maximum likelihood estimator (MLE), the usual estimator for this problem; we study the asymptotic properties of M $$\phi $$ Es, showing that they share the same asymptotic distribution as the MLE. To compare the efficiency of the M $$\phi $$ Es when the sample size is not big enough to apply the asymptotic results, we have carried out an extensive simulation study; from this study, we conclude that there are estimators in this family that are competitive with the MLE. Next, we deal with the problem of testing whether a LCM for multinomial data fits a data set; again, $$\phi $$ -divergence measures can be used to generate a family of test statistics generalizing both the classical likelihood ratio test and the chi-squared test statistics. Finally, we treat the problem of choosing the best model out of a sequence of nested LCMs; as before, $$\phi $$ -divergence measures can handle the problem and we derive a family of $$\phi $$ -divergence test statistics based on them; we study the asymptotic behavior of these test statistics, showing that it is the same as the classical test statistics. A simulation study for small and moderate sample sizes shows that there are some test statistics in the family that can compete with the classical likelihood ratio and the chi-squared test statistics.

On divergence tests for composite hypotheses under composite likelihood

It is well-known that in some situations it is not easy to compute the likelihood function as the datasets might be large or the model is too complex. In that contexts composite likelihood, derived by multiplying the likelihoods of subjects of the variables, may be useful. The extension of the classical likelihood ratio test statistics to the framework of composite likelihoods is used as a procedure to solve the problem of testing in the context of composite likelihood. In this paper we introduce and study a new family of test statistics for composite likelihood: Composite $$\phi $$ -divergence test statistics for solving the problem of testing a simple null hypothesis or a composite null hypothesis. To do that we introduce and study the asymptotic distribution of the restricted maximum composite likelihood estimate.

A Necessary Power Divergence Type Family Tests of Multivariate Normality

In a recent article, Cardoso de Oliveira and Ferreira have proposed a multivariate extension of the univariate chi-squared normality test, using a known result for the distribution of quadratic forms in normal variables. In this article, we propose a family of power divergence type test statistics for testing the hypothesis of multinormality. The proposed family of test statistics includes as a particular case the test proposed by Cardoso de Oliveira and Ferreira. We assess the performance of the new family of test statistics by using Monte Carlo simulation. In this context, the type I error rates and the power of the tests are studied, for important family members. Moreover, the performance of significant members of the proposed test statistics are compared with the respective performance of a multivariate normality test, proposed recently by Batsidis and Zografos. Finally, two well-known data sets are used to illustrate the method developed in this article as well as the specialized test of multivariate normality proposed by Batsidis and Zografos.

Robust confidence regions for multinomial probabilities

Generally, confidence regions for the probabilities of a multinomial population are constructed based on the Pearson χ2 statistic. Morales et al. (Bootstrap confidence regions in multinomial sampling. Appl Math Comput. 2004;155:295–315) considered the bootstrap and asymptotic confidence regions based on a broader family of test statistics known as power-divergence test statistics. In this study, we extend their work and propose penalized power-divergence test statistics-based confidence regions. We only consider small sample sizes where asymptotic properties fail and alternative methods are needed. Both bootstrap and asymptotic confidence regions are constructed. We consider the percentile and the bias corrected and accelerated bootstrap confidence regions. The latter confidence region has not been studied previously for the power-divergence statistics much less for the penalized ones. Designed simulation studies are carried out to calculate average coverage probabilities. Mean absolute deviation between actual and nominal coverage probabilities is used to compare the proposed confidence regions.

Hypothesis testing in a generic nesting framework for general distributions

Nested parameter spaces, either in the null or alternative hypothesis, often enable an improvement in the performance of the tests. In this context, order restricted inference has not been studied in detail. Divergence based measures provide a flexible tool for proposing some useful test statistics, which usually contain the likelihood ratio-test statistics as a special case. The existing literature on hypothesis testing under inequality constraints, based on phi-divergence measures, is concentrated on specific models with multinomial sampling. In this paper the existing results are extended and unified through new families of test-statistics that are valid for nested parameter spaces containing either equality or inequality constraints and general distributions for either single or multiple populations.

Two classes of divergence statistics for testing uniform association

The problem of testing uniform association in cross-classifications having ordered categories is considered. Two families of test statistics, both based on divergences between certain functions of the observed data, are studied and compared. Our theoretical study is based on asymptotic properties. For each family, two consistent approximations to the null distribution of the test statistic are studied: the asymptotic null distribution and a bootstrap estimator; all the tests considered are consistent against fixed alternatives; finally, we do a local power study. Surprisingly, both families detect the same local alternatives. The finite sample performance of the tests in these two classes is numerically investigated through some simulation experiments. In the light of the obtained results, some practical recommendations are given.

A necessary power divergence-type family of tests for testing elliptical symmetry

This paper presents a family of power divergence-type test statistics for testing the hypothesis of elliptical symmetry. We assess the performance of the new family of test statistics, using Monte Carlo simulation. In this context, the type I error rate as well as the power of the tests are studied. Specifically, for selected alternatives, we compare the power of the proposed procedure with that proposed by Schott [Testing for elliptical symmetry in covariance-matrix-based analyses, Stat. Probab. Lett. 60 (2002), pp. 395–404]. This last test statistic is an easily computed one with a tractable null distribution and very good power for various alternatives, as it has established in previous published simulations studies [F. Huffer and C. Park, A test for elliptical symmetry, J. Multivariate Anal. 98 (2007), pp. 256–281; L. Sakhanenko, Testing for ellipsoidal symmetry: A comparison study, Comput. Stat. Data Anal. 53 (2008), pp. 565–581]. Finally, a well-known real data set is used to illustrate the method developed in this paper.

A Powerful Method of Assessing the Fit of the Lognormal Distribution

When faced with the problem of goodness-of-fit to the Lognormal distribution, testing methods typically reduce to comparing the empirical distribution function of the corresponding logarithmic data to that of the normal distribution. In this article, we consider a family of test statistics which make use of the moment structure of the Lognormal law. In particular, a continuum of moment conditions is employed in the construction of a new statistic for this distribution. The proposed test is shown to be consistent against fixed alternatives, and a simulation study shows that it is more powerful than several classical procedures, including those utilizing the empirical distribution function. We conclude by applying the proposed method to some, not so typical, data sets.

Rao’s statistic for constant and proportional hazard models

In this paper we introduce a new family of measures of divergence for the analysis of the degree of departure from a model with a constant hazard function and also for comparing if two models have proportional hazard rates. Our family of measures is based on the family of divergences introduced by Burbea and Rao (see [J. Burbea, C.R. Rao, On the convexity of higher order Jensen differences based on entropy functions, IEEE Transactions on Information Theory 28 (1982) 961–963]). Some well-known sets of data are reanalyzed using the new families of test statistics and confidence intervals introduced in this paper.

New families of estimators and test statistics in log-linear models

In this paper we consider categorical data that are distributed according to a multinomial, product-multinomial or Poisson distribution whose expected values follow a log-linear model and we study the inference problem of hypothesis testing in a log-linear model setting. The family of test statistics considered is based on the family of ϕ -divergence measures. The unknown parameters in the log-linear model under consideration are also estimated using ϕ -divergence measures: Minimum ϕ -divergence estimators. A simulation study is included to find test statistics that offer an attractive alternative to the Pearson chi-square and likelihood-ratio test statistics.

Minimum Φ-Divergence Estimator and Φ-Divergence Statistics in Generalized Linear Models with Binary Data

In this paper, we assume that the data are distributed according to a binomial distribution whose probabilities follow a generalized linear model. To fit the data the minimum φ-divergence estimator is studied as a generalization of the maximum likelihood estimator. We use the minimum φ-divergence estimator, which is the basis of some new statistics, for solving the problems of testing in a generalized linear model with binary data. A wide simulation study is carried out for studying the behavior of the new family of estimators as well as of the new family of test statistics.

Goodness-of-fit tests via phi-divergences

A unified family of goodness-of-fit tests based on O-divergences is introduced and studied. The new family of test statistics S n (s) includes both the supremum version of the Anderson-Darling statistic and the test statistic of Berk and Jones [Z. Wahrsch. Verw. Gebiete 47 (1979) 47-59] as special cases (s = 2 and s = 1, resp.). We also introduce integral versions of the new statistics. We show that the asymptotic null distribution theory of Berk and Jones [Z. Wahrsch. Verw. Gebiete 47 (1979) 47-59] and Wellner and Koltchinskii [High Dimensional Probability III (2003) 321-332. Birkhauser, Basel] for the Berk-Jones statistic applies to the whole family of statistics S n (s) with s ∈ [-1, 2]. On the side of power behavior, we study the test statistics under fixed alternatives and give extensions of the Poisson boundary phenomena noted by Berk and Jones for their statistic. We also extend the results of Donoho and Jin [Ann. Statist. 32 (2004) 962-994] by showing that all our new tests for s ∈ [- 1, 2] have the same optimal detection boundary for normal shift mixture alternatives as Tukey's higher-criticism statistic and the Berk-Jones statistic.