One-sample location tests for multilevel data

In this paper, we consider testing the location parameter with multilevel (or hierarchical) data. A general family of weighted test statistics is introduced. This family includes extensions to the case of multilevel data of familiar procedures like the t, the sign and the Wilcoxon signed-rank tests. Under mild assumptions, the test statistics have a null limiting normal distribution which facilitates their use. An investigation of the relative merits of selected members of the family of tests is achieved theoretically by deriving their asymptotic relative efficiency (ARE) and empirically via a simulation study. It is shown that the performance of a test depends on the clusters configurations and on the intracluster correlations. Explicit formulas for optimal weights and a discussion of the impact of omitting a level are provided for 2 and 3-level data. It is shown that using appropriate weights can greatly improve the performance of the tests. Finally, the use of the new tests is illustrated with a real data example.     

Tests for the total time on test transform order

In this paper we consider two test statistics for testing the strict TTT transform order between two life distributions of interest. We give their asymptotic distributions and compare our tests with some other related tests in terms of Pitman's asymptotic efficiency. Also we present some results to show the performance and the asymptotic normality of our tests.     

Distribution-free test for symmetry based on Bonferroni's measure

We propose a test based on Bonferroni's measure of skewness. The test detects the asymmetry of a distribution function about an unknown median. We study the asymptotic distribution of the given test statistic and provide a consistent estimate of its variance. The asymptotic relative efficiency of the proposed test is computed along with Monte Carlo estimates of its power. This allows us to perform a comparison of the test based on Bonferroni's measure with other tests for symmetry.     

Intermediate efficiency by shifting alternatives and evaluation of power

We study a modification of the notion of asymptotic intermediate efficiency of statistical tests by defining it in terms of shifting alternatives. We prove a theorem providing conditions for its existence and show that this modification is closely related to the original Kallenberg's asymptotic intermediate efficiency in a quite general setting. Next, we find estimates for differences between powers of the Neyman–Pearson test under original alternatives and that of a given test under shifted alternatives. We also present some simulation results. They attest to consistency of theoretical results with observed empirical powers for quite small sample sizes.     

Moment inequalities for HNBRUE with hypothesis testing application

The paper deals with life distributions which are harmonic new better than renewal used in expectation (HNBRUE). The goal is to derive moment inequalities and use them for further analysis of the class HNBRUE. We establish new characterization of exponentiality versus HNBRUE. Pitman’s asymptotic relative efficiency is employed to assess the performance of the proposed test with respect to other available tests. Finally, we carried out numerical simulation to produce table for the critical values of the test.     

Guarded Weights of Evidence and Acceptability Profiles Based on Signs: II. Asymptotic Results

The concepts of guarded weights of evidence and acceptability profiles have been extended to the distribution-free setting in Dollinger, Kulinskaya & Staudte (1999). In that first of two parts the advantages of these concepts relative to traditional ones such as p -values and confidence intervals derived from hypothesis tests are emphasized for small samples. Here in Part II asymptotic expressions are found for guarded weights of evidence for hypothesesregarding the median of a symmetric distribution and related acceptability profiles for the median. It is also seen that for local alternatives the efficacy and Pitman asymptotic relative efficiency of the sign statistic for testing hypotheses carries over to the more general setting of guarded weights of evidence.     

Optimal testing in a three way anova model

We consider likelihood ratio, score and Wald tests for a three-way random effects ANOVA model. Competitor tests are compared using criteria such as small sample power, asymptotic relative efficiency, and convenient null distribution. The final choice is between a new test and two tests long used in practice.     

Generalized information matrix tests for copulas

We propose a family of goodness-of-fit tests for copulas. The tests use generalizations of the information matrix (IM) equality of White and so relate to the copula test proposed by Huang and Prokhorov. The idea is that eigenspectrum-based statements of the IM equality reduce the degrees of freedom of the test’s asymptotic distribution and lead to better size-power properties, even in high dimensions. The gains are especially pronounced for vine copulas, where additional benefits come from simplifications of score functions and the Hessian. We derive the asymptotic distribution of the generalized tests, accounting for the nonparametric estimation of the marginals and apply a parametric bootstrap procedure, valid when asymptotic critical values are inaccurate. In Monte Carlo simulations, we study the behavior of the new tests, compare them with several Cramer–von Mises type tests and confirm the desired properties of the new tests in high dimensions.     

Testing hypotheses about the common mean of two normal populations

Two simple tests which allow for unequal sample sizes are considered for testing hypothesis for the common mean of two normal populations. The first test is an exact test of size a based on two available t-statistics based on single samples made exact through random allocation of α among the two available t-tests. The test statistic of the second test is a weighted average of two available t-statistics with random weights. It is shown that the first test is more efficient than the available two t-tests with respect to Bahadur asymptotic relative efficiency. It is also shown that the null distribution of the test statistic in the second test, which is similar to the one based on the normalized Graybill-Deal test statistic, converges to a standard normal distribution. Finally, we compare the small sample properties of these tests, those given in Zhou and Mat hew (1993), and some tests given in Cohen and Sackrowitz (1984) in a simulation study. In this study, we find that the second test performs better than the tests given in Zhou and Mathew (1993) and is comparable to the ones given in Cohen and Sackrowitz (1984) with respect to power..     

Asymptotic properties of a goodness-of-fit test based on maximum correlations

We study the efficiency properties of the goodness-of-fit test based on the Q _n statistic introduced in Fortiana and Grané [Goodness-of-fit tests based on maximum correlations and their orthogonal decompositions, J. R. Stat. Soc. B 65 (2003), pp. 115–126] using the concepts of Bahadur asymptotic relative efficiency and Bahadur asymptotic optimality. We compare the test based on this statistic with those based on the Kolmogorov–Smirnov, the Cramér-von Mises criterion and the Anderson–Darling statistics. We also describe the distribution families for which the test based on Q _n is locally asymptotically optimal in the Bahadur sense and, as an application, we use this test to detect the presence of hidden periodicities in a stationary time series.     

Optimally weighted,fixed sequence and gatekeeper multiple testing procedures

We consider a class of closed multiple test procedures indexed by a fixed weight vector. The class includes the Holm weighted step-down procedure, the closed method using the weighted Fisher combination test, and the closed method using the weighted version of Simes’ test. We show how to choose weights to maximize average power, where “average power” is itself weighted by importance assigned to the various hypotheses.Numerical computations suggest that the optimal weights for the multiple test procedures tend to certain asymptotic configurations. These configurations offer numerical justification for intuitive multiple comparisons methods, such as downweighting variables found insignificant in preliminary studies, giving primary variables more emphasis, gatekeeping test strategies, pre-determined multiple testing sequences, and pre-determined sequences of families of tests. We establish that such methods fall within the envelope of weighted closed testing procedures, thus providing a unified view of fixed sequences, fixed sequences of families, and gatekeepers within the closed testing paradigm. We also establish that the limiting cases control the familywise error rate (or FWE), using well-known results about closed tests, along with the dominated convergence theorem.     

On the Usefulness or Lack Thereof of Optimality Criteria for Structural Change Tests

Elliott and Müller (2006) considered the problem of testing for general types of parameter variations, including infrequent breaks. They developed a framework that yields optimal tests, in the sense that they nearly attain some local Gaussian power envelop. The main ingredient in their setup is that the variance of the process generating the changes in the parameters must go to zero at a fast rate. They recommended the so-called qL?L test, a partial sums type test based on the residuals obtained from the restricted model. We show that for breaks that are very small, its power is indeed higher than other tests, including the popular sup-Wald (SW) test. However, the differences are very minor. When the magnitude of change is moderate to large, the power of the test is very low in the context of a regression with lagged dependent variables or when a correction is applied to account for serial correlation in the errors. In many cases, the power goes to zero as the magnitude of change increases. The power of the SW test does not show this non-monotonicity and its power is far superior to the qL?L test when the break is not very small. We claim that the optimality of the qL?L test does not come from the properties of the test statistics but the criterion adopted, which is not useful to analyze structural change tests. Instead, we use fixed-break size asymptotic approximations to assess the relative efficiency or power of the two tests. When doing so, it is shown that the SW test indeed dominates the qL?L test and, in many cases, the latter has zero relative asymptotic efficiency.     

Augmenting lehmann alternatives for improving asymptotic efficiency of rank tests

Locally most powerful tests for augmented simple Lehmann alternatives are obtained. These tests turn out to be linearcombinations of the Savage and Mann-Whitney-Wilcoxon test criteria. We study their performance in terms of the asymptotic efficiency relative to their parametric competitors against location and scale alternatives. For small sample sizes, critical points of a couple of test procedures are given.     

Testing Covariate Effects in Aalen's Linear Hazard Model

The performance of tests in Aalen's linear regression model is studied using asymptotic power calculations and stochastic simulation. Aalen's original least squares test is compared to two modifications: a weighted least squares test with correct weights and a test where the variance is re-estimated under the null hypothesis. The test with re-estimated variance provides the highest power of the tests for the setting of this paper, and the gain is substantial for covariates following a skewed distribution like the exponential. It is further shown that Aalen's choice for weight function with re-estimated variance is optimal in the one-parameter case against proportional alternatives.     

Robust analysis of variance for a randomized block design

In this paper we study the asymptotic theory of M-estimates and their associated tests for a one-factor experiment in a randomized block design. In this case one natural asymptotic theory corresponds to leaving the number of treatments fixed and letting the number of blocks tend to infinity. The classic asymptotic theory of M-estimates does not apply here, because the number of parameters and the number of observations are of the same order. In this paper we prove the consistency and asymptotic normality of the estimators of the treatment effects. It turns out that the asymptotic covariance matrix of the treatment effects estimators differs from the one derived from the classic theory of M-estimates for the linear model with a fixed number of parameters. We also study a test for treatment effects derived from M-estimates and we compare by Monte Carlo simulation the efficiency of this test with respect to the F-test, the Friedman test and the test based on aligned ranks.     

Evaluating the Accuracy of Small P‐Values In Genetic Association Studies Using Edgeworth Expansions

The asymptotic distributions of many classical test statistics are normal. The resulting approximations are often accurate for commonly used significance levels, 0.05 or 0.01. In genome‐wide association studies, however, the significance level can be as low as 1×10⁻⁷, and the accuracy of the p‐values can be challenging. We study the accuracies of these small p‐values are using two‐term Edgeworth expansions for three commonly used test statistics in GWAS. These tests have nuisance parameters not defined under the null hypothesis but estimable. We derive results for this general form of testing statistics using Edgeworth expansions, and find that the commonly used score test, maximin efficiency robust test and the chi‐squared test are second order accurate in the presence of the nuisance parameter, justifying the use of the p‐values obtained from these tests in the genome‐wide association studies.     

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.     

Non parametric portmanteau tests for detecting non linearities in high dimensions

ABSTRACT

We propose two non parametric portmanteau test statistics for serial dependence in high dimensions using the correlation integral. One test depends on a cutoff threshold value, while the other test is freed of this dependence. Although these tests may each be viewed as variants of the classical Brock, Dechert, and Scheinkman (BDS) test statistic, they avoid some of the major weaknesses of this test. We establish consistency and asymptotic normality of both portmanteau tests. Using Monte Carlo simulations, we investigate the small sample properties of the tests for a variety of data generating processes with normally and uniformly distributed innovations. We show that asymptotic theory provides accurate inference in finite samples and for relatively high dimensions. This is followed by a power comparison with the BDS test, and with several rank-based extensions of the BDS tests that have recently been proposed in the literature. Two real data examples are provided to illustrate the use of the test procedure.     

Parametric bootstrap and approximate tests for two Poisson variates

The parametric bootstrap tests and the asymptotic or approximate tests for detecting difference of two Poisson means are compared. The test statistics used are the Wald statistics with and without log-transformation, the Cox F statistic and the likelihood ratio statistic. It is found that the type I error rate of an asymptotic/approximate test may deviate too much from the nominal significance level α under some situations. It is recommended that we should use the parametric bootstrap tests, under which the four test statistics are similarly powerful and their type I error rates are all close to α. We apply the tests to breast cancer data and injurious motor vehicle crash data.     

Smooth goodness of fit tests for the weibull distribution with singly censored data

The smooth goodness of fit tests are generalized to singly censored data and applied to the problem of testing Weibull (or extreme value) fit. Smooth tests, Pearson-type tests, and the spacings tests proposed by Mann, Schemer, and Fertig (1973) are compared on the basis of local asymptotic relative efficiency with respect to the asymptotic best test against generalized gamma alternatives, The smooth test of order one Is found to be most efficient for the generalized gamma alternatives.