Adaptive positive and negative runs test

As a nonparametric randomness test, the positive and negative runs test is widely used in practice due to the simplicity of its procedures. The test can lose efficiency if the alternative distribution is symmetrical at 0.5. In addition, the test can only be applied to test the randomness of a sequence from the uniform distribution. In this paper, we introduce an adaptive positive and negative runs test method to maximize the power function by choosing the optimal cut point. Also, the test is extended to check the randomness of a sequence generated from any other given distributions. Furthermore, we derive the exact distribution and obtain the asymptotical critical values of the proposed test statistics. Compared with the existed test, the efficiency of the proposed adaptive positive and negative runs test is competitive through simulation study.     

A simple nonparametric test for bivariate symmetry about a line

Over the years many researchers have dealt with testing the hypotheses of symmetry in univariate and multivariate distributions in the parametric and nonparametric setup. In a multivariate setup, there are several formulations of symmetry, for example, symmetry about an axis, joint symmetry, marginal symmetry, radial symmetry, symmetry about a known point, spherical symmetry, and elliptical symmetry among others. In this paper, for the bivariate case, we formulate a concept of symmetry about a straight line passing through the origin in a plane and accordingly develop a simple nonparametric test for testing the hypothesis of symmetry about a straight line. The proposed test is based on a measure of deviance between observed counts of bivariate samples in suitably defined pairs of sets. The exact null distribution and non-null distribution, for specified classes of alternatives, of the test statistics are obtained. The null distribution is tabulated for sample size from n=5 up to n=30. The null mean, null variance and the asymptotic null distributions of the proposed test statistics are also obtained. The empirical power of the proposed test is evaluated by simulating samples from the suitable class of bivariate distributions. The empirical findings suggest that the test performs reasonably well against various classes of asymmetric bivariate distributions. Further, it is advocated that the basic idea developed in this work can be easily adopted to test the hypotheses of exchangeability of bivariate random variables and also bivariate symmetry about a given axis which have been considered by several authors in the past.     

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 nonparametric test of symmetry based on the overlapping coefficient

In this paper, we introduce a new nonparametric test of symmetry based on the empirical overlap coefficient using kernel density estimation. Our investigation reveals that the new test is more powerful than the runs test of symmetry proposed by McWilliams [31]. Intensive simulation is conducted to examine the power of the proposed test. Data from a level I Trauma center are used to illustrate the procedures developed in this paper.     

An approximate entropy test for randomness

We develop an entropy-based test for randomness of binary time series of finite length. The test uses the frequencies of contiguous blocks of different lengths. A simple condition ib the block lengths and the length of the time series enables one to estimate the entropy rate for the data, and this information is used to develop a statistic to test the hypothesis of randomness. This static measures the deviation of the estimated entropy of the observed data from the theoretical maximum under the randomness hypothesis. This test offers a real alternative to the conventional runs test. Critical percentage points, based on simulations, are provided for testing the hypothesis of randomness. Power calculations using dependent data show that the proposed test has higher power against the runs test for short series, and it is similar to the runs test for long series. The test is applied to two published data sets that wree investigated by others with respect to their randomness.     

Some count-based nonparametric tests for circular symmetry of a bivariate distribution

In this paper, we revisit the problem of testing of the hypothesis of circular symmetry of a bivariate distribution. We propose some nonparametric tests based on sector counts. These include tests based on chi-square goodness-of-fit test, the classical likelihood ratio, mean deviation, and the range. The proposed tests are easy to implement and the exact null distributions for small sample sizes of the test statistics are obtained. Two examples with small and large data sets are given to illustrate the application of the tests proposed. For small and moderate sample sizes, the performances of the proposed tests are evaluated using empirical powers (empirical sizes are also reported). Also, we evaluate the performance of these count-based tests with adaptations of several well-known tests such as the Kolmogorov–Smirnov-type tests, tests based on kernel density estimator, and the Wilcoxon-type tests. It is observed that among the count-based tests the likelihood ratio test performs better.     

A rule of thumb for testing symmetry about an unknown median against a long right tail

A rule of thumb for testing symmetry of an unknown univariate continuous distribution against the alternative of a long right tail is proposed. Our proposed test is based on the concept of exceedance statistic and is ad hoc in nature. Exact performances of the proposed rule are investigated in detail. Some results from an asymptotic point of view are also provided. We compare our proposed test with several classical tests which are practically applicable and are known to be exact or nearly distribution free. We see that the proposed rule is better than most of the existing tests for symmetry and can be applied with ease. An illustration with real data is provided.     

The difference-sign runs length distribution in testing for serial independence

We investigate the sequence of difference-sign runs length of a time series in the context of non-parametric tests for serial independence. This sequence is, under suitable conditioning, a stationary sequence and we prove that the normalized correlation of two consecutive runs length is small (≈0.0427). We use this result in a test based on the relative entropy of the empirical distribution of the runs length. We investigate the performance of the test in simulated series and test serial independence of cardiac data series in atrial fibrillation.     

A test statistic for weighted runs

A new test statistic based on runs of weighted deviations is introduced. Its use for observations sampled from independent normal distributions is worked out in detail. It supplements the classic χ² test which ignores the ordering of observations and provides additional sensitivity to local deviations from expectations. The exact distribution of the statistic in the non-parametric case is derived and an algorithm to compute p-values is presented. The computational complexity of the algorithm is derived employing a novel identity for integer partitions.     

A simple test for equally likely random responses in a sequence of bernoulli trials

A simple versatile combinatorial test is given for the null hypothesis that the two kinds of responses in a sequence of binary random variables are both equiprobable and randomly distributed. It is sensitive to a preponderance of either kind of response in either of two pre-selected halves of the positions of the sequence, e.g., the right half vs. the left, or the center vs. both ends. It combines the nice features of being, on the one hand, relatively assumption free and easy to apply and, on the other hand, sensitive to basic patterns underlying a poor fit of the empirical distribution to a theoretical model.     

A characterization of and new consistent tests for symmetry

Let X and Y be independent and identically distributed random variables having a continuous distribution function. We study new consistent tests for symmetry around a known median based on the fact that the distribution of X is symmetric around 0 if, and only if, |X| and |max(X,Y)| have the same distribution.     

A modified one-sample test for goodness-of-fit

This paper introduces a modified one-sample test of goodness-of-fit based on the cumulative distribution function. Damico [A new one-sample test for goodness-of-fit. Commun Stat – Theory Methods. 2004;33:181–193] proposed a test for testing goodness-of-fit of univariate distribution that uses the concept of partitioning the probability range into n intervals of equal probability mass 1/n and verifies that the hypothesized distribution evaluated at the observed data would place one case into each interval. The present paper extends this notion by allowing for m intervals of probability mass r/n, where r≥1 and n=m×r. A simulation study for small and moderate sample sizes demonstrates that the proposed test for two observations per interval under various alternatives is more powerful than the test proposed by Damico (2004).     

A test for bivariate symmetry based on the empirical distribution function

Hollander (1970) proposed a conditionally distribution-free test of bivariate symmetry based on the empirical distribution function. In this paper Hollander’s test statistic is examined In greater detail: in particular; its conditional asymptotic distribution is derived under the null hypothesis as well as under a sequence of local alternatives. Percentage points of the asymptotic distribution are presented; a power comparison between Hollander’s statistic and the likelihood ratio criterion in testing a variant of the sphericity hypothesis in multivariate analysis is made.     

A robust test for the homogeneity of scales

Many robust tests for the equality of variances have been proposed recently. Brown and Forsythe (1974) and Layard (1973) review some of the well-known procedures and compare them by simulation methods. Brown and Forsythe’s alternative formulation of Levene’s test statistic is found to be quite robust under certain nonnormal distributions. The performance of the methods, however, suffers in the presence of heavy tailed distributions such as the Cauchy distribution.

In this paper, we propose and study a simple robust test. The results obtained from the Monte Carlo study compare favorably with those of the existing procedures.     

Conditional tests for elliptical symmetry using robust estimators

This paper presents a procedure for testing the hypothesis that the underlying distribution of the data is elliptical when using robust location and scatter estimators instead of the sample mean and covariance matrix. Under mild assumptions that include elliptical distributions without first moments, we derive the test statistic asymptotic behavior under the null hypothesis and under special alternatives. Numerical experiments allow to compare the behavior of the tests based on the sample mean and covariance matrix with that based on robust estimators, under various elliptical distributions and different alternatives. We also provide a numerical comparison with other competing tests.     

Testing circular symmetry

The author addresses the problem of testing circular data for reflective symmetry about an unknown central direction and proposes a simple omnibus test based on the sample second sine moment about an estimation of this direction. Under quite general conditions, for an underlying distribution which is reflectively symmetric, the large‐sample asymptotic distribution of the test statistic is standard normal. Randomization and bootstrap variants of the test are also introduced, and the operating characteristics of different versions of the test are investigated in a Monte Carlo study. The large‐sample and bootstrap versions of the test are applied in the analysis of two illustrative examples drawn from the circular statistics literature.