Two-sample Location Problem Research Articles

New nonparametric tests based on data depth for multivariate multi-sample location problem

Data depth based nonparametric tests for multivariate multisample location problem are provided which extend the depth based nonparametric test for multivariate two-sample location problem by Shirke and Khorate (2018) [Shirke, D.T. and Khorate, S.D., 2018. Two-sample nonparametric test for testing equality of locations based on data depth. Journal of the Indian Society for Probability and Statistics, 19, pp. 9-23.] to the multi-sample setting. Proposed tests are based on the correlations among the depths of data points. Power comparison of proposed tests with the existing depth based nonparametric tests is provided using a simulation study based on a permutation test. Simulation study reveals that the proposed tests give comparable powers compared to the existing data depth based nonparametric tests. Proposed tests are illustrated with real life data sets.

A control chart based on data depth for monitoring the variability in a multivariate process

Majority of the existing statistical procedures for monitoring the covariance matrix of the multivariate processes are based on normality or other distributional assumption. In order to isolate the change in the process variation, these procedures are less powerful due to distributional assumptions. In this article, we study the control charts for monitoring covariance matrix critically. We provide a distribution-free two-sample test for multivariate scale problem called as modified rank test based on data depth. A control chart is proposed based on the data depth based modified rank test. A modified rank test based on data depth is the extension of the modified rank test for univariate two-sample location and scale problem. Data depth based center outward ranking is used for defining test statistic. The performance study of the proposed control charts reveals that the proposed control chart is a better alternative to traditional control charts for variability when a distribution of the process is not known but some historical in-control data is available. The control chart is illustrated with the real-life CSTR process data.

A combined bootstrap test for the two-sample location problem

The comparison of two samples is a problem very frequently encountered in practice. There is active interest in researching for tests that are powerful also when the data are not compatible with the assumptions of normality and equal variances – as it is common in many fields – while controlling their type-one error rate. The Yuen test is a familiar method based on trimmed means. There is no agreement in the literature about the preferable rate of trimming. This paper has two aims: to study the power of many Yuen tests with different trimming rates and propose a bootstrap test based on the combination of Yuen tests with different trimming rates. It is shown that the various Yuen tests have very different power for different distributions because the best rate of trimming depends on distribution tailweight. Conversely, the combined test is powerful irrespective to the underlying distribution.

Permutation Tests for Two-sample Location Problem Under Extreme Ranked Set Sampling

In this paper, permutation test of comparing two-independent samples is investigated in the context of extreme ranked set sampling (ERSS). Three test statistics are proposed. The statistical power of these new test statistics are evaluated numerically. The results are compared with the statistical power of the classical independent two-sample $t$-test, Mann-Whitney $U$ test, and the usual two-sample permutation test under simple random sampling (SRS). In addition, the method of computing a confidence interval for the two-sample permutation problem under ERSS is explained. The performance of this method is compared with the intervals obtained by SRS and Mann-Whitney procedures in terms of empirical coverage probability and expected length. The comparison shows that the proposed statistics outperform their counterparts. Finally, the application of the proposed statistics is illustrated using a real life example.

On Tests for Special Two-Sample Location Problem Based on Subsample Minimum and Median

Shetty and Umarani(2005) considered a special type of two-sample location problem which has some potential applications, particularly in comparing the performances of two packing machines. In this paper, a new class of test statistics is proposed which is based on sub-sample minima of first sample and sub-sample median of second sample. The performance of members of the new class is evaluated in terms of Pitman asymptotic relative efficiency in comparision with Shetty and Umarani(2005). It can be concluded that the performance of the new proposed class of tests is better for medium and heavy tailed distributions such as Normal, Logistic, Laplace and Cauchy. For light tailed distributions the performance of new testis better in specific cases. It is proved that the test is consistent for one-sided alternatives.

A Comparative Study of the Power of Parametric and Permutation Tests for a Multidimensional Two-sample Location Problem

Objective: A comparison of multidimensional populations is a very interesting and common statistical problem. It most often involves verifying a hypothesis about the equality of mean vectors in two populations. The classical test for verification of this hypothesis is the Hotelling’s T 2 test. Another solution is to use simulation and randomization methods to test the significance of differences between the studied populations. Permutation tests are to enable statistical inference in situations where it is not possible to use classical parametric tests. These tests are supposed to provide comparable power to parametric tests with a simultaneous reduction of assumptions, e.g. regarding the sample size taken or the distribution of the tested variable in the population. The purpose of this study is a comparative analysis of the parametric test, the (usual) permutation test, and the nonparametric permutation procedure using two-stage ASL determination. Research Design & Methods: The study considered the analysis of multivariate data. The paper presents theoretical considerations and refers to the Monte Carlo simulation. Findings: The article presents a permutational, complex procedure for assessing the overall ASL (achieved significance level) value. The applied nonparametric statistical inference procedure uses combining functions. A simulation study was carried out to determine the size and power of the test under normality. A Monte Carlo simulation made it possible to compare the empirical power of this test with that of Hotelling’s T 2 test. The most powerful test was the permutation test based on a two-stage ASL determination method using the Fisher combining function. Implications/Recommendations: The advantage of the proposed method is that it can be used even when samples are taken from any type of continuous distributions in a population. Contribution: The proposed test can be used in the analysis of multidimensional economic phenomena.

A high-dimensional spatial rank test for two-sample location problems

In high-dimensional situations, the traditional multivariate sign- or rank-based procedures for the two-sample location testing problems are ineffective, since in the construction of the test statistics, the scatter matrix to be inverted is singular. To solve this problem, many high-dimensional spatial sign or rank tests have been proposed, some of which are very efficient. However, most of these existing tests no longer work in very high dimensional situations, which only allows the dimension of variables to be the square of the sample sizes at most, hence are restrictive for practical applications. On this ground, a new high-dimensional spatial rank test is proposed in this paper, which is invariant under scalar transformations, maintains the efficiency advantage of spatial-rank-based testing methods, and could even allow the dimension to grow almost exponentially with the sample sizes. The theoretical results of the proposed test are established, followed by some convincing numerical results and two real data analyses.

Empirical Likelihood-Based Inference for the Difference of Two Location Parameters Using Smoothed M-Estimators

We consider one of the classical problems in statistics: the inference for the two-sample location problem. In this paper, we present a new empirical likelihood (EL) method for the difference of two smoothed M-estimators. To deal with additional nuisance scale parameters, we use the plug-in empirical likelihood, and we establish asymptotic properties of the new estimators. For the empirical study, we consider the important case of the smoothed Huber M-estimator. Our empirical results show that the new method is a competitive alternative to the classical procedures regarding inference about the difference of two location parameters. The software implementation for the new empirical likelihood method is based on the R package EL, which has been developed for related two-sample problems.

Constructions of New Classes of One- and Two-Sample Nonparametric Location Tests

In this paper, we examine the role of nonparametric test statistics with constructions analogous to those introduced in Mukhopadhyay and Chattopadhyay (Stat Pap 54:827–837 2013) and Mukhopadhyay and Chattopadhyay (Sri Lankan J Appl Stat 15:71–80 2014) in the context of one-sample and two-sample location problems. In the case of a one-sample location problem, we focus on the customary sign test and its appropriate modifications. In the case of a two-sample location problem, we focus on Mann-Whitney test when the population distribution F remains unknown. Using large sample approximations, we propose new versions of tests and compare their performances with those of the customary sign and Mann-Whitney tests. We do so on the basis of asymptotic power, asymptotic efficiency, and robustness. Our present treatment substantially broadens the coverage found in Walsh (Ann Math Stat 17:360–361 1946, Ann Math Stat 20:64–81 1949) and Ylvisaker (J Am Stat Assoc 72:551–556 1977. In summary, we have come up with (i) a test that is more efficient than a sign test, and (ii) a limited optimality of Mann-Whitney test within our class of newly constructed tests.

Adaptive Group-combined P-values Test for Two-sample Location Problem with Applications to Microarray Data

The purpose of this article is to propose a test for two-sample location problem in high-dimensional data. In general highdimensional case, the data dimension can be much larger than the sample size and the underlying distribution may be far from normal. Existing tests requiring explicit relationship between the data dimension and sample size or designed for multivariate normal distributions may lose power significantly and even yield type I error rates strayed from nominal levels. To overcome this issue, we propose an adaptive group p-values combination test which is robust against both high dimensionality and normality. Simulation studies show that the proposed test controls type I error rates correctly and outperforms some existing tests in most situations. An Ageing Human Brain Microarray data are used to further exemplify the method.

Rank-based two-sample tests for paired data with missing values.

Two-sample location problem is one of the most encountered problems in statistical practice. The two most commonly studied subtypes of two-sample location problem involve observations from two populations that are either independent or completely paired, but a third subtype can oftentimes occur in practice when some observations are paired and some are not. Partially paired two-sample problems, also known as paired two-sample problems with missing data, often arise in biomedical fields when it is difficult for some invasive procedures to collect data from an individual at both conditions we are interested in comparing. Existing rank-based two-sample comparison procedures for partially paired data, however, do not make efficient use of all available data. In order to improve the power of testing procedures for this problem, we propose several new rank-based test statistics and study their asymptotic distributions and, when necessary, exact variances. Through extensive numerical studies, we show that the best overall power come from the proposed tests based on weighted linear combinations of the test statistics comparing paired data and the test statistics comparing independent data, using weights inversely proportional to their variances. We illustrate the proposed methods with a real data example from HIV research for prevention.

A new nonparametric scheme for simultaneous monitoring of bivariate processes and its application in monitoring service quality

We consider a new nonparametric control chart for monitoring bivariate processes based on a distance measure. This statistic was originally introduced in context of a strictly nonparametric bivariate test for two-sample location problem and was shown to be more powerful than several competitors including Hotelling’s T2 in a large number of situations. Our proposed scheme is useful in simultaneous monitoring of the event frequency and magnitude, call centre response times and its service quality or time between the two major earthquakes and the magnitude in Richter scale, etc. Unlike most of the traditional approaches for simultaneous monitoring of event frequency and magnitude that consider event frequency and magnitude are independent, our proposed chart takes into account the dependence between two variables. We consider that the frequencies are large enough and the two variables are continuous, but we make no assumption about the mathematical form of the process distribution. We consider the availability of a Phase-I data. We discuss the design and implementation procedure in details. We conduct some Monte-Carlo simulation to examine various properties of the proposed scheme. We also illustrate our proposed scheme with a real data.

SOME CLASSES OF NONPARAMETRIC TESTS FOR SPECIAL TWO-SAMPLE LOCATION PROBLEM BASED ON SUBSAMPLE EXTREMES

The special two-sample location problem is an important problem which is useful in comparing the performance of two measuring instruments. The problem of comparing the performances of two packing machines in which one machine may underfill the packets and the other may overfill the packets on an average, fits into special twosample location setup wherein one wishes to test for the point of symmetry versus an appropriate alternative. The only test available in the literature to the best of our knowledge is the class of tests due to Shetty and Umarani [13] which is based on U-statistics. In this paper, two classes of test statistics are proposed which are based on extremes of subsamples. The performances of the proposed classes of tests are

A new test for two-sample location problem based on empirical distribution function

ABSTRACTWe propose a new test for testing the equality of location parameter of two populations based on empirical distribution function (ECDF). The test statistics is obtained as a power divergence between two ECDFs. The test is shown to be distribution free, and its null distribution is obtained. We conducted empirical power comparison of the proposed test with several other available tests in the literature. We found that the proposed test performs better than its competitors considered here under several population structures. We also used two real datasets to illustrate the procedure.

A Class of Nonparametric Tests for the Two-Sample Location Problem

The two-sample location problem is one of the fundamental problems encountered in Statistics. In many applications of Statistics, two-sample problems arise in such a way as to lead naturally to the formulations of the null hypothesis to the effect that the two samples come from identical populations. A class of nonparametric test statistics is proposed for two-sample location problem based on U-statistic with the kernel depending on a constant ’a’ when the underlying distribution is symmetric. The optimal choice of ’a’ for different underlying distributions is determined. An alternative expression for the class of test statistics is established. Pitman asymptotic relative efficiencies indicate that the proposed class of test statistics does well in comparison with many of the test statistics available in the literature. The small sample performance is also studied through Monte-Carlo Simulation technique

Fast tests for the two-sample problem based on the empirical characteristic function

A class of tests for the two-sample problem whose test statistic is an L2 norm of the difference of the empirical characteristic functions of the samples is considered. The null distribution can be estimated by means of bootstrap or permutation procedures. Although very easy to implement, such procedures can become computationally expensive as the sample size or the dimension of the data increase. This paper proposes to approximate the null distribution through a weighted bootstrap. The method is studied both theoretically and numerically. It provides a consistent estimator of the null distribution. The asymptotic properties are similar to those of the bootstrap and permutation estimators but, from a computational point of view, the weighted bootstrap estimator is more efficient. The proposed approach is also applied to the two-sample location problem and to the k-sample problem.

Multivariate-Sign-Based High-Dimensional Tests for the Two-Sample Location Problem

ABSTRACTThis article concerns tests for the two-sample location problem when data dimension is larger than the sample size. Existing multivariate-sign-based procedures are not robust against high dimensionality, producing tests with Type I error rates far away from nominal levels. This is mainly due to the biases from estimating location parameters. We propose a novel test to overcome this issue by using the “leave-one-out” idea. The proposed test statistic is scalar-invariant and thus is particularly useful when different components have different scales in high-dimensional data. Asymptotic properties of the test statistic are studied. Compared with other existing approaches, simulation studies show that the proposed method behaves well in terms of sizes and power. Supplementary materials for this article are available online.

Distribution-Free Tests for Two-Sample Location Problems Based on Subsamples

Nonparametric tests for location problems have received much attention in the literature. Many nonparametric tests have been proposed for one, two and several samples location problems. In this paper a class of test statistics is proposed for two sample location problem when the underlying distributions of the samples are symmetric. The class of test statistics proposed is linear combination of U-statistics whose kernel is based on subsamples extrema. The members of the new class are shown to be asymptotically normal. The performance of the proposed class of tests is evaluated using Pitman Asymptotic Relative Efficiency. It is observed that the members of the proposed class of tests are better than the existing tests in the literature.

A note on high-dimensional two-sample test

We propose a new scalar and shift transform invariant test statistic for the high-dimensional two-sample location problem. Theoretical results and simulation studies show the good performance of our test under certain circumstances.

Tests for Two-Sample Location Problem Based on Subsample Quantiles

This paper presents a new class of test procedures for two-sample location problem based on subsample quantiles. The class includes Mann-Whitney test as a special case. The asymptotic normality of the class of tests proposed is established. The asymptotic relative performance of the proposed class of test with respect to the optimal member of Xie and Priebe (2000) is studied in terms of Pitman efficiency for various underlying distributions.