Levene type tests for the ratio of two scales

Tests for the equality of variances are of interest in many areas such as quality control, agricultural production systems, experimental education, pharmacology, biology, as well as a preliminary to the analysis of variance, dose–response modelling or discriminant analysis. The literature is vast. Traditional non-parametric tests are due to Mood, Miller and Ansari–Bradley. A test which usually stands out in terms of power and robustness against non-normality is the W50 Brown and Forsythe [Robust tests for the equality of variances, J. Am. Stat. Assoc. 69 (1974), pp. 364–367] modification of the Levene test [Robust tests for equality of variances, in Contributions to Probability and Statistics, I. Olkin, ed., Stanford University Press, Stanford, 1960, pp. 278–292]. This paper deals with the two-sample scale problem and in particular with Levene type tests. We consider 10 Levene type tests: the W50, the M50 and L50 tests [G. Pan, On a Levene type test for equality of two variances, J. Stat. Comput. Simul. 63 (1999), pp. 59–71], the R-test [R.G. O'Brien, A general ANOVA method for robust tests of additive models for variances, J. Am. Stat. Assoc. 74 (1979), pp. 877–880], as well as the bootstrap and permutation versions of the W50, L50 and R tests. We consider also the F-test, the modified Fligner and Killeen [Distribution-free two-sample tests for scale, J. Am. Stat. Assoc. 71 (1976), pp. 210–213] test, an adaptive test due to Hall and Padmanabhan [Adaptive inference for the two-sample scale problem, Technometrics 23 (1997), pp. 351–361] and the two tests due to Shoemaker [Tests for differences in dispersion based on quantiles, Am. Stat. 49(2) (1995), pp. 179–182; Interquantile tests for dispersion in skewed distributions, Commun. Stat. Simul. Comput. 28 (1999), pp. 189–205]. The aim is to identify the effective methods for detecting scale differences. Our study is different with respect to the other ones since it is focused on resampling versions of the Levene type tests, and many tests considered here have not ever been proposed and/or compared. The computationally simplest test found robust is W50. Higher power, while preserving robustness, is achieved by considering the resampling version of Levene type tests like the permutation R-test (recommended for normal- and light-tailed distributions) and the bootstrap L50 test (recommended for heavy-tailed and skewed distributions). Among non-Levene type tests, the best one is the adaptive test due to Hall and Padmanabhan.     

Improved likelihood inference in beta regression

We consider the issue of performing accurate small-sample likelihood-based inference in beta regression models, which are useful for modelling continuous proportions that are affected by independent variables. We derive small-sample adjustments to the likelihood ratio statistic in this class of models. The adjusted statistics can be easily implemented from standard statistical software. We present Monte Carlo simulations showing that inference based on the adjusted statistics we propose is much more reliable than that based on the usual likelihood ratio statistic. A real data example is presented.     

A Method for Designing Three-Hypothesis Test Problems and Sequential Schemes

In applications, a two-sided hypothesis test problem sometimes needs to be changed to a three-hypothesis one with the two alternative hypotheses properly selected. In this article, we obtain the hypothesis design and the three-hypothesis sequential test scheme under the Koopman–Darmois distribution by solving a system of equations that meet requirements on the error rates and average sample number. This method provides a useful guide for practitioners to design hypotheses in multihypothesis test problems with controlled error rates and sampling cost. Formulas of the scheme's error rates and average sample number are obtained using numerical quadrature for the discrete-time situation.     

A NEW TEST FOR EQUALITY OF VARIANCES FOR k NORMAL POPULATIONS

ABSTRACT

A simple test based on Gini's mean difference is proposed to test the hypothesis of equality of population variances. Using 2000 replicated samples and empirical distributions, we show that the test compares favourably with Bartlett's and Levene's test for the normal population. Also, it is more powerful than Bartlett's and Levene's tests for some alternative hypotheses for some non-normal distributions and more robust than the other two tests for large sample sizes under some alternative hypotheses. We also give an approximate distribution to the test statistic to enable one to calculate the nominal levels and P-values.     

Comparing two dependent groups via quantiles

This paper considers two general ways dependent groups might be compared based on quantiles. The first compares the quantiles of the marginal distributions. The second focuses on the lower and upper quantiles of the usual difference scores. Methods for comparing quantiles have been derived that typically assume that sampling is from a continuous distribution. There are exceptions, but generally, when sampling from a discrete distribution where tied values are likely, extant methods can perform poorly, even with a large sample size. One reason is that extant methods for estimating the standard error can perform poorly. Another is that quantile estimators based on a single-order statistic, or a weighted average of two-order statistics, are not necessarily asymptotically normal. Our main result is that when using the Harrell–Davis estimator, good control over the Type I error probability can be achieved in simulations via a standard percentile bootstrap method, even when there are tied values, provided the sample sizes are not too small. In addition, the two methods considered here can have substantially higher power than alternative procedures. Using real data, we illustrate how quantile comparisons can be used to gain a deeper understanding of how groups differ.     

Inference for the common mean of several Birnbaum–Saunders populations

The Birnbaum–Saunders distribution is a widely used distribution in reliability applications to model failure times. For several samples from possible different Birnbaum–Saunders distributions, if their means can be considered as the same, it is of importance to make inference for the common mean. This paper presents procedures for interval estimation and hypothesis testing for the common mean of several Birnbaum–Saunders populations. The proposed approaches are hybrids between the generalized inference method and the large sample theory. Some simulation results are conducted to present the performance of the proposed approaches. The simulation results indicate that our proposed approaches perform well. Finally, the proposed approaches are applied to analyze a real example on the fatigue life of 6061-T6 aluminum coupons for illustration.     

基于软件无线电的LTE-Advanced实时测试平台同步的设计与实现

基于GNU Radio和USRP以软件无线电方式构建了一个移动通信技术实时测试平台.在此平台上，利用LTE无线帧中的主/辅同步信号实现了符合LTE-Advanced物理层规范的实时通信.该测试平台为真实物理环境下移动通信技术的设计、测试及验证提供了技术手段.     

The Effects of Misconceptions on the Properties of Friedman's Test

Friedman's test is a widely used rank-based alternative to the analysis of variance (ANOVA) F-test for identifying treatment differences in a randomized complete block design. Many texts provide incomplete or misleading information about when Friedman's test may be appropriately applied. We discuss the assumptions needed for the test and common misconceptions. We show via simulation that when the variance or skew of the treatment distributions differ, application of Friedman's test to detect differences in treatment location can result in Type I error probabilities larger than the nominal α, and even when α is unaffected, the power of the test can be less than expected.     

To Pool or Not to Pool: The Quality Bank Case

Drawing on a recent hearing before the Federal Energy Regulatory Commission, this article illustrates how two statisticians on opposing sides, using two different approaches to the problem, arrived at the same estimator. The differences between the two approaches are highlighted and a proof for the equivalence of the two estimators is given.