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.     

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 for randomness based on a complexity measure

A test for randomness based on a statistic related to the complexity of finite sequences is presented. Simulation of binary sequences under different stochastic models provides estimates of the power of the test. The results show that the test is sensitive to a variety of alternatives to randomness and suggest that the proposed test statistic is a reasonable measure of the stochastic complexity of a finite sequence of discrete random variables.     

A likelihood ratio test of quasi-independence for sparse two-way contingency tables

We consider a likelihood ratio test of independence for large two-way contingency tables having both structural (non-random) and sampling (random) zeros in many cells. The solution of this problem is not available using standard likelihood ratio tests. One way to bypass this problem is to remove the structural zeroes from the table and implement a test on the remaining cells which incorporate the randomness in the sampling zeros; the resulting test is a test of quasi-independence of the two categorical variables. This test is based only on the positive counts in the contingency table and is valid when there is at least one sampling (random) zero. The proposed (likelihood ratio) test is an alternative to the commonly used ad hoc procedures of converting the zero cells to positive ones by adding a small constant. One practical advantage of our procedure is that there is no need to know if a zero cell is structural zero or a sampling zero. We model the positive counts using a truncated multinomial distribution. In fact, we have two truncated multinomial distributions; one for the null hypothesis of independence and the other for the unrestricted parameter space. We use Monte Carlo methods to obtain the maximum likelihood estimators of the parameters and also the p-value of our proposed test. To obtain the sampling distribution of the likelihood ratio test statistic, we use bootstrap methods. We discuss many examples, and also empirically compare the power function of the likelihood ratio test relative to those of some well-known test statistics.     

A CONTINUOUSLY ADAPTIVE RANK TEST FOR SHIFT IN LOCATION

This paper considers the problem of testing for shift in location when the symmetry of the underlying distribution is in doubt. Various adaptive test procedures have been suggested in the literature; they are mainly based on a preliminary test or measure of asymmetry, and then choosing between the sign or the Wilcoxon tests accordingly. However, as this paper demonstrates, there are some disadvantages with such procedures. This paper develops a test that does not suffer from such disadvantages. The proposed test is based on modifying the Wilcoxon scores according to the evidence of asymmetry of the distribution present in the data as indicated by the magnitude of the P‐value from a preliminary test of symmetry. A simulation study investigates and compares the performance of the proposed test and other known adaptive procedures in terms of power and attainment of the nominal size. The performance of a suitable bootstrap procedure for the situation under consideration is also studied. In most cases under consideration, the proposed test is found to be superior to the other tests.     

Distribution-Free Runs Test for Conditional Symmetry

A distribution-free runs test for conditional symmetry is proposed. The null distribution of the test statistics is derived. Intensive simulation is conducted to examine the power of the proposed test for different sample sizes and different alternatives. Data on the bilirubin level of babies in neonatal intensive care is used to illustrate the method.     

Tests for randomness of observations that involve two factors

Tests for randomness of observations that involve one factor have been considered by many authors among them are Mosteller [4], Bateman [21], Barton and David [1] and Shaughnessy [7]. However, on many occasions, data involve two different factors such as time and location, temperature and pressure, or levels of doses and responses of patients and so on. In this paper, we consider tests for randomness of observations that involve two factors for which data are given in a matrix form. Some new definitions of runs of a matrix of data are defined and discussed. A special kind of run is proposed for the test for randomness. Distributions and properties of this run are studied. Some critical regions are tabulated.     

Methods for testing subblock patterns

A number of statistical tests have been recommended over the last twenty years for assessing the randomness of long binary strings used in cryptographic algorithms. Several of these tests include methods of examining subblock patterns. These tests are the uniformity test, the universal test and the repetition test. The effectiveness of these tests are compared based on the subblock length, the limitations on data requirements, and on their power in detecting deviations from randomness. Due to the complexity of the test statistics, the power functions are estimated by simulation methods. The results show that for small subblocks the uniformity test is more powerful than the universal test, and that there is some doubt about the parameters of the hypothesised distribution for the universal test statistic. For larger subblocks the results show that the repetition test is the most effective test, since it requires far less data than either of the other two tests and is an efficient test in detecting deviations from randomness in binary strings.     

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.     

Asymptotic hypothesis test to compare likelihood ratios of multiple diagnostic tests in unpaired designs

The accuracy of a binary diagnostic test is usually measured in terms of its sensitivity and its specificity. Other measures of the performance of a diagnostic test are the positive and negative likelihood ratios, which quantify the increase in knowledge about the presence of the disease through the application of a diagnostic test, and which depend on the sensitivity and specificity of the diagnostic test. In this article, we construct an asymptotic hypothesis test to simultaneously compare the positive and negative likelihood ratios of two or more diagnostic tests in unpaired designs. The hypothesis test is based on the logarithmic transformation of the likelihood ratios and on the chi-square distribution. Simulation experiments have been carried out to study the type I error and the power of the constructed hypothesis test when comparing two and three binary diagnostic tests. The method has been extended to the case of multiple multi-level diagnostic tests.     

A comparison of approximations to the distribution of average kendall tau

The average Kendall tau test for the null hypothesis of compLete randomness in n judges' rankings of r objects was introduced by Ehrenberg (1952). The superior efficiency of this test. as measured by approximate Bahadur slope has been established by Alva, Cabilio, and Feigin (1982).

The purpose of this paper is first to provide further justification for the use of this test, and second to compare the accuracy of the asymptotic distribution to other app roxtmations to the null distribution for small values of r and n. In the process we generate tables for this exact dis tribution for r=3 and n=3(1)19, r=4 and n=3(1)9, r=5 and n=3(1)5.     

A comparison of several adaptive tests for paired data

In the last few years, two adaptive tests for paired data have been proposed. One test proposed by Freidlin et al. [On the use of the Shapiro–Wilk test in two-stage adaptive inference for paired data from moderate to very heavy tailed distributions, Biom. J. 45 (2003), pp. 887–900] is a two-stage procedure that uses a selection statistic to determine which of three rank scores to use in the computation of the test statistic. Another statistic, proposed by O'Gorman [Applied Adaptive Statistical Methods: Tests of Significance and Confidence Intervals, Society for Industrial and Applied Mathematics, Philadelphia, 2004], uses a weighted t-test with the weights determined by the data. These two methods, and an earlier rank-based adaptive test proposed by Randles and Hogg [Adaptive Distribution-free Tests, Commun. Stat. 2 (1973), pp. 337–356], are compared with the t-test and to Wilcoxon's signed-rank test. For sample sizes between 15 and 50, the results show that the adaptive test proposed by Freidlin et al. and the adaptive test proposed by O'Gorman have higher power than the other tests over a range of moderate to long-tailed symmetric distributions. The results also show that the test proposed by O'Gorman has greater power than the other tests for short-tailed distributions. For sample sizes greater than 50 and for small sample sizes the adaptive test proposed by O'Gorman has the highest power for most distributions.     

A Comparison of the Runs Test for Volatility Forecastability and the LM Test for GARCH Using Aggregated Returns

Christoffersen and Diebold (2000) have introduced a runs test for forecastable volatility in aggregated returns. In this note, we compare the size and power of their runs test and the more conventional LM test for GARCH by Monte Carlo simulation. When the true daily process is GARCH, EGARCH, or stochastic volatility, the LM test has better power than the runs test for the moderate-horizon returns considered by Christoffersen and Diebold. For long-horizon returns, however, the tests have very similar power. We also consider a qualitative threshold GARCH model. For this process, we find that the runs test has greater power than the LM test. Theresults support the use of the runs test with aggregated returns.     

Generalized run tests for statistical process control

ABSTRACT

In a sequence of elements, a run is defined as a maximal subsequence of like elements. The number of runs or the length of the longest run has been widely used to test the randomness of an ordered sequence. Based on two different sampling methods and two types of test statistics used, run tests can be classified into one of four cases. Numerous researchers have derived the probability distributions in many different ways, treating each case separately. In the paper, we propose a unified approach which is based on recurrence arguments of two mutually exclusive sub-sequences. We also consider the sequence of nominal data that has more than two classes. Thus, the traditional run tests for a binary sequence are special cases of our generalized run tests. We finally show that the generalized run tests can be applied to many quality management areas, such as testing changes in process variation, developing non-parametric multivariate control charts, and comparing the shapes and locations of more than two process distributions.     

A nonparametric test for current status data with unequal censoring

Nonparametric tests for the comparison of different treatments based on current status data are proposed. For this problem, most methods proposed in the literature require that observation times on all subjects follow the same distribution. In other words, censoring distributions are identical between the treatment groups. In this paper, we focus on the situation where the censoring distributions may be different for subjects in different treatment groups and the test that can take this unequal censoring into account is given. The asymptotic distribution of the test proposed is derived. The method proposed is applied to data arising from a tumorigenicity experiment.     

An adaptive test for the one-way layout

An adaptive test is proposed for the one-way layout. This test procedure uses the order statistics of the combined data to obtain estimates of percentiles, which are used to select an appropriate set of rank scores for the one-way test statistic. This test is designed to have reasonably high power over a range of distributions. The adaptive procedure proposed for a one-way layout is a generalization of an existing two-sample adaptive test procedure. In this Monte Carlo study, the power and significance level of the F-test, the Kruskal-Wallis test, the normal scores test, and the adaptive test were evaluated for the one-way layout. All tests maintained their significance level for data sets having at least 24 observations. The simulation results show that the adaptive test is more powerful than the other tests for skewed distributions if the total number of observations equals or exceeds 24. For data sets having at least 60 observations the adaptive test is also more powerful than the F-test for some symmetric distributions.     

Bivariate Sign Test for One-Sample Bivariate Location Model Using Ranked Set Sample

In this article, we introduce a bivariate sign test for the one-sample bivariate location model using a bivariate ranked set sample (BVRSS). We show that the proposed test is asymptotically more efficient than its counterpart sign test based on a bivariate simple random sample (BVSRS). The asymptotic null distribution and the non centrality parameter are derived. The asymptotic distribution of the vector of sample median as an estimator of the locations of the bivariate model is introduced. Theoretical and numerical comparisons of the asymptotic efficiency of the BVRSS sign test with respect to the BVSRS sign test are also given.     

A modified runs test for symmetry

We propose a modification of a Modarres–Gastwirth test for the hypothesis of symmetry about a known center. By means of a Monte Carlo Study we show that the modified test overtakes the original Modarres–Gastwirth test for a wide spectrum of asymmetrical alternatives coming from the lambda family and for all assayed sample sizes. We also show that our test is the best runs test among the runs tests we have compared.     

Data-Transformation and Test of Fit for the Generalized Pareto Hypothesis

In this article, we consider Crámer–von Mises type goodness-of-fit statistics for the Generalized Pareto law. The tests involve a certain transformation of the original observations, which, at least in the case of completely specified null distribution, may be viewed as transforming to uniformity and comparing the resulting moments of arbitrary positive order to those of a uniform distribution. The method is shown to be consistent, and the asymptotic null distribution of the test statistic is derived. Simulation results indicate that the proposed test compares well with standard methods based on the empirical distribution function.     

Bootstrap tests: how many bootstraps?

In practice, bootstrap tests must use a finite number of bootstrap samples. This means that the outcome of the test will depend on the sequence of random numbers used to generate the bootstrap samples, and it necessarily results in some loss of power. We examine the extent of this power loss and propose a simple pretest procedure for choosing the number of bootstrap samples so as to minimize experimental randomness. Simulation experiments suggest that this procedure will work very well in practice.