Resampling schemes with low resampling intensity and their applications in testing hypotheses

The paper explores statistical features of different resampling schemes under low resampling intensity. The original sample is considered in a very general framework of triangular arrays, without independence or equally distributed assumptions, although improvements under such conditions are also provided. We show that low resampling schemes have very interesting and flexible properties, providing new insights into the performance of widely used resampling methods, including subsampling, two-sample unbalanced permutation statistics or wild bootstrap. It is shown that, under regularity assumptions, resampling tests with critical values derived by the appertaining low resampling procedures are asymptotically valid and there is no loss of power compared with the power function of an ideal (but unfeasible) parametric family of tests. Moreover we show that in several contexts, including regression models, they may act as a filter for the normal part of a limit distribution, turning down the influence of outliers.     

Tests for statistical significance of a treatment effect in the presence of hidden sub-populations

For testing the statistical significance of a treatment effect, we often compare between two parts of a population; one is exposed to the treatment, and the other is not exposed to it. Standard parametric or nonparametric two-sample tests are commonly used for this comparison. But direct applications of these tests can yield misleading results, especially when the population has some hidden sub-populations, and the effect of this sub-population difference on the response dominates the treatment effect. This problem becomes more evident if these sub-populations have widely different proportions of representatives in the samples obtained from these two parts. In this article, we propose some simple methods to overcome these limitations. These proposed methods first use a suitable clustering algorithm to find the hidden sub-populations, and then they eliminate the sub-population effect by using a suitable transformation of the data. Standard two-sample tests, when they are applied on the transformed data, usually yield better results. We analyze some simulated and real data sets to demonstrate the utility of these proposed methods.     

A two-sample empirical likelihood ratio test based on samples entropy

Powerful entropy-based tests for normality, uniformity and exponentiality have been well addressed in the statistical literature. The density-based empirical likelihood approach improves the performance of these tests for goodness-of-fit, forming them into approximate likelihood ratios. This method is extended to develop two-sample empirical likelihood approximations to optimal parametric likelihood ratios, resulting in an efficient test based on samples entropy. The proposed and examined distribution-free two-sample test is shown to be very competitive with well-known nonparametric tests. For example, the new test has high and stable power detecting a nonconstant shift in the two-sample problem, when Wilcoxon’s test may break down completely. This is partly due to the inherent structure developed within Neyman-Pearson type lemmas. The outputs of an extensive Monte Carlo analysis and real data example support our theoretical results. The Monte Carlo simulation study indicates that the proposed test compares favorably with the standard procedures, for a wide range of null and alternative distributions.     

Nonparametric test for the homogeneity of the overall variability

In this paper, we propose a nonparametric test for homogeneity of overall variabilities for two multi-dimensional populations. Comparisons between the proposed nonparametric procedure and the asymptotic parametric procedure and a permutation test based on standardized generalized variances are made when the underlying populations are multivariate normal. We also study the performance of these test procedures when the underlying populations are non-normal. We observe that the nonparametric procedure and the permutation test based on standardized generalized variances are not as powerful as the asymptotic parametric test under normality. However, they are reliable and powerful tests for comparing overall variability under other multivariate distributions such as the multivariate Cauchy, the multivariate Pareto and the multivariate exponential distributions, even with small sample sizes. A Monte Carlo simulation study is used to evaluate the performance of the proposed procedures. An example from an educational study is used to illustrate the proposed nonparametric test.     

Nonparametric Bayes Stochastically Ordered Latent Class Models

Latent class models (LCMs) are used increasingly for addressing a broad variety of problems, including sparse modeling of multivariate and longitudinal data, model-based clustering, and flexible inferences on predictor effects. Typical frequentist LCMs require estimation of a single finite number of classes, which does not increase with the sample size, and have a well-known sensitivity to parametric assumptions on the distributions within a class. Bayesian nonparametric methods have been developed to allow an infinite number of classes in the general population, with the number represented in a sample increasing with sample size. In this article, we propose a new nonparametric Bayes model that allows predictors to flexibly impact the allocation to latent classes, while limiting sensitivity to parametric assumptions by allowing class-specific distributions to be unknown subject to a stochastic ordering constraint. An efficient MCMC algorithm is developed for posterior computation. The methods are validated using simulation studies and applied to the problem of ranking medical procedures in terms of the distribution of patient morbidity.     

Bayesian prediction with a random sample size for the burr lifetime distribution

This paper is concerned with the problem of deriving Bayesian prediction bounds for the Burr distribution when the sample size is a random variable. Prediction bounds for both the future observations (the case of two-sample prediction) and the remaining observations in the same sample (the case of one-sample prediction) will be derived. The analysis will depend mainly on assuming that the size of the sample is a random variable having the Poisson distribution. Finally, numerical examples are given to illustrate the results.     

A note on the nonparametric test based on theL 1-version of the Cramér-von Mises statistic

We consider the test based on theL ₁-version of the Cramér-von Mises statistic for the nonparametric two-sample problem. Some quantiles of the exact distribution under H₀ of the test statistic are computed for small sample sizes. We compare the test in terms of power against general alternatives to other two-sample tests, namely the Wilcoxon rank sum test, the Smirnov test and the Cramér-von Mises test in the case of unbalanced small sample sizes. The computation of the power is rather complicated when the sample sizes are unequal. Using Monte Carlo power estimates it turns out that the Smirnov test is more sensitive to non stochastically ordered alternatives than the new test. And under location-contamination alternatives the power estimates of the new test and of the competing tests are equal.     

Combination of distribution-free tests for the general two-sample problem with application to the social sciences

ABSTRACT

The problem of detecting any differences between the distributions of two populations is addressed within the non parametric permutation framework of combined tests. Combined testing has been very useful to address the location, the scale, and the location/scale problems. The aim of the paper is to see whether combined testing is useful also for the general two-sample problem. The framework of combined testing for the general two-sample problem is presented and some tests are proposed. These tests are valid even when a non random sample of units is randomized into two groups. Type 1 error rate and power characteristics of the new tests are investigated and compared to former tests. It is shown that the new tests compare favorably with the former ones. An application to a very important socioeconomic problem is discussed.     

Testing nonparametric statistical functionals with applications to rank tests

The present paper discusses how nonparametric tests can be deduced from statistical functionals. Efficient and asymptotically most powerful maximin tests are derived. Their power function is calculated under implicit alternatives given by the functional for one – and two – sample testing problems. It is shown that the asymptotic power function does not depend on the special implicit direction of the alternatives but only on quantities of the functional. The present approach offers a nonparametric principle how to construct common rank tests as the Wilcoxon test, the log rank test, and the median test from special two-sample functionals. In addition it is shown that studentized permutation tests yield asymptotically valid tests for certain extended null hypotheses given by functionals which are strictly larger than the common i.i.d. null hypothesis. As example tests concerning the von Mises functional and the Wilcoxon two-sample test are treated.     

A note on the multivariate multisample rank sum and median tests

The muitivariate nonparametric tests analogous to the univar-iate rank sum test and median test are contained in Puri and Sen (1970). These tests provided a practical alternative for the analysis of multivariate data when the assumptions of parametric methods are not satisfied.

In this paper maximum values for L_Nthe asymptotic chi-Square test statistic for both the Multivariate Multisample Rank Sum Test (MMRST) and the Multivariate Multisample Median Test (MMMT) are developed.     

Two sample nonparametric tests based on subsamples

The paper introduces a general class of nonparametric tests for the two-sample location problem based on subsamples. Includ- ed in this class is the Mann-Whitney (or the Wilcoxon rank sum) test. General formulas for the Pitman efficacy for different methods of subsampling are derived. A small sample power simu- lation compares the performance of members of this class     

Comparison of Nonparametric and Parametric Methods in Repeated Measures Designs - A Simulation Study

This study investigates the performance of parametric and nonparametric tests to analyze repeated measures designs. Both multivariate normal and exponential distributions were simulated for varying values of the correlation and ten or twenty subjects within each cell. For multivariate normal distributions, the type I error rates were lower than the usual 0.05 level for nonparametric tests, whereas the parametric tests without the Greenhouse-Geisser or the Huynh-Feldt adjustment produced slightly higher type I error rates. Type I error rates for nonparametric tests, for multivariate exponential distributions, were more stable than parametric, Greenhouse-Geisser or Huynh-Feldt adjusted tests. For ten subjects within each cell, the parametric tests were more powerful than nonparametric tests. For twenty subjects per cell, the power of the nonparametric and parametric tests was comparable.     

Šidák-type tests for the two-sample problem based on precedence and exceedance statistics

This paper deals with a class of nonparametric two-sample tests for ordered alternatives. The test statistics proposed are based on the number of observations from one sample that precede or exceed a threshold specified by the other sample, and they are extensions of ?idák's test. We derive their exact null distributions and also discuss a large-sample approximation. We then study their power properties exactly against the Lehmann alternative and make some comparative comments. Finally, we present an example to illustrate the proposed tests.     

Estimating curves and derivatives with parametric penalized spline smoothing

Accurate estimation of an underlying function and its derivatives is one of the central problems in statistics. Parametric forms are often proposed based on the expert opinion or prior knowledge of the underlying function. However, these strict parametric assumptions may result in biased estimates when they are not completely accurate. Meanwhile, nonparametric smoothing methods, which do not impose any parametric form, are quite flexible. We propose a parametric penalized spline smoothing method, which has the same flexibility as the nonparametric smoothing methods. It also uses the prior knowledge of the underlying function by defining an additional penalty term using the distance of the fitted function to the assumed parametric function. Our simulation studies show that the parametric penalized spline smoothing method can obtain more accurate estimates of the function and its derivatives than the penalized spline smoothing method. The parametric penalized spline smoothing method is also demonstrated by estimating the human height function and its derivatives from the real data.     

Nonparametric predictive inference for stock returns

In finance, inferences about future asset returns are typically quantified with the use of parametric distributions and single-valued probabilities. It is attractive to use less restrictive inferential methods, including nonparametric methods which do not require distributional assumptions about variables, and imprecise probability methods which generalize the classical concept of probability to set-valued quantities. Main attractions include the flexibility of the inferences to adapt to the available data and that the level of imprecision in inferences can reflect the amount of data on which these are based. This paper introduces nonparametric predictive inference (NPI) for stock returns. NPI is a statistical approach based on few assumptions, with inferences strongly based on data and with uncertainty quantified via lower and upper probabilities. NPI is presented for inference about future stock returns, as a measure for risk and uncertainty, and for pairwise comparison of two stocks based on their future aggregate returns. The proposed NPI methods are illustrated using historical stock market data.     

Nonparametric Simultaneous Tests for Location and Scale Testing: A Comparison of Several Methods

The two-sample location-scale problem arises in many situations like climate dynamics, bioinformatics, medicine, and finance. To address this problem, the nonparametric approach is considered because in practice, the normal assumption is often not fulfilled or the observations are too few to rely on the central limit theorem, and moreover outliers, heavy tails and skewness may be possible. In these situations, a nonparametric test is generally more robust and powerful than a parametric test. Various nonparametric tests have been proposed for the two-sample location-scale problem. In particular, we consider tests due to Lepage, Cucconi, Podgor-Gastwirth, Neuhäuser, Zhang, and Murakami. So far all these tests have not been compared. Moreover, for the Neuhäuser test and the Murakami test, the power has not been studied in detail. It is the aim of the article to review and compare these tests for the jointly detection of location and scale changes by means of a very detailed simulation study. It is shown that both the Podgor–Gastwirth test and the computationally simpler Cucconi test are preferable. Two actual examples within the medical context are discussed.     

Semiparametric Analysis of Truncated Data

Randomly truncated data are frequently encountered in many studies where truncation arises as a result of the sampling design. In the literature, nonparametric and semiparametric methods have been proposed to estimate parameters in one-sample models. This paper considers a semiparametric model and develops an efficient method for the estimation of unknown parameters. The model assumes that K populations have a common probability distribution but the populations are observed subject to different truncation mechanisms. Semiparametric likelihood estimation is studied and the corresponding inferences are derived for both parametric and nonparametric components in the model. The method can also be applied to two-sample problems to test the difference of lifetime distributions. Simulation results and a real data analysis are presented to illustrate the methods.     

Robust nonparametric tests for the two-sample location problem

We construct and investigate robust nonparametric tests for the two-sample location problem. A test based on a suitable scaling of the median of the set of differences between the two samples, which is the Hodges-Lehmann shift estimator corresponding to the Wilcoxon two-sample rank test, leads to higher robustness against outliers than the Wilcoxon test itself, while preserving its efficiency under a broad range of distributions. The good performance of the constructed test is investigated under different distributions and outlier configurations and compared to alternatives like the two-sample t-, the Wilcoxon and the median test, as well as to tests based on the difference of the sample medians or the one-sample Hodges-Lehmann estimators.     

Nearly exact tests in factorial experiments using the aligned rank transform

A procedure is studied that uses rank-transformed data to perform exact and estimated exact tests, which is an alternative to the commonly used F-ratio test procedure. First, a common parametric test statistic is computed using rank-transformed data, where two methods of ranking-ranks taken for the original observations and ranks taken after aligning the observations-are studied. Significance is then determined using either the exact permutation distribution of the statistic or an estimate of this distribution based on a random sample of all possible permutations. Simulation studies compare the performance of this method with the normal theory parametric F-test and the traditional rank transform procedure. Power and nominal type I error rates are compared under conditions when normal theory assumptions are satisfied, as well as when these assumptions are violated. The method is studied for a two-factor factorial arrangement of treatments in a completely randomized design and for a split-unit experiment. The power of the tests rivals the parametric F-test when normal theory assumptions are satisfied, and is usually superior when normal theory assumptions are not satisfied. Based on the evidence of this study, the exact aligned rank procedure appears to be the overall best choice for performing tests in a general factorial experiment.     

Nonparametric tests for left-truncated and interval-censored data

This paper considers two-sample nonparametric comparison of survival function when data are subject to left truncation and interval censoring. We propose a class of rank-based tests, which are generalization of weighted log-rank tests for right-censored data. Simulation studies indicate that the proposed tests are appropriate for practical use.