A Comparative Study of Model-based Tests of Independence for Ordinal Data Using the Bootstrap

When analyzing categorical data using loglinear models in sparse contingency tables, asymptotic results may fail. In this paper the empirical properties of three commonly used asymptotic tests of independence, based on the uniform association model for ordinal data, are investigated by means of Monte Carlo simulation. Five different bootstrapped tests of independence are presented and compared to the asymptotic tests. The comparisons are made with respect to both size and power properties of the tests. Results indicate that the asymptotic tests have poor size control. The test based on the estimated association parameter is severely conservative and the two chi-squared tests (Pearson, likelihood-ratio) are both liberal. The bootstrap tests that either use a parametric assumption or are based on non-pivotal test statistics do not perform better than the asymptotic tests in all situations. The bootstrap tests that are based on approximately pivotal statistics provide both adjustment of size and enhancement of power. These tests are therefore recommended for use in situations similar to those included in the simulation study.     

Size distributions reconsidered

We consider tests of the hypothesis that the tail of size distributions decays faster than any power function. These are based on a single parameter that emerges from the Fisher–Tippett limit theorem, and discriminate between leading laws considered in the literature without requiring fully parametric models/specifications. We study the proposed tests taking into account the higher order regular variation of the size distribution that can lead to catastrophic distortions. The theoretical bias corrections realign successfully nominal and empirical test behavior, and inform a sensitivity analysis for practical work. The methods are used in an examination of the size distribution of cities and firms.     

Analysis of seasonal level shift (SLS) detection in SARIMA models

This study aims at exploring correct identification of seasonal outliers using most commonly applied test statistics. We evaluate the performance of seasonal level shift (SLS) by means of empirical level of significance, power of the test for sensitivity in detecting changes, and the vulnerability to masking of outliers by misspecification frequencies. We observe that the size of SLS affects the sampling distribution of η_SLS (test statistics for SLS detection) in case of SAR (1) and SMA (1) model. The empirical critical values for 1%, 5%, and 10% upper percentiles are higher than the usual cut off points and the empirical level of significance is inversely related to sample size and the model coefficients. The empirical power of the test statistics is not satisfactory at small sample size, and for large model coefficient. η_SLS gets confused with IO. The potential list of types of outliers should retain both IO and SLS as a part of outlier detection procedure for most efficient results. We apply the method suggested by Kaiser and Maravall with five possible types of outliers, that is, AO, IO, LS, TC, and SLS, to a number of quarterly and monthly time series data from Pakistan.     

A wide review on exponentiality tests and two competitive proposals with application on reliability

In this paper, two new statistics based on comparison of the theoretical and empirical distribution functions are proposed to test exponentiality. Critical values are determined by means of Monte Carlo simulations for various sample sizes and different significance levels. Through an extensive simulation study, 50 selected exponentiality tests are studied for a wide collection of alternative distributions. From the empirical power study, it is concluded that, firstly, one of our proposals is preferable for IFR (increasing failure rate) and UFR (unimodal failure rate) alternatives, whereas the other one is preferable for DFR (decreasing failure rate) and BFR (bathtub failure rate) alternatives and, secondly, the new tests can be considered serious and powerful competitors to other existing proposals, since they have the same (or higher) level of performance than the best tests in the statistical literature.     

Estimation of the parameters of the Gompertz distribution under the first failure-censored sampling plan

This article presents the goodness-of-fit tests for the Laplace distribution based on its maximum entropy characterization result. The critical values of the test statistics estimated by Monte Carlo simulations are tabulated for various window and sample sizes. The test statistics use an entropy estimator depending on the window size; so, the choice of the optimal window size is an important problem. The window sizes for yielding the maximum power of the tests are given for selected sample sizes. Power studies are performed to compare the proposed tests with goodness-of-fit tests based on the empirical distribution function. Simulation results report that entropy-based tests have consistently higher power than EDF tests against almost all alternatives considered.     

Simulation Study on Improved Shapiro–Wilk Tests for Normality

The article concerns tests for normality based on the Shapiro–Wilk W statistic. The constants in the test statistic are recalculated as those given in Shapiro and Wilk are incorrect. The empirical significance levels and power of improved tests have been evaluated in simulation study and compared to original ones. The improved tests were also applied to the multivariate case. In this case, we consider two implementations of the W statistic, the first one proposed by Srivastava and Hui and the other by Hanusz and Tarasinska. Empirical size of tests and their power have been compared to the Henze–Zirkler test.     

Testing independence based on Bernstein empirical copula and copula density

In this paper we provide three nonparametric tests of independence between continuous random variables based on the Bernstein copula distribution function and the Bernstein copula density function. The first test is constructed based on a Cramér-von Mises divergence-type functional based on the empirical Bernstein copula process. The two other tests are based on the Bernstein copula density and use Cramér-von Mises and Kullback–Leibler divergence-type functionals, respectively. Furthermore, we study the asymptotic null distribution of each of these test statistics. Finally, we consider a Monte Carlo experiment to investigate the performance of our tests. In particular we examine their size and power which we compare with those of the classical nonparametric tests that are based on the empirical distribution function.     

Estimating and testing conditional sums of means in high dimensional multivariate binary data

We study the estimation of the strength of signals corresponding to the high valued observations in multivariate binary data. These problems can arise in a variety of areas, such as mass spectrometry or function magnetic resonance imaging (fMRI), where the underlying signals could be interpreted as a proxy for biochemical or physiological response to a condition or treatment. More specifically, the problem we consider involves estimating the sum of a collection of binomial probabilities corresponding to large values of the associated binomial random variables. We emphasize the case where the dimension is much greater than the sample size, and most of the probabilities of the events of interest are close to zero. Two estimation approaches are proposed: conditional maximum likelihood and nonparametric empirical Bayes. We use these estimators to construct a test of homogeneity for two groups of high dimensional multivariate binary data. Simulation studies on the size and power of the proposed tests are given, and the tests are demonstrated using mass spectrometry data from a breast cancer study.     

A Comparison of Analysis of Covariate-Adjusted Residuals and Analysis of Covariance

Various methods to control the influence of a covariate on a response variable are compared. These methods are ANOVA with or without homogeneity of variances (HOV) of errors and Kruskal–Wallis (K–W) tests on (covariate-adjusted) residuals and analysis of covariance (ANCOVA). Covariate-adjusted residuals are obtained from the overall regression line fit to the entire data set ignoring the treatment levels or factors. It is demonstrated that the methods on covariate-adjusted residuals are only appropriate when the regression lines are parallel and covariate means are equal for all treatments. Empirical size and power performance of the methods are compared by extensive Monte Carlo simulations. We manipulated the conditions such as assumptions of normality and HOV, sample size, and clustering of the covariates. The parametric methods on residuals and ANCOVA exhibited similar size and power when error terms have symmetric distributions with variances having the same functional form for each treatment, and covariates have uniform distributions within the same interval for each treatment. In such cases, parametric tests have higher power compared to the K–W test on residuals. When error terms have asymmetric distributions or have variances that are heterogeneous with different functional forms for each treatment, the tests are liberal with K–W test having higher power than others. The methods on covariate-adjusted residuals are severely affected by the clustering of the covariates relative to the treatment factors when covariate means are very different for treatments. For data clusters, ANCOVA method exhibits the appropriate level. However, such a clustering might suggest dependence between the covariates and the treatment factors, so makes ANCOVA less reliable as well.     

Simple Robust Tests for Autocorrelated Errors in Time Series Design Intervention Models

A simple, robust test for the autocorrelation parameter in an intervention time-series model (AB design) is proposed. It is analogous to the traditional tests and can easily be computed by using the freeware R. In the same way as traditional tests of autocorrelation are based on least squares (LS) fits of a linear model, our robust test is based on the highly efficient Wilcoxon fit of the linear model. We present the results of a Monte Carlo study which show that our robust test inherits the good efficiency properties of this Wilcoxon fit. Its empirical power is only slightly less than the empirical power of the least squares test over situations with normally distributed errors while it exhibited much more power over situations with error distributions having tails heavier than those of a normal distribution. It also showed robustness of validity over all null situations simulated. We also present the results of the application of our test to a real data set which illustrates the robustness of our test.     

On Testing Equality of Distributions of Technical Efficiency Scores

The challenge of the econometric problem in production efficiency analysis is that the efficiency scores to be analyzed are unobserved. Statistical properties have recently been discovered for a type of estimator popular in the literature, known as data envelopment analysis (DEA). This opens up a wide range of possibilities for well-grounded statistical inference about the true efficiency scores from their DEA estimates. In this paper we investigate the possibility of using existing tests for the equality of two distributions in such a context. Considering the statistical complications pertinent to our context, we consider several approaches to adapting the Li test to the context and explore their performance in terms of the size and power of the test in various Monte Carlo experiments. One of these approaches shows good performance for both the size and the power of the test, thus encouraging its use in empirical studies. We also present an empirical illustration analyzing the efficiency distributions of countries in the world, following up a recent study by Kumar and Russell (2002), and report very interesting results.     

The finite-sample performance of robust unit root tests

This paper investigates the relative small sample performance of several robust unit root tests by means of a simulation study. It is confirmed that the traditional least-squares based Dickey-Fuller test has substantially lower power than several robust alternatives if the error distribution is fat-tailed while its power gain is small at the normal model. Particularly good results are achieved by a quasi-maximum likelihood test. However, all robust tests under consideration exhibit severe size distortions if the disturbances follow a skewed distribution. Moreover, under additive outliers, robust tests fail to produce stable sizes and good power properties. Consequently, the value of using robust unit root tests depends heavily of the type of nonnormality at hand.     

A two-sample test for mean functions with increasing number of projections

We propose a test for equality of two means when data are functions and obtain the asymptotic properties of the test statistic as data dimension increases with the sample size. We also derive the asymptotic power of the test under some local alternatives and show that the test statistic is root-n consistent. A simulation study is conducted to evaluate the performance of the test numerically and to compare the proposed test with other existing four popular tests.     

Bootstrap methods for single structural change tests: power versus corrected size and empirical illustration

This paper assesses the performance of tests for a single structural change at unknown date when regressors are stationary, trending and when they have a break in mean. Size and power of the test procedures are compared in a simulation setup particularly aimed at autoregressive models using their limiting distribution and some bootstrap approximations. The comparisons are performed using graphical methods, namely P value discrepancy plots and size–power curves. The simulation study gives some interesting insights to the test procedures. Indeed, it documents that tests based on the conventional asymptotic distribution are oversized in small samples. The size correction is achieved by some bootstrap methods which appear to possess reasonable size properties. For the power study, the proposed bootstrap method improves on the asymptotic approximations of some tests for heteroskedastic regression errors especially when there is a mean-shift in the regressors. This result has not been found for the case of i.i.d. errors where the bootstrap tests have the same power properties as the tests based on the asymptotic approximations. We finally study the relationship between two monthly US interest rates. The results show that such relationship has been altered by a regime-shift located in May 1981.     

Effect size for comparing two or more normal distributions based on maximal contrasts in outcomes

Effect size is a concept that can be especially useful in bioequivalence and studies designed to find important and not just statistically significant differences among responses to treatments based on independent random samples. We develop and explore a new effect size related to a maximal superiority ordering for assessing the separation among two or more normal distributions, possibly having different means and different variances. Confidence intervals and tests of hypothesis for this effect size are developed using a p value obtained by averaging over a distribution on variances. Since there is almost always some difference among treatments, instead of the usual hypothesis test of exactly no effect, researchers should consider testing that an appropriate effect size has at least, or at most, some meaningful magnitude, when one is available, possibly established using the framework developed here. A simulation study of type I error rate, power and interval length is presented. R-code for constructing the confidence intervals and carrying out the tests here can be downloaded from Author’s website.     

Empirical power study of a multi-sample test of exponentiality based on spacings

We present the results of an empirical power study of a new multi-sample test of exponentiality due to (1980)and show that this test is on the whole considerably more powerful than the other prominent tests considered by Dyer and Harbin (1981).     

Recent tests of exponentiality against IFR alternatives: a survey

Testing of various classes of life distributions has been a subject of investigation for more than four decades. In this study, we restrict ourselves to the problem of testing exponentiality (which essentially means no aging) against positive aging, which is captured by the class of increasing failure rate alternatives. Recent tests are discussed and compared. The empirical size of the tests is obtained by simulation. Power computations, using simulations, are done for each test procedure. These comparisons are done both for small and large sample sizes. Suggestions are made regarding the choice of the test when a particular alternative is suspected.     

A class of robust stepwise alternatives to Hotelling's T 2 tests

Hotelling's T 2 test is known to be optimal under multivariate normality and is reasonably validity-robust when the assumption fails. However, some recently introduced robust test procedures have superior power properties and reasonable type I error control with non-normal populations. These, including the tests due to Tiku & Singh (1982), Tiku & Balakrishnan (1988) and Mudholkar & Srivastava (1999b, c), are asymptotically valid but are useful with moderate size samples only if the population dimension is small. A class of B-optimal modifications of the stepwise alternatives to Hotellings T 2 introduced by Mudholkar & Subbaiah (1980) are simple to implement and essentially equivalent to the T 2 test even with small samples. In this paper we construct and study the robust versions of these modified stepwise tests using trimmed means instead of sample means. We use the robust one- and two-sample trimmed- t procedures as in Mudholkar et al. (1991) and propose statistics based on combining them. The results of an extensive Monte Carlo experiment show that the robust alternatives provide excellent type I error control and a substantial gain in power.     

A simple investigation of the Granger-causality test in integrated-cointegrated VAR systems

The size and power of various generalization tests for the Granger-causality in integrated-cointegrated VAR systems are considered. By using Monte Carlo methods, properties of eight versions of the test are studied in two different forms, the standard form and the modified form by Dolado & Lütkepohl (1996) in a study confined to properties of the Wald test only. In their study as well as in ours, both the standard and the modified Wald tests are shown to perform badly especially in small samples. We find, however, that the corrected LR tests exhibit correct size even in small samples. The power of the test is higher when the true VAR(2) model is estimated, and the modified test loses information by estimating the extra coefficients. The same is true when considering the power results in the VAR(3) model, and the power of the tests is somewhat lower than those in the VAR(2).     

Measuring Nonlinear Granger Causality in Mean

We propose model-free measures for Granger causality in mean between random variables. Unlike the existing measures, ours are able to detect and quantify nonlinear causal effects. The new measures are based on nonparametric regressions and defined as logarithmic functions of restricted and unrestricted mean square forecast errors. They are easily and consistently estimated by replacing the unknown mean square forecast errors by their nonparametric kernel estimates. We derive the asymptotic normality of nonparametric estimator of causality measures, which we use to build tests for their statistical significance. We establish the validity of smoothed local bootstrap that one can use in finite sample settings to perform statistical tests. Monte Carlo simulations reveal that the proposed test has good finite sample size and power properties for a variety of data-generating processes and different sample sizes. Finally, the empirical importance of measuring nonlinear causality in mean is also illustrated. We quantify the degree of nonlinear predictability of equity risk premium using variance risk premium. Our empirical results show that the variance risk premium is a very good predictor of risk premium at horizons less than 6 months. We also find that there is a high degree of predictability at the 1-month horizon, that can be attributed to a nonlinear causal effect. Supplementary materials for this article are available online.