Nonparametric Test Procedure Research Articles

A Two-Sample Conditional Distribution Test Using Conformal Prediction and Weighted Rank Sum

We consider the problem of testing the equality of conditional distributions of a response variable given a vector of covariates between two populations. Such a hypothesis testing problem can be motivated from various machine learning and statistical inference scenarios, including transfer learning and causal predictive inference. We develop a nonparametric test procedure inspired from the conformal prediction framework. The construction of our test statistic combines recent developments in conformal prediction with a novel choice of conformity score, resulting in a weighted rank-sum test statistic that is valid and powerful under general settings. To our knowledge, this is the first successful attempt of using conformal prediction for testing statistical hypotheses beyond exchangeability. Our method is suitable for modern machine learning scenarios where the data has high dimensionality and large sample sizes, and can be effectively combined with existing classification algorithms to find good conformity score functions. The performance of the proposed method is demonstrated in various numerical examples.

A nonparametric test for the two-sample problem based on order statistics

We study a nonparametric test procedure based on order statistics for testing the null hypothesis of equality of two continuous distributions. The exact null distribution of the proposed test statistic is obtained using an enumeration method and a novel combinatorial argument. A recurrence relation for the probability generating function and a sequential approach for computing the mean and variance of the distribution are given. Critical values and characteristics of the distribution for selected small sample sizes are presented. For the Lehmann alternative family, the exact power function of the new test is derived, and its power performance is examined. We also study the power performance of the proposed test under the location-shift and scale-shift alternatives using Monte Carlo simulations and observe its superior performance when compared to commonly used nonparametric tests under various scenarios. A generalization of the proposed procedure for unequal sample sizes is discussed. An illustrative example and some concluding remarks are provided.

Some first inferential tools for spatial regression with differential regularization

Spatial regression with differential regularization is an innovative approach at the crossroad between functional data analysis and spatial data analysis. These models have been shown to be numerically efficient and capable to handle complex applied problems. On the other hand, their theoretical properties are still largely unexplored. Here we consider the discrete estimators in spatial regression models with differential regularization, obtained after numerical discretization, using an expansion on a finite element basis. We study the consistency and the asymptotic normality of these discrete estimators. We also propose a nonparametric test procedure for the linear part of the models, based on random sign-flipping of the score components. The test exploits an appropriate decomposition of the smoothing matrix, in order to reduce the effect of the spatial dependence, without any parametric assumption on the form of the correlation structure. The proposed test is shown to be superior to parametric alternatives.

Analysis of Means by Ranks: Modification and Application in Contract Acquisition Analysis

Analysis of Means by Ranks is a nonparametric statistical test procedure that was developed in Bakir (1989) but has rarely been applied in practice. This paper modifies and applies Analysis of Means by Ranks to a case study data involving the comparison of three contract proposals. For comparison purposes, we analyze the same data using the well-known Analysis of Variance, Analysis of Means, and the Kruskal-Wallis test. Analysis of Variance and Analysis of Means are two parametric (assume data to be samples from normal populations) test procedures whereas Kruskal-Wallis and Analysis of Means by Ranks are two nonparametric (or distribution-free) procedures. This paper shows that the parametric tests fail to detect a significant difference among three contract proposals, while the nonparametric tests do. The conclusions of the parametric tests are in doubt because a descriptive statistics analysis indicates that the required normality assumption is in doubt; the nonparametric conclusions are more trustful because the normality assumption is not required by nonparametric procedures.

Variance Homogeneity Test Based on Cumulative Wavelet Coefficients

The aim of this paper is to address the problem of variance break detection in time series in wavelet domain. The maximal overlapped discrete wavelet transform (MODWT) decomposes the series variance across scales into components known as the wavelet variances. We introduce all scale wavelet coefficients based test statistic that allows detecting a break in the homogeneity of the variance of a series through changes in the mean of wavelet variances. The statistic makes use of the traditional CUSUM (cumulative sum) based test designed to test for a break in the mean and constructed using cumulative sums of the square of wavelet coefficients. Under moments and mixing conditions, the test statistic satisfies the functional central limit theorem (FCLT) for a broad class of time series models. The overall performance of our test statistic is compared to the traditional Inclan [8] test statistic. The effectiveness of our statistic is supported by good performances reported in simulations and is as reliable as the traditional statistic. Our method provides a nonparametric test procedure that can be applied to a large class of linear and non linear models. We illustrate the practical use of our test procedure with the quarterly percentage changes in the Americans personal savings data set over the period 1970-2016. Both statistics detect a break in the variance in the second quarter of 2001.

Symbolic transfer entropy test for causality in longitudinal data

In this study, we use multiple-unit symbolic dynamics and transfer entropy to develop a non-parametric Granger causality test procedure for longitudinal data. Monte Carlo simulations show that our test exhibits the correct size and a high power in situations where linear panel data causality tests fail, such as (1) when the linearity assumption does not hold, (2) when the data generating process is heterogeneous across the cross-section units or presents structural breaks, (3) when there are extreme observations in some of the cross-section units, (4) when the process exhibits causal dependence on the conditional variance, or (5) when the analysis involves qualitative data. We illustrate the usefulness of our proposed procedure by analyzing the dynamic causal relationships between public expenditure and GDP, between firm productivity and firm size in US manufacturing sectors, and among sovereign credit ratings, growth, and interest rates.

Nonparametric multiple comparisons.

Nonparametric multiple comparisons are a powerful statistical inference tool in psychological studies. In this paper, we review a rank-based nonparametric multiple contrast test procedure (MCTP) and propose an improvement by allowing the procedure to accommodate various effect sizes. In the review, we describe relative effects and show how utilizing the unweighted reference distribution in defining the relative effects in multiple samples may avoid the nontransitive paradoxes. Next, to improve the procedure, we allow the relative effects to be transformed by using the multivariate delta method and suggest a log odds-type transformation, which leads to effect sizes similar to Cohen's d for easier interpretation. Then, we provide theoretical justifications for an asymptotic strong control of the family-wise error rate (FWER) of the proposed method. Finally, we illustrate its use with a simulation study and an example from a neuropsychological study. The proposed method is implemented in the 'nparcomp' R package via the 'mctp' function.

Distribution Free Test for Unequal Observation of Panel Count Data with Application in Medical Follow-Up Study

Objectives: In medical follow-up study, patients are treated with different treatments may have different follow-up schedule and the number of patients assigned in each treatment group may not be balanced. To address this, a nonparametric test procedure is proposed. Methods/Statistical Analysis: The proposed test statistics is constructed based on the integrated weighted differences between the mean cumulative function of the recurrences event with condition on treatment group. For performance evaluation, the empirical power of proposed test statistics are evaluated via Monte Carlo simulation study conducted using R statistical software. Findings: Based on simulation results, the proposed method gives a good power for both identical and unequal follow-up processes, even when the sample sizes are imbalanced. Applications/Improvement: The proposed test procedure is also applied to a set of panel count data arises from the National Cooperative Gallstone Study to compare the treatment efficiency. The proposed test procedure able to detect treatment differences and is in line with earlier research.

Massively parallel processing of recursive multi-period portfolio models

A recursive portfolio decision system is extended with parallel processing capability monitored by the Genetic Hybrid Algorithm (GHA). Massively parallel portfolio efficiency testing is conducted using stochastic simulation. Genuine out-of-sample forecasts are generated for all titles in the universe using fast cutting-edge time series algorithms. The computation of dynamic optimal portfolio weights is done within an affine multi-period setting. The terminal wealth within the planning horizon forms a moving target as the system evolves through time and only current transactions are carried out. Fixed and variable transaction costs are recognized without increasing computational complexity. We show using a recursive multi-period portfolio framework (RMP) that robust grid search and stochastic simulation with thousands of parallel processors can be conducted to provide evidence on portfolio efficiency. The downside risk of the RMP-strategy is significantly lower than that of the corresponding buy-and-hold strategy. The upside potential of RMP is much better than that of buy-and-hold. The non-parametric test procedure is independent of the underlying model and hence completely general. The modular structure of the system allows new forecasting techniques and optimization formulations to be introduced and tested in future development efforts.

Jumps in equilibrium prices and asymmetric news in foreign exchange markets

In this paper we examine the intraday effects of surprises from scheduled macroeconomic announcements and unscheduled event news on six major exchange rate excess returns (jumps) using a Tobit model with conditionally heteroskedastic errors that we extend so as to account for asymmetries. Besides this novel model, our approach embodies several important features: we perform Lee and Mykland's (2012) non-parametric test procedure to filter out microstructure noise from observed exchange rates and extract jumps as the significant “equilibrium” returns; various categories of information news from different geographical regions are exploited; the hypothesis of a leverage effect on foreign exchange jumps due to asymmetric volatility shocks is examined. We found that the most influential scheduled macroeconomic news are globally related to the US job markets, output growth indicators and public debt, whereas significant event news include announcements of bank failures and government rescue plans. Surprises impact Forex jumps for about one third as a result of rather pessimistic forecasts due to the crisis period analyzed. For most of the currencies the hypothesis that negative volatility shocks have a greater impact on volatility than positive shocks of the same magnitude is validated, reflecting markets' concern about the costs implied by central bank's stabilization policies. Our findings provide evidence that the major foreign exchange markets are not efficient.

A New Non-parametric Test Procedure for the Median of an Asymmetrical Population

This paper proposes and investigates the performance of a new non-parametric test procedure for the median of a non-normal population when the symmetry assumption is suspected. The new test procedure uses the Yeo-Johnson family of power transformations and the Shapiro-Wilk test of normality to modify the classical normal scores test. Under skewed models, simulation results show that the proposed test procedure is superior to all competitor tests under consideration in terms of preserving the empirical size of the test at its nominal level and also having higher empirical powers.

A Nonparametric Test Procedure Based on Quantiles for the Right Censored Data

In this study, we propose nonparametric tests using the several quantile statistics simultaneously for the right censored data. First of all, we consider statistics of the quadratic form with estimated covariance matrices. Then we derive the limiting distribution using the large sample approximation theory. Also we consider different forms of statistics such as the maximal and summing types with their limiting distributions. Then we illustrate our procedure with examples and compare performance among tests with empirical powers through a simulation study. Also we comment briefly on some interesting features including re-sampling methods as concluding remarks. Finally in Appendices, we provide proofs for the theoretic results needed for the derivation of the limiting distributions of the proposed test statistics.

The Effectiveness of a Meaning-Centred Intervention in Protecting the Well-Being HIV/AIDS Health Educators

This study investigated whether teaching HIV/AIDS Health Educators about Viktor Frankl’s theory of finding meaning in life, results in a significant improvement in their psychological well-being. The participants were 24 volunteers from the Information, Education and Communication (IEC) cluster of a Non-Governmental Organization (NGO) in South Africa (age range = 20–39 years, females = 67 %, majority ethnicity = 58 % Sotho speaking). The study was both quantitative and qualitative. Quantitative data was collected using Purpose in Life Test (PIL) and Beck Depression Inventory (BDI-IA). In addition, qualitative data was collected on the participants’ subjective experience of development in creative potential. A one group pretest–posttest design was employed. The quantitative data was analysed using a non-parametric test procedure for small sample size designs. The qualitative data was analysed by means of themes. The researcher discovered that a sense of meaning intervention seems to result in significant improvement in levels of meaning, and a decrease in depressive symptoms among participants. Consequently, the findings indicate that it is possible to support front line health workers in their search for meaning by means of a meaning-centred intervention, and that this helps to alleviate depressive symptoms. Similarly, meaning-centred intervention seems to be a promising intervention for instilling resilience amongst front line health workers, who provide a valuable voluntary service in the community. Participants indicated they could apply the theory to their personal lives. More importantly, they would use the skills they have acquired in their duties.

A mutual information-based k-sample test for discrete distributions

The two-sample problem and its extension to the k-sample problem are well known in the statistical literature. However, the discrete version of the k-sample problem is relatively less explored. Here in this work we suggest a k-sample non-parametric test procedure for discrete distributions based on mutual information. A detailed power study with comparison with other alternatives is provided. Finally, a comparison of some English soccer league teams based on their goal-scoring pattern is discussed.

Goodness-of-fit test for noisy directional data

We consider spherical data $X_i$ noised by a random rotation $\varepsilon_i\in$ SO(3) so that only the sample $Z_i=\varepsilon_iX_i$, $i=1,\dots, N$ is observed. We define a nonparametric test procedure to distinguish $H_0:$ ''the density $f$ of $X_i$ is the uniform density $f_0$ on the sphere'' and $H_1:$ ''$\|f-f_0\|_2^2\geq \C\psi_N$ and $f$ is in a Sobolev space with smoothness $s$''. For a noise density $f_\varepsilon$ with smoothness index $\nu$, we show that an adaptive procedure (i.e. $s$ is not assumed to be known) cannot have a faster rate of separation than $\psi_N^{ad}(s)=(N/\sqrt{\log\log(N)})^{-2s/(2s+2\nu+1)}$ and we provide a procedure which reaches this rate. We also deal with the case of super smooth noise. We illustrate the theory by implementing our test procedure for various kinds of noise on SO(3) and by comparing it to other procedures. Applications to real data in astrophysics and paleomagnetism are provided.

On the Study for the Simultaneous Test

In this study, we propose a nonparametric simultaneous test procedure for the location translation and scale parameters. We consider the Wilcoxon rank sum test for the location translation parameter and the Mood test for the scale parameter with the quadratic and maximal types of combining functions. Then we derive the limiting null distributions of the combining functions. We illustrate our procedure with an example and compare efficiency by obtaining the empirical powers through a simulation study. Finally, we discuss some interesting features related to the nonparametric simultaneous tests.

Differential evolution and differential ant-stigmergy on dynamic optimisation problems

Many real-world optimisation problems are of dynamic nature, requiring an optimisation algorithm which is able to continuously track a changing optimum over time. To achieve this, we propose two population-based algorithms for solving dynamic optimisation problems (DOPs) with continuous variables: the self-adaptive differential evolution algorithm (jDE) and the differential ant-stigmergy algorithm (DASA). The performances of the jDE and the DASA are evaluated on the set of well-known benchmark problems provided for the special session on Evolutionary Computation in Dynamic and Uncertain Environments. We analyse the results for five algorithms presented by using the non-parametric statistical test procedure. The two proposed algorithms show a consistently superior performance over other recently proposed methods. The results show that both algorithms are appropriate candidates for DOPs.

Small is not that beautiful after all: measuring the scale efficiency of Tunisian high schools using a DEA-bootstrap method

Allocation of resources to schools in a centrally managed state system, as the Tunisian one, should depend on the performance of the individual institutions. The optimal size is of crucial importance in this context and we need accurate measurement for sound policies. This article discusses and implements a nonparametric statistical test procedure for organization scale efficiency. This procedure allows us to test whether the observed scale efficiency is optimal or not, using a smooth bootstrap methodology for efficiency measures estimated using Data Envelopment Analysis (DEA) methods. Because school principals do not control for the size of their institution, i.e. the capital available at decision time, the scale efficiency measures are defined so as to include quasi-fixed inputs. The results show that scale efficiency measures are subject to sampling variation. We also found that the schools that are scale efficient are usually mid-sized and large schools, when size is measured by the number of students. This contradicts the largely shared view among decision makers that small schools were optimal.

A new class of generalized log rank tests for interval-censored failure time data

This paper discusses nonparametric comparison of survival functions when one observes only interval-censored failure time data (Peto and Peto, 1972; Sun, 2006; Zhao et al., 2008). For the problem, a few procedures have been proposed in the literature. However, most of the existing test procedures determine the test results or p-values based on ad hoc methods or the permutation approach. Furthermore for the test procedures whose asymptotic distributions have been derived, the results are only for the null hypothesis. In other words, no nonparametric test procedure exists that has a known asymptotic distribution under the alternative hypothesis and thus can be employed to carry out the power and test size calculation. In this paper, a new class of generalized log-rank tests is proposed and their asymptotic distributions are derived under both null and alternative hypotheses. A simulation study is conducted to assess their performance for finite sample situations and an illustrative example is provided.

Multivariate Test based on the Multiple Testing Approach

In this study, we propose a new nonparametric test procedure for the multivariate data. In order to accommodate the generalized alternatives for the multivariate case, we construct test statistics via-values with some useful combining functions. Then we illustrate our procedure with an example and compare efficiency among the combining functions through a simulation study. Finally we discuss some interesting features related with the new nonparametric test as concluding remarks.