On a test of independence for contingency tables

Using the concept of distributional distance, a test statistic is proposed FOR the hypothesis of independence in multidimensional contingency tables. A Monte Carlo Study is done to empirically compare the power of the proposed test to the Pearson x² and the likelihood ratio test- Further, the nonnull distribution under various spike alternatives is tabulated     

Bayesian prediction for small areas using sur models

Sample surveys are usually designed and analyzed to produce estimates for larger areas and/or populations. Nevertheless, sample sizes are often not large enough to give adequate precision for small area estimates of interest. To circumvent such difficulties, borrowing strength from related small areas via modeling becomes essential. In line with this, we propose a hierarchical multivariate Bayes prediction method for small area estimation based on the seemingly unrelated regressions (SUR) model. The performance of the proposed method was evaluated through simulation studies.     

A likelihood ratio test of quasi-independence for sparse two-way contingency tables

We consider a likelihood ratio test of independence for large two-way contingency tables having both structural (non-random) and sampling (random) zeros in many cells. The solution of this problem is not available using standard likelihood ratio tests. One way to bypass this problem is to remove the structural zeroes from the table and implement a test on the remaining cells which incorporate the randomness in the sampling zeros; the resulting test is a test of quasi-independence of the two categorical variables. This test is based only on the positive counts in the contingency table and is valid when there is at least one sampling (random) zero. The proposed (likelihood ratio) test is an alternative to the commonly used ad hoc procedures of converting the zero cells to positive ones by adding a small constant. One practical advantage of our procedure is that there is no need to know if a zero cell is structural zero or a sampling zero. We model the positive counts using a truncated multinomial distribution. In fact, we have two truncated multinomial distributions; one for the null hypothesis of independence and the other for the unrestricted parameter space. We use Monte Carlo methods to obtain the maximum likelihood estimators of the parameters and also the p-value of our proposed test. To obtain the sampling distribution of the likelihood ratio test statistic, we use bootstrap methods. We discuss many examples, and also empirically compare the power function of the likelihood ratio test relative to those of some well-known test statistics.     

Power analysis of independence testing for three-way contingency tables of small sizes

The first aim of this paper is to introduce a modular test for the three-way contingency table (TT). The second aim is to describe the procedure of generating TT using the bar method. The third aim is on the one hand to suggest the measure of untruthfulness of H₀ and on the other hand to compare the quality of independence tests by using their power. Critical values for analyzed statistics were determined by simulating the Monte Carlo method.     

A modified chi-square test of independence against a class of ordered alternatives in an r x c contingency table

A modified chi-square test statistic is constructed for testing the hypothesis of independence in a two-way contingency table against a class of ordered alternatives defined in terms of pooled cross-product ratios. The test procedure can also be used to test for positive quadrant dependence in a two-way contingency table. The asymptotic distribution of the test statistic under the null hypothesis is obtained. Some power comparisons with known test procedures are given. A numerical example is given to illustrate the use of this test.     

A fast distance‐based approach for determining the number of components in mixtures

The authors propose a procedure for determining the unknown number of components in mixtures by generalizing a Bayesian testing method proposed by Mengersen & Robert (1996). The testing criterion they propose involves a Kullback‐Leibler distance, which may be weighted or not. They give explicit formulas for the weighted distance for a number of mixture distributions and propose a stepwise testing procedure to select the minimum number of components adequate for the data. Their procedure, which is implemented using the BUGS software, exploits a fast collapsing approach which accelerates the search for the minimum number of components by avoiding full refitting at each step. The performance of their method is compared, using both distances, to the Bayes factor approach.     

A Bayesian test of independence in a two-way contingency table using surrogate sampling

We consider a Bayesian approach to the study of independence in a two-way contingency table which has been obtained from a two-stage cluster sampling design. If a procedure based on single-stage simple random sampling (rather than the appropriate cluster sampling) is used to test for independence, the p-value may be too small, resulting in a conclusion that the null hypothesis is false when it is, in fact, true. For many large complex surveys the Rao–Scott corrections to the standard chi-squared (or likelihood ratio) statistic provide appropriate inference. For smaller surveys, though, the Rao–Scott corrections may not be accurate, partly because the chi-squared test is inaccurate. In this paper, we use a hierarchical Bayesian model to convert the observed cluster samples to simple random samples. This provides surrogate samples which can be used to derive the distribution of the Bayes factor. We demonstrate the utility of our procedure using an example and also provide a simulation study which establishes our methodology as a viable alternative to the Rao–Scott approximations for relatively small two-stage cluster samples. We also show the additional insight gained by displaying the distribution of the Bayes factor rather than simply relying on a summary of the distribution.     

A nonparametric specification test for the volatility functions of diffusion processes

This paper develops a new test for the parametric volatility function of a diffusion model based on nonparametric estimation techniques. The proposed test imposes no restriction on the functional form of the drift function and has an asymptotically standard normal distribution under the null hypothesis of correct specification. It is consistent against any fixed alternatives and has nontrivial asymptotic power against a class of local alternatives with proper rates. Monte Carlo simulations show that the test performs well in finite samples and generally has better power performance than the nonparametric test of Li (2007 Li, F. (2007). Testing the parametric specification of the diffusion function in a diffusion process. Econometric Theory 23(2):221–250.[Crossref], [Web of Science ®] , [Google Scholar]) and the stochastic process-based tests of Dette and Podolskij (2008 Dette, H., Podolskij, M. (2008). Testing the parametric form of the volatility in continuous time diffusion models–a stochastic process approach. Journal of Econometrics 143(1):56–73.[Crossref], [Web of Science ®] , [Google Scholar]). When applying the test to high frequency data of EUR/USD exchange rate, the empirical results show that the commonly used volatility functions fit more poorly when the data frequency becomes higher, and the general volatility functions fit relatively better than the constant volatility function.     

Bayesian latent variable model for mixed continuous and ordinal responses with possibility of missing responses

A general framework is proposed for joint modelling of mixed correlated ordinal and continuous responses with missing values for responses, where the missing mechanism for both kinds of responses is also considered. Considering the posterior distribution of unknowns given all available information, a Markov Chain Monte Carlo sampling algorithm via winBUGS is used for estimating the posterior distribution of the parameters. For sensitivity analysis to investigate the perturbation from missing at random to not missing at random, it is shown how one can use some elements of covariance structure. These elements associate responses and their missing mechanisms. Influence of small perturbation of these elements on posterior displacement and posterior estimates is also studied. The model is illustrated using data from a foreign language achievement study.     

A test of fit for a continuous distribution based on the empirical convex conditional mean function

Gupta and Kirmani (2008 Gupta, R.C., Kirmani, S.N.U.A. (2008). Characterization based on convex conditional mean function. J. Stat. Plann Inference. 138:964–970.[Crossref], [Web of Science ®] , [Google Scholar]) showed that the convex conditional mean function (CCMF) characterizes the distribution function completely. In this paper, we introduce a consistent estimator of CCMF and call it empirical convex conditional mean function (ECCMF). Then we construct a simple consistent test of fit based on the integrated squared difference between ECCMF and CCMF. The theoretical and asymptotic properties of the estimator ECCMF and the proposed test statistic are studied. The performance of the constructed test is investigated under different distributions using simulations.     

A Bayesian predictive approach to determining the number of components in a mixture distribution

This paper describes a Bayesian approach to mixture modelling and a method based on predictive distribution to determine the number of components in the mixtures. The implementation is done through the use of the Gibbs sampler. The method is described through the mixtures of normal and gamma distributions. Analysis is presented in one simulated and one real data example. The Bayesian results are then compared with the likelihood approach for the two examples.