A bivariate signed rank test for two sample location problem

An affine-invariant signed rank test for the difference in location between two symmetric populations is proposed. The proposed test statistic is compared with Hotelling's T2 test statistic, Mardia's(1967)test statistic, Peters-Randles(1991) test statistic and Wilcoxon's rank sum test statistic using a Monte Carlo Study. It performs better than Mardia's test statistic under almost all populations considered. Under the bivariate normal distribution, it performs better than other test statistics compared for small differences in location between two populations except Hotelling's T2. It performs better than all statistics, including Hotelling's T , for sample size 15 when samples are drawn from Pearson type.     

A weighted test of internal symmetry

This study examines extensions of McNemar's Test with multinomial responses, and proposes a linear weighting scheme, based on the distance of the response change, that is applied to one of these extensions (Bowker's test). This weighted version of Bowker's test is then appropriate for ordinal response variables. A Monte Carlo simulation was conducted to examine the Type I error rate of the weighted Bowker's test for a cross-classification table based on a five-category ordinal response scale. The weighted Bowker's test was also applied to a data set involving change in student attitudes towards mathematics. The results of the weighted Bowker's test were compared with the results of Bowker's test applied to the same set of data.     

A new revised version of McNemar's test for paired binary data

By considering separately B and C, the frequencies of individuals who consistently gave positive or negative answers in before and after responses, a new revised version of McNemar's test is derived. It improves upon Lu's revised formula, which considers B and C together. When both B and C are 0, the new revised version produces the same results as McNemar's test. When one of B and C is 0, the new revised test produces the same results as Lu's version. Compared to Lu's version, the new revised test is a more complete revision of McNemar's test.     

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.     

Generalisation of Mood's Test to Factorial Designs to Detect Heterogeneity of Variances

Mood's test, which is a relatively old test (and the oldest non‐parametric test among those tests in its class) for determining heterogeneity of variance, is still being widely used in different areas such as biometry, biostatistics and medicine. Although it is a popular test, it is not suitable for use on a two‐way factorial design. In this paper, Mood's test is generalised to the 2 × 2 factorial design setting and its performance is compared with that of Klotz's test. The power and robustness of these tests are examined in detail by means of a simulation study with 10,000 replications. Based on the simulation results, the generalised Mood's and Klotz's tests can especially be recommended in settings in which the parent distribution is symmetric. As an example application we analyse data from a multi‐factor agricultural system that involves chilli peppers, nematodes and yellow nutsedge. This example dataset suggests that the performance of the generalised Mood test is in agreement with that of the generalised Klotz's test.     

Separated hypotheses testing for autoregressive models with non-negative residuals

Normal residual is one of the usual assumptions in autoregressive model but sometimes in practice we are faced with non-negative residuals. In this paper, we have derived modified maximum likelihood estimators of parameters of the residuals and autoregressive coefficient. Also asymptotic distribution of modified maximum likelihood estimators in both stationary and non-stationary models are computed. So that, we can derive asymptotic distribution of unit root, Vuong's and Cox's tests statistics in stationary situation. Using simulation, it shows that Akaike information criterion and Vuong's test work to select the optimal autoregressive model with non-negative residuals. Sometimes Vuong's test select two competing models as equivalent models. These models may be suitable or unsuitable equivalent models. So we consider Cox's test to make inference after model selection. Kolmogorov–Smirnov test confirms our results. Also we have computed tracking interval for competing models to choosing between two close competing models when Vuong's test and Cox's test cannot detect the differences.     

On the independence problem and kendall's tau

A modification of Kendall's test for independence is described which allows one to test for association in a bivariate distribution as measured by Kendall's tau, a property not shared by Kendall's procedure. The proposed procedure, however, still provides an exactly distribution-free test of independence. The test procedure is inverted to obtain a confidence interval for tau which has distinct advantages over the currently employed confidence interval.     

The union-intersection principle applied to the test of dimensionality

In the first section Anderson-Rao-Fujikoshi's test statistics for testing the hypothesis of dimensionality are reviewed and then Olkin-Tomsky's generalized union-intersection principle is applied to show that a new class of test statistics for testing the hypothesis of dimensionality are derived which includes the likelihood ratio test statistics, the trace test statistics and a version of ROY'S maximum root test statistics.     

A test for bivariate two sample location problem

A consistent test for difference in locations between two bivariate populations is proposed, The test is similar as the Mann-Whitney test and depends on the exceedances of slopes of the two samples where slope for each sample observation is computed by taking the ratios of the observed values. In terms of the slopes, it reduces to a univariate problem, The power of the test has been compared with those of various existing tests by simulation. The proposed test statistic is compared with Mardia's(1967) test statistics, Peters-Randies(1991) test statistic, Wilcoxon's rank sum test. statistic and Hotelling' T² test statistic using Monte Carlo technique. It performs better than other statistics compared for small differences in locations between two populations when underlying population is population 7(light tailed population) and sample size 15 and 18 respectively. When underlying population is population 6(heavy tailed population) and sample sizes are 15 and 18 it performas better than other statistic compared except Wilcoxon's rank sum test statistics for small differences in location between two populations. It performs better than Mardia's(1967) test statistic for large differences in location between two population when underlying population is bivariate normal mixture with probability p=0.5, population 6, Pearson type II population and Pearson type VII population for sample size 15 and 18 .Under bivariate normal population it performs as good as Mardia' (1967) test statistic for small differences in locations between two populations and sample sizes 15 and 18. For sample sizes 25 and 28 respectively it performs better than Mardia's (1967) test statistic when underlying population is population 6, Pearson type II population and Pearson type VII population     

Testing Wagner's Law in India: A cointegration and causality analysis

The present study empirically analyzes the validity of Wagner's Law for Indian economy. With the use of annual time series data from 1970–71 to 2013–14, all the six versions of Wagner's Law have been analyzed to test the relationship between government expenditure and gross domestic product. Wagner's Law states that the economic growth is the causative factor of the growth of the public expenditure. The study applied the unit root test and cointegration test to find the long-run relationship between government expenditure and gross domestic product. The present study employed the various econometric techniques such as unit root test, cointegration, and causality analysis for empirical analysis. The empirical analysis under study inferred mixed results of Wagner's Law for Indian economy, where four versions, namely Peacock, Gupta, Guffman, and Musgrave, found valid for Indian economy. The study concluded that the Wagner's Law is valid for the Indian economy except the Pryor and Mann Versions of the Wagner's Law.     

Nonparametric Applications of Shaffer's Extension of Dennett's Procedure

Shaffer's extensions and generalization of Dunnett's procedure are shown to be applicable in several nonparametric data analyses. Applications are considered within the context of the Kruskal-Wallis one-way analysis of variance (ANOVA) test for ranked data, Friedman's two-way ANOVA test for ranked data, and Cochran's test of change for dichotomous data.     

On the optimality of rao's statistic

This paper establishes the asymptotic optimality of Rao's

test within a very wide class of tests that Includes the likeli hood ratio test and Wald's test. An expression for the defici¬ency of the tests in this class relative to Rao's test has also been obtained     

A complete solution to the problem of robustness of Grubbs's test

Srivastava (1980) showed that Grubbs's test for detecting a univariate outlier is robust against the effect of intraclass correlation structure. Young, Pavur, and Marco (1989) extended this result by proving that both the significance level and the power of Grubbs's test remain unchanged within a wider family of dispersion matrices, introduced by Baldessari (1966) in a different context. In this note, we derive a complete solution of the problem by establishing that the characteristics of Grubbs's test are invariant with respect to a given dispersion matrix if and only if it has Baldessari's structure.     

A circular-cone test for testing homogeneity against a simple tree order

Several methods exist for the problem of testing the equality of several treatments against the one-sided alternative that the treatments are better than the control. These methods include Dunnett's test, Bartholomew's likelihood-ratio test, the Abelson-Tukey-Schaafsma-Smid optimal-contrast test, and the multiple-contrast test of Mukerjee, Robertson, and Wright. A new test is proposed based on an approximation of the likelihood-ratio test of Bartholomew. This test involves using a circular cone in place of the alternative-hypothesis cone. The circular-cone test has excellent power characteristics similar to those of Bartholomew's test. Moreover, it has the advantages of being simpler to compute and may be used with unequal sample sizes.     

A note on testing the regression functions via nonparametric smoothing

The authors present a consistent lack‐of‐fit test in nonlinear regression models. The proposed procedure possesses some nice properties of Zheng's test such as the consistency, the ability to detect any local alternatives approaching the null at rates slower than the parametric rate. What's more, for a predetermined kernel function, the proposed test is more powerful than Zheng's test and the validity of these findings is confirmed by the simulation studies and a real data example. In addition, the authors find out a close connection between the choices of normal kernel functions and the bandwidths. The Canadian Journal of Statistics 39: 108–125; 2011 © 2011 Statistical Society of Canada     

Some modifications of levene's test of variance homogeneity

Various modifications of Levene's test of homogeneity of variance are proposed and evaluated, including the use of (i) Satterthwaite's method for correcting degrees of freedom, (ii) data-based power transformations, and (iii) computer simulation. Satterthwaite's correction is shown to be effective in controlling the slightly liberal behaviour of Levene's test in small samples. The use of power transformation turns out to make the test extremely liberal and is not recommended. Modifications which employ computer simulation are exact under normality, and one version, at least, is asymptotically robust of nonnormality. They also posses excellent small-sample properties.     

The generalized inference on the sign testing problem about the normal variances

For the sign testing problem about the normal variances, we develop the heuristic testing procedure based on the concept of generalized test variable and generalized p-value. A detailed simulation study is conducted to empirically investigate the performance of the proposed method. Through the simulation study, especially in small sample sizes, the proposed test not only adequately controls empirical size at the nominal level, but also uniformly more powerful than likelihood ratio test, Gutmann's test, Li and Sinha's test and Liu and Chan's test, showing that the proposed method can be recommended in practice. The proposed method is illustrated with the published data.     

A NOTE ON MULTIVARIATE LINEAR MODELS WITH NON-ADDITIVITY

Tukey's non-additivity test in an analysis of variance model is extended to a multivariate linear model with covariates. If non-additivity is found to exist, a Wilks's Lambda test for the dimensionality of the matrix of the non-additivity parameters is derived and the Lambda criterion is then factorized into two independent test criteria to test meaningful hypotheses concerning the multivariate model.     

A Note on an Unbiased Test of Homogeneity of Variances

A necessary and sufficient condition for unbiasedness of the test of homogeneity of variances in normal samples is derived in a convenient form. In the case of two samples, it is shown that Bartlett's test is the only unbiased test of homogeneity of variances. A simple alternative proof of the unbiasedness of Bartlett's test in the general case is also provided.     

Some pitfalls of tests of separate families of hypotheses: normality vs. lognormality

It is often desirable to test non-nested hypotheses. Cox (1961, 1962) proposed forming a log-likelihood ratio from their maxima and then comparing this value to its expected value under the null hypothesis. Pitfalls exists when we apply Cox's test to the special case of testing normality versus lognormality. Pesaran (1981) and Kotz (1973) pointed out the slow convergence rate of the Cox's test. In this paper, this fact has been reemphasized; moreover, we propose an alternative likelihood ratio test which remedies problems arising from negative estimates of the asymptotic variance of Cox's test statistic and is uniformly more powerful than most commonly used tests.