One-sample Location Research Articles

A Shewhart-type nonparametric multivariate depth-based control chart for monitoring location

Parametric multivariate control charts like Hotelling’s rely on certain distributional assumptions that if these assumptions are not met, it leads us to an excessive number of false alarms and reduces the effectiveness of the monitoring strategy. In contrast, nonparametric multivariate control charts, specifically data-depth-based charts and related inferences, do not require any distributional assumptions and no constraints on data dimensions. In this paper, based on a family of multivariate affine invariant distribution-free data-depth-based tests for the multivariate one-sample location problem, a Shewhart-type nonparametric multivariate phase-II control chart is proposed. The performance of the proposed control chart is evaluated with a Monte Carlo simulation study using the average run length measure and is compared to that of competitors. It is observed that the proposed control chart maintained a desirable performance and leads to quicker detection of shifts. It is also a close competitor to Hotelling’s chart when the underlying distribution of the process is bivariate normal. Moreover, it performs better than its nonparametric competitor under the bivariate non-normal elliptical direction class of distributions, specifically heavy-tailed, light-tailed, and t-distributions, in a variety of settings. The control chart procedure is illustrated as an application to real-life data.

One-sample location test based on the sign and Wilcoxon signed-rank tests

One-sample testing problem is one of the most important topics in statistics. The sign and Wilcoxon tests are commonly used for testing the location parameter. An extension of the sign and Wilcoxon signed-rank tests has been proposed. However, when the sample size increases, the exact critical value of the statistic becomes difficult to determine. Therefore, in this study, we used a saddlepoint approximation with continuity correction to approximate the distribution of the statistic. Our numerical results illustrate that the saddlepoint approximation to the statistic is more suitable than the normal approximation. We also derived the asymptotic power under the alternative hypothesis and investigated the unbiasedness and biasedness of the test conducted in this study.

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.

Affine invariant depth-based tests for the multivariate one-sample location problem

A multivariate affine invariant family of depth-based tests is proposed for the one-sample location problem. Suitable outlyingness functions which are formulated using depth functions are used to construct the proposed tests. The asymptotic null distribution and the asymptotic relative efficiency of the tests are discussed under the class of centrally and elliptically symmetric distributions, respectively. Furthermore, a conditional distribution-free property of the tests is shown. The performance of the proposed tests is evaluated using a Monte Carlo study as well as asymptotic relative efficiencies and is compared to that of several competitors. It is observed that such tests yield a better performance as compared to their competitors for a wide spectrum of alternatives.

Some high-dimensional one-sample tests based on functions of interpoint distances

The multivariate one-sample location problem is well studied in the literature, and several tests are available for it. But most of the existing one-sample tests perform poorly for high-dimensional data, and many of them are not even applicable when the dimension of the data exceeds the sample size. In this article, we develop and investigate some nonparametric one-sample tests based on functions of interpoint distances. These proposed tests can be conveniently used in high dimension, low sample size (HDLSS) situations, and good power properties of these tests for HDLSS data have been established using theoretical as well as numerical results.

A one-sample location test based on weighted averaging of two test statistics when the dimension and the sample size are large

ABSTRACTWe discuss a one-sample location test that can be used when the dimension and the sample size are large. It is well-known that the power of Hotelling’s test decreases when the dimension is close to the sample size. To address this loss of power, some non exact approaches were proposed, e.g., Dempster (1958, 1960), Bai and Saranadasa (1996), and Srivastava and Du (2008). In this article, we focus on Hotelling’s test and Dempster’s test. The comparative merits and demerits of these two tests vary according to the local parameters. In particular, we consider the situation where it is difficult to determine which test should be used, that is, where the two tests are asymptotically equivalent in terms of local power. We propose a new statistic based on the weighted averaging of Hotelling’s T2-statistic and Dempster’s statistic that can be applied in such a situation. Our weight is determined on the basis of the maximum local asymptotic power on a restricted parameter space that induces local asymptotic equivalence between Hotelling’s test and Dempster’s test. Numerical results show that our test is more stable than Hotelling’s T2-statistic and Dempster’s statistic in most parameter settings.

일표본 위치검정에서의 표본크기 결정

이 논문에서는 일표본 위치검정에서 표본크기를 결정하는데 일반적으로 사용되고 있는 근사분포방법의 문제점에 대해 알아본다. 모의실험 결과 이들 근사분포방법은 검정력의 명목수준에 도달하지 않는 것으로 나타났으며 이에 대한 개선이 필요한 것으로 나타났다. 이 논문에서는 검정통계량의 정확분포를 이용한 표본크기 결정에 대해 알아보고 이를 통한 표본크기의 검정력이 명목수준을 충족하는 것을 보인다. We study problems of sample size determination for one-sample location tests. A simulation study shows that sample size calculations based on approximated distribution do not achieve the nominal level of power. We investigate sample size determinations based on exact distribution and with a power that attains the nominal level.

On some exact distribution-free one-sample tests for high dimension low sample size data

Several rank-based tests for the multivariate one-sample problem are available in the literature. But, unlike univariate rank-based tests, most of these multivariate tests are not distribution-free. Moreover, many of them are not ap- plicable when the dimension of the data exceeds the sample size. We develop and investigate some distribution-free tests for the one-sample location problem, which can be conveniently used in high dimension low sample size (HDLSS) situations. Under some appropriate regularity conditions, we prove the consistency of these tests when the sample size remains fixed and the dimension grows to infinity. Some simulations and data sets are analyzed to compare their performance with popular one-sample tests.

Identification of Significant Features by the Global Mean Rank Test

With the introduction of omics-technologies such as transcriptomics and proteomics, numerous methods for the reliable identification of significantly regulated features (genes, proteins, etc.) have been developed. Experimental practice requires these tests to successfully deal with conditions such as small numbers of replicates, missing values, non-normally distributed expression levels, and non-identical distributions of features. With the MeanRank test we aimed at developing a test that performs robustly under these conditions, while favorably scaling with the number of replicates. The test proposed here is a global one-sample location test, which is based on the mean ranks across replicates, and internally estimates and controls the false discovery rate. Furthermore, missing data is accounted for without the need of imputation. In extensive simulations comparing MeanRank to other frequently used methods, we found that it performs well with small and large numbers of replicates, feature dependent variance between replicates, and variable regulation across features on simulation data and a recent two-color microarray spike-in dataset. The tests were then used to identify significant changes in the phosphoproteomes of cancer cells induced by the kinase inhibitors erlotinib and 3-MB-PP1 in two independently published mass spectrometry-based studies. MeanRank outperformed the other global rank-based methods applied in this study. Compared to the popular Significance Analysis of Microarrays and Linear Models for Microarray methods, MeanRank performed similar or better. Furthermore, MeanRank exhibits more consistent behavior regarding the degree of regulation and is robust against the choice of preprocessing methods. MeanRank does not require any imputation of missing values, is easy to understand, and yields results that are easy to interpret. The software implementing the algorithm is freely available for academic and commercial use.

The p value line: a way to choose the tuning constant in tests based on the Huber $${\varvec{M}}$$ M -estimator

In this paper, we propose a method to choose a test result from several competing ones. The choice is based on the p value under the alternative, i.e., the probability of obtaining an extreme or a more extreme result than the one actually observed, assuming that the alternative hypothesis is true. The function obtained, by moving the alternative, is called the p value line. Because this combines the power of a test and its sensitivity to outliers, it is considered to be the basis for the selection between several classical and/or robust test results. When we change the trimming fraction in the trimmed mean, or the tuning constant when we consider a test based on the Huber \(M\)-estimator, we obtain competing robust test results. For this reason, we can use the p value line to choose the trimming fraction or the tuning constant in an objective way. In this article, we shall focus on the second class of estimators. Because computing the distribution of the test statistic under the alternative could be a very hard problem, a linear approximation of the tail probability functional based on the von Mises expansion can be used to compute the p value line. Applications of the proposal to the one-sample location problem, generalized linear models and generalized additive models are considered in the article. We conclude the article with the random p value line to decide if the observed differences between p value lines are significant or not.

An optimal sign test for one-sample bivariate location model using an alternative bivariate ranked-set sample

The aim of this paper is to find an optimal alternative bivariate ranked-set sample for one-sample location model bivariate sign test. Our numerical and theoretical results indicated that the optimal designs for the bivariate sign test are the alternative designs with quantifying order statistics with labels {((r+1)/2, (r+1)/2)}, when the set size r is odd and {(r/2+1, r/2), (r/2, r/2+1)} when the set size r is even. The asymptotic distribution and Pitman efficiencies of these designs are derived. A simulation study is conducted to investigate the power of the proposed optimal designs. Illustration using real data with the Bootstrap algorithm for P-value estimation is used.

One-sample location tests for multilevel data

In this paper, we consider testing the location parameter with multilevel (or hierarchical) data. A general family of weighted test statistics is introduced. This family includes extensions to the case of multilevel data of familiar procedures like the t, the sign and the Wilcoxon signed-rank tests. Under mild assumptions, the test statistics have a null limiting normal distribution which facilitates their use. An investigation of the relative merits of selected members of the family of tests is achieved theoretically by deriving their asymptotic relative efficiency (ARE) and empirically via a simulation study. It is shown that the performance of a test depends on the clusters configurations and on the intracluster correlations. Explicit formulas for optimal weights and a discussion of the impact of omitting a level are provided for 2 and 3-level data. It is shown that using appropriate weights can greatly improve the performance of the tests. Finally, the use of the new tests is illustrated with a real data example.

One-Sample Likelihood Ratio Tests for Mixed Data

We revisit a hypothesis-testing problem recently investigated by de Leon and Carrière (2000). Specifically, we obtain exact likelihood ratio tests of one-sample location hypotheses for multivariate mixed data modeled according to the general location model (Olkin and Tate, 1961). The tests generalize those previously proposed by de Leon and Carrière (2000) for the case of mixed bivariate data. Optimal properties of the tests are briefly studied. Simulations show that the tests are reasonably powerful in detecting differences between the true and hypothesized populations. We illustrate the tests with a few examples, including one concerning data on academic achievement.

A weighted spatial median for clustered data

A weighted spatial median is proposed for the multivariate one-sample location problem with clustered data. Its limiting distribution is derived under mild conditions (no moment assumptions) and it is shown to be multivariate normal. Asymptotic as well as finite sample efficiencies and breakdown properties are considered, and the theoretical results are supplied with illustrative examples. It turns out that there is a potential for meaningful gains in estimation efficiency: the weighted spatial median has superior efficiency to the unweighted spatial median particularly when the cluster sizes are widely disparate and in the presence of strong intracluster correlation. The unweighted spatial median for clustered data was considered earlier by Nevalainen et al. (Can J Statist, in press, 2007). The proposed weighted estimators provide companion estimates to the weighted affine invariant sign test proposed recently by Larocque et al. (Biometrika, in press, 2007). An affine equivariant weighted spatial median is discussed in parallel.

Power of One-Sample Location Tests Under Distributions with Equal Lévy Distance

In this article, we study the power of one-sample location tests under classical distributions and two supermodels which include the normal distribution as a special case. The distributions of the supermodels are chosen in such a way that they have equal distance to the normal as the logistic, uniform, double exponential, and the Cauchy, respectively. As a measure of distance we use the Lévy metric. The tests considered are two parametric tests, the t-test and a trimmed t-test, and two nonparametric tests, the sign test and the Wilcoxon signed-rank tests. It turns out that the power of the tests, first of all, does not depend on the Lévy distance but on the special chosen supermodel.

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 Nonparametric Test for the One-Sample Bivariate Location Problem

SYNOPTIC ABSTRACTA nonparametric test is proposed for the bivariate one-sample location problem. The proposed test is robust in nature. The asymptotic distribution of the test statistic is computed as Chi-square. The power of the proposed test statistic is compared with Hotelling's T2 test statistic, Mardia's (1967) test statistic, Blumen's (1958) test statistic, and Wilcoxon's rank sum test statistic using Monte Carlo techniques. The proposed test statistic performs better than its competitors considered here under almost all the underlying distributions used for the power study. The application of the test is illustrated using a bivariate one sample data.

Robustness to Non-Normality of Various Tests For the One-Sample Location Problem

This paper studies the effect of the normal distribution assumption on the power and size of the sign test, Wilcoxon's signed rank test and the t-test when used in one-sample location problems. Power functions for these tests under various skewness and kurtosis conditions are produced for several sample sizes from simulated data using the g-and-k distribution of MacGillivray and Cannon [5].

Multivariate Nonparametric Tests

Multivariate nonparametric statistical tests of hypotheses are described for the one-sample location problem, the several-sample location problem and the problem of testing independence between pairs of vectors. These methods are based on affine-invariant spatial sign and spatial rank vectors. They provide affine-invariant multivariate generalizations of the univariate sign test, signed-rank test, Wilcoxon rank sum test, Kruskal–Wallis test, and the Kendall and Spearman correlation tests. While the emphasis is on tests of hypotheses, certain references to associated affine-equivariant estimators are included. Pitman asymptotic efficiencies demonstrate the excellent performance of these methods, particularly in heavy-tailed population settings. Moreover, these methods are easy to compute for data in common dimensions.

Robustness to non-normality of various tests for the one-sample location problem

This paper studies the effect of the normal distribution assumption on the power and size of the sign test, Wilcoxon's signed rank test and the t-test when used in one-sample location problems. Power functions for these tests under various skewness and kurtosis conditions are produced for several sample sizes from simulated data using the g-and-k distribution of MacGillivray and Cannon [5].