Data depth‐based nonparametric scale tests

Liu and Singh (1993, 2006) introduced a depth‐based d‐variate extension of the nonparametric two sample scale test of Siegel and Tukey (1960). Liu and Singh (2006) generalized this depth‐based test for scale homogeneity of k ≥ 2 multivariate populations. Motivated by the work of Gastwirth (1965), we propose k sample percentile modifications of Liu and Singh's proposals. The test statistic is shown to be asymptotically normal when k = 2, and compares favorably with Liu and Singh (2006) if the underlying distributions are either symmetric with light tails or asymmetric. In the case of skewed distributions considered in this paper the power of the proposed tests can attain twice the power of the Liu‐Singh test for d ≥ 1. Finally, in the k‐sample case, it is shown that the asymptotic distribution of the proposed percentile modified Kruskal‐Wallis type test is χ² with k ? 1 degrees of freedom. Power properties of this k‐sample test are similar to those for the proposed two sample one. The Canadian Journal of Statistics 39: 356–369; 2011 © 2011 Statistical Society of Canada     

Accurate tests for the equality of coefficients of variation

ABSTRACT

A frequently encountered statistical problem is to determine if the variability among k populations is heterogeneous. If the populations are measured using different scales, comparing variances may not be appropriate. In this case, comparing coefficient of variation (CV) can be used because CV is unitless. In this paper, a non-parametric test is introduced to test whether the CVs from k populations are different. With the assumption that the populations are independent normally distributed, the Miller test, Feltz and Miller test, saddlepoint-based test, log likelihood ratio test and the proposed simulated Bartlett-corrected log likelihood ratio test are derived. Simulation results show the extreme accuracy of the simulated Bartlett-corrected log likelihood ratio test if the model is correctly specified. If the model is mis-specified and the sample size is small, the proposed test still gives good results. However, with a mis-specified model and large sample size, the non-parametric test is recommended.     

DETERMINATION OF DOMAINS OF ATTRACTION BASED ON A SEQUENCE OF MAXIMA

Suppose that the maximum of a random sample from a distribution F(x) may be obtained in each of k equally spaced observation periods. This paper proposes a test to determine the domain of attraction of F(x), and investigates the properties when the sample size is very large and perhaps unknown and k is fixed and small. The test statistic is a function of the spacings between the order statistics based on the sequence of maxima and is suggested by reference to one studied previously when inference was based on the largest k observations of a random sample. A Monte Carlo study shows that the proposed test is more powerful than its main competitor. The test is illustrated by two examples.     

Inference for random coefficient INAR(k) with the occasional level shift random noise based on dual empirical likelihood

The INAR(k) model has been widely used in various kinds of fields. However, there are little discussions about the INAR(k) model with the occasional level shift random noise. In this paper, the maximum likelihood estimation of parameter based on martingale difference sequence is given, the log empirical likelihood ratio test statistic is obtained and the test statistic converges to chi-square distribution, we prove that the confidence region of the parameter is convex. Furthermore, the numerical simulation of the proposed INAR(k) model is given, which illustrates the effectiveness of the model. Then, the proofs of asymptotic results are given in the Appendix.     

An Adaptive Test for the Two-Sample Location Problem Based on U-Statistics

For the two-sample location problem with continuous data we consider a general class of tests, all members of it are based on U-statistics. The asymptotic efficacies are investigated in detail. We construct an adaptive test where all statistics involved are suitably chosen U-statistics. It is shown that the proposed adaptive test has good asymptotic and finite sample power properties.     

A new generalized p-value approach for testing equality of coefficients of variation in k normal populations

A new generalized p-value method is proposed for testing the equality of coefficients of variation in k normal populations. Simulation studies show that the type I error probabilities are close to the nominal level. The proposed test is also compared with likelihood ratio test, modified Bennett's test and score test through Monte Carlo simulation, the results demonstrate that the generalized p-value method has satisfactory performance in terms of sizes and powers.     

Further Results on the Limiting Distribution of GMM Sample Moment Conditions

In this article, we examine the limiting behavior of generalized method of moments (GMM) sample moment conditions and point out an important discontinuity that arises in their asymptotic distribution. We show that the part of the scaled sample moment conditions that gives rise to degeneracy in the asymptotic normal distribution is T-consistent and has a nonstandard limiting distribution. We derive the appropriate asymptotic (weighted chi-squared) distribution when this degeneracy occurs and show how to conduct asymptotically valid statistical inference. We also propose a new rank test that provides guidance on which (standard or nonstandard) asymptotic framework should be used for inference. The finite-sample properties of the proposed asymptotic approximation are demonstrated using simulated data from some popular asset pricing models.     

A permutation test approach to the choice of size k for the nearest neighbors classifier

The k nearest neighbors (k-NN) classifier is one of the most popular methods for statistical pattern recognition and machine learning. In practice, the size k, the number of neighbors used for classification, is usually arbitrarily set to one or some other small numbers, or based on the cross-validation procedure. In this study, we propose a novel alternative approach to decide the size k. Based on a k-NN-based multivariate multi-sample test, we assign each k a permutation test based Z-score. The number of NN is set to the k with the highest Z-score. This approach is computationally efficient since we have derived the formulas for the mean and variance of the test statistic under permutation distribution for multiple sample groups. Several simulation and real-world data sets are analyzed to investigate the performance of our approach. The usefulness of our approach is demonstrated through the evaluation of prediction accuracies using Z-score as a criterion to select the size k. We also compare our approach to the widely used cross-validation approaches. The results show that the size k selected by our approach yields high prediction accuracies when informative features are used for classification, whereas the cross-validation approach may fail in some cases.     

Asymptotic Distribution for Symmetric Spacing Statistics

A new procedure for testing the H ₀: μ₁ = ··· = μ _k against the alternative H _u :μ₁ ≥ ··· ≥μ _r  ≤ ··· ≤ μ _k with at least one strict inequality, where μ _i is the location parameter of the ith two-parameter exponential distribution, i = 1,…, k, is proposed. Exact critical constants are computed using a recursive integration algorithm. Tables containing these critical constants are provided to facilitate the implementation of the proposed test procedure. Simultaneous confidence intervals for certain contrasts of the location parameters are derived by inverting the proposed test statistic. In comparison to existing tests, it is shown, by a simulation study, that the new test statistic is more powerful in detecting U-shaped alternatives when the samples are derived from exponential distributions. As an extension, the use of the critical constants for comparing Pareto distribution parameters is discussed.     

A simulation study of the Bennett test and Miller test for coefficients of variation

In the article, properties of the Bennett test and Miller test are analyzed. Assuming that the sample size is the same for each sample and considering the null hypothesis that the coefficients of variation for k populations are equal against the hypothesis that k ? 1 coefficients of variation are the same but differ from the coefficient of variation for the kth population, the empirical significance level and the power of the test are studied. Moreover, the dependence of the test statistic and the power of the test on the ratio of coefficients of variation are considered. The analyses are performed on simulated data.     

A note on determining the number of outliers in an exponential sample by least squares procedure

In this paper, we suggest a least squares procedure for the determination of the number of upper outliers in an exponential sample by minimizing sample mean squared error. Moreover, the method can reduce the masking or “swamping” effects. In addition, we have also found that the least squares procedure is easy and simple to compute than test test procedure T _k suggested by Zhang (1998) for determining the number of upper outliers, since Zhang (1998) need to use the complicated null distribution of T _k . Moreover, we give three practical examples and a simulated example to illustrate the procedures. Further, simulation studies are given to show the advantages of the proposed method. Finally, the proposed least squares procedure can also determine the number of upper outliers in other continuous univariate distributions (for example, Pareto, Gumbel, Weibull, etc.). Received: May 10, 1999; revised version: June 5, 2000     

A new distribution-free k-sample test: Analysis of kernel density functionals

A novel distribution-free k-sample test of differences in location shifts based on the analysis of kernel density functional estimation is introduced and studied. The proposed test parallels one-way analysis of variance and the Kruskal–Wallis (KW) test aiming at testing locations of unknown distributions. In contrast to the rank (score)-transformed non-parametric approach, such as the KW test, the proposed F-test uses the measurement responses along with well-known kernel density estimation (KDE) to estimate the locations and construct the test statistic. A practical optimal bandwidth selection procedure is also provided. Our simulation studies and real data example indicate that the proposed analysis of kernel density functional estimate (ANDFE) test is superior to existing competitors for fat-tailed or heavy-tailed distributions when the k groups differ mainly in location rather than shape, especially with unbalanced data. ANDFE is also highly recommended when it is unclear whether test groups differ mainly in shape or location. The Canadian Journal of Statistics 48: 167–186; 2020 © 2019 Statistical Society of Canada     

A CLASS OF NONPARAMETRIC TESTS FOR THE ONE-SAMPLE LOCATION PROBLEM

A class of “optimal”U-statistics type nonparametric test statistics is proposed for the one-sample location problem by considering a kernel depending on a constant a and all possible (distinct) subsamples of size two from a sample of n independent and identically distributed observations. The “optimal” choice of a is determined by the underlying distribution. The proposed class includes the Sign and the modified Wilcoxon signed-rank statistics as special cases. It is shown that any “optimal” member of the class performs better in terms of Pitman efficiency relative to the Sign and Wilcoxon-signed rank statistics. The effect of deviation of chosen a from the “optimal” a on Pitman efficiency is also examined. A Hodges-Lehmann type point estimator of the location parameter corresponding to the proposed “optimal” test-statistics is also defined and studied in this paper.     

An extension of the Anderson–Darling k-sample test to arbitrary sample space partition sizes

In this paper we first show that the k-sample Anderson–Darling test is basically an average of Pearson statistics in 2?×?k contingency tables that are induced by observation-based partitions of the sample space. As an extension, we construct a family of rank test statistics, indexed by c?∈??, which is based on similarly constructed c?×?k partitions. An extensive simulation study, in which we compare the new test with others, suggests that generally very high powers are obtained with the new tests. Finally we propose a decomposition of the test statistic in interpretable components.     

Testing for equivalence of variances using Hartley's ratio

Hartley's test for homogeneity of k normal‐distribution variances is based on the ratio between the maximum sample variance and the minimum sample variance. In this paper, the author uses the same statistic to test for equivalence of k variances. Equivalence is defined in terms of the ratio between the maximum and minimum population variances, and one concludes equivalence when Hartley's ratio is small. Exact critical values for this test are obtained by using an integral expression for the power function and some theoretical results about the power function. These exact critical values are available both when sample sizes are equal and when sample sizes are unequal. One related result in the paper is that Hartley's test for homogeneity of variances is no longer unbiased when the sample sizes are unequal. The Canadian Journal of Statistics 38: 647–664; 2010 © 2010 Statistical Society of Canada     

Sign test based on k-tuple general ranked set samples

ABSTRACT

The sign test based on the k-tuple ranked set samples is discussed here. We first derive the distribution of the k-tuple ranked set sample sign test statistic, and then the asymptotic distribution is also obtained. We then compare its performance with its counterparts based on simple random sample and classical ranked set sample. The asymptotic relative efficiency and the power are then derived. Finally, the effect of imperfect ranking on the procedure is assessed.     

Testing the proportional odds model for interval-censored data

This paper discusses the goodness-of-fit test for the proportional odds model for K-sample interval-censored failure time data, which frequently occur in, for example, periodic follow-up survival studies. The proportional odds model has a feature that allows the ratio of two hazard functions to be monotonic and converge to one and provides an important tool for the modeling of survival data. To test the model, a procedure is proposed, which is a generalization of the method given in Dauxois and Kirmani [Dauxois JY, Kirmani SNUA (2003) Biometrika 90:913–922]. The asymptotic distribution of the procedure is established and its properties are evaluated by simulation studies     

A test for the equality of k medians against the simple tree alternative under right censorship

We consider a test for the equality of k population medians, θ_i i=1,2,….,k, when it is believed a priori, that θ _i: The observations are subject to right censorhip. The distributions of the censoring variables for each population are assumed to be equal. This test is compared with the general k-sample test proposed by Breslow     

Small-sample comparison of the exact and asymptotic upper tail probabilities of chi-squared goodness-of-fit statistics: the binomila and the mixture binomial

The exact and asymptotic upper tail probabilities (α = .10, .05, .01, .001) of the three chi-squared goodness-of-fit statistics Pearson's X ², likelihood ratioG ², and powerdivergence statisticD ²(λ), with λ= 2/3 are compared by complete enumeration for the binomial and the mixture binomial. For the two-component mixture binomial, three cases have been distinguished. 1. Both success probabilities and the mixing weights are unknwon. 2. One of the two success probabilities is known. And 3., the mixing weights are known. The binomial was investigated for the number of cellsk, being between 3 and 6 with sample sizes between 5 and 100, for k = 7 with sample sizes between 5 and 45, and for k = 10 with sample sizes ranging from 5 to 20. For the mixture binomial, solely k = 5 cells were considered with sample sizes from 5 to 100 and k = 8 cells with sample sizes between 4 and 20. Rating the relative accuracy of the chi-squared approximation in terms of ±10% and ±20% intervals around α led to the following conclusions for the binomial: 1. Using G2 is not recommendable. 2. At the significance levels α=.10 and α=.05X ² should be preferred over D ²; D ² is the best choice at α = .01. 3. Cochran's (1954; Biometrics, 10, 417-451) rule for the minimum expectation when using X ² seems to generalize to the binomial for G ² and D ² ; as a compromise, it gives a rather strong lower limit for the expected cell frequencies in some circumstances, but a rather liberal in others. To draw similar conclusions concerning the mixture binomial was not possible, because in that case, the accuracy of the chi-squared approximation is not only a function of the chosen test statistic and of the significance level, but also heavily depends on the numerical value of theinvolved unknown parameters and on the hypothesis to be tested. Thereto, the present study may give rise only to warnings against the application of mixture models to small samples.     

A power study of k-linear-r-ahead recursive residuals test for change-point in finite sequences

A change-point problem in finite sequences is considered along with, so-called, k-linear-r-ahead recursive residuals and a test procedure proposed by ?o?a¸d? et al. [?o?a¸d?, J.A., Szkutnik, Z., Majerczak, J. and Duda, K. 1998, Detection of change point in oxygen uptake during an incremental exercise test using recursive residuals: relationship to the plasma lactate accumulation and blood acid base balance. European Journal of Applied Physiology, 78, 369–377.]. Theoretical significance levels of that (conservative) test are compared with its simulated sizes. Numerical approximations to the powers against various alternatives are given. Properties of the k-linear-r-ahead recursive residuals are described and the consistency of the test is proved, when the noise level goes to zero.