Theory Of U-statistics Research Articles

Testing the equality of distributions using integrated maximum mean discrepancy

Comparing and testing for the homogeneity of two independent random samples is a fundamental statistical problem with many applications across various fields. However, existing methods may not be effective when the data is complex or high-dimensional. We propose a new method that integrates the maximum mean discrepancy (MMD) with a Gaussian kernel over all one-dimensional projections of the data. We derive the closed-form expression of the integrated MMD and prove its validity as a distributional similarity metric. We estimate the integrated MMD with the U-statistic theory and study its asymptotic behaviors under the null and two kinds of alternative hypotheses. We demonstrate that our method has the benefits of the MMD, and outperforms existing methods on both synthetic and real datasets, especially when the data is complex and high-dimensional.

A novel test of missing completely at random: U-statistics-based approach

In this paper, a novel test for testing whether data are missing completely at random is proposed. Asymptotic properties of the test are derived utilizing the theory of non-degenerate U-statistics. It is shown that the novel test statistic coincides with the well-known Little's d 2 statistic in the case of a multivariate data that has only one variable susceptible to missingness. Then, the extensive simulation study is conducted to examine the performance of the test in terms of the preservation of type I error and in terms of power. Various underlying distributions, dimensions of the data, sample sizes and alternatives are used. Performance of the Little's MCAR test is used as a benchmark for the comparison. The novel test shows better performance in all of the studied scenarios, better preserving the type I error and having higher empirical powers for every studied alternative.

On Point Estimators for Gamma and Beta Distributions

Let X 1 , … , X n be a random sample from the gamma distribution with density f ( x ) = λ α x α − 1 e − λ x / Γ ( α ) , x > 0, where both α > 0 (the shape parameter) and λ > 0 (the reciprocal scale parameter) are unknown. The main result shows that the uniformly minimum variance unbiased estimator (UMVUE) of the shape parameter, α, exists if and only if n ≥ 4 ; moreover, it has finite variance if and only if n ≥ 6 . More precisely, the form of the UMVUE is given for all parametric functions α, λ, 1 / α , and 1 / λ . Furthermore, a highly efficient estimating procedure for the two-parameter beta distribution is also given. This is based on a Stein-type covariance identity for the beta distribution, followed by an application of the theory of U-statistics and the delta-method.

Testing homogeneity in high dimensional data through random projections

Testing for homogeneity of two random vectors is a fundamental problem in statistics. In the past two decades, numerous efforts have been made to detect heterogeneity when the random vectors are multivariate or even high dimensional. Due to the “curse of dimensionality”, existing tests based on Euclidean distance may fail to capture the overall homogeneity in high-dimensional settings while can only capture the moment discrepancy. To address this issue, we propose a fully nonparametric test for homogeneity of two random vectors. Our method involves randomly selecting two subspaces consisting of components of the vectors, projecting the subspaces onto one-dimensional spaces, respectively, and constructing the test statistic using the Cramér–von Mises distance of the projections. To enhance the performance, we repeatedly implement this procedure to construct the final test statistic. Theoretically, if the replication time tends to infinity, we can avoid potential power loss caused by lousy directions. Owing to the U-statistic theory, the asymptotic null distribution of our proposed test is standard normal, regardless of the parent distributions of the random samples and the relationship between data dimensions and sample sizes. As a result, no re-sampling procedure is needed to determine critical values. The empirical size and power of the proposed test are demonstrated through numerical simulations.

A model-free-based control chart for batch process using U-statistics

Batch processes are known to generate time series of successive measurements of many process variables in each run and the main challenge is to capture and accommodate the variability in the batch domain (batch-to-batch variability) and in the time domain (data with serial correlation). Classical approaches are focused on the first goal by applying multivariate techniques in the columns of a data matrix, in which each row has the data for the entire batch run. Those approaches are grounded in the traditional Multiway Principal Component Analysis (MPCA). Recent approaches are focused on the balance between the two sources, taking into account the time series nature of the data mainly by using time series models and neural network methods to directly accommodate the variability in the time domain. This paper proposes a model-free approach based on U-statistics theory. Through this theory, we can build a group of control charts (named V charts) capable of monitoring batch processes considering both sources of variability. In general, the new batch under monitoring is evaluated according to a group of reference in-control samples, based on the distance measures from those batches. The proposed approach is really flexible since we avoid model identification, parameter estimates issues and it is able to accommodate the within and between batch variability. Additionally, our method is suitable to deal with a wide range of batch data features, since under weak conditions, the distribution of the V statistics remains well known. It makes the V-based charts able to be used in different kinds of processes. We are focused here on controlling changes caused by data drifts (including trends) and by data serial correlation (data dynamics). Through the simulated and real batch process, we show that the V charts work well, including a scenario with very few reference batches available.

A new goodness of fit test for gamma distribution with censored observations

In the present paper, we develop a new goodness of fit test for gamma distribution based on the fixed point characterization. We use U-Statistics theory to derive the test statistic. We discuss how the right censored observations are incorporated into the proposed test. The asymptotic properties of the test statistics in censored and uncensored cases are studied. We carry out a Monte Carlo simulation study to validate the finite sample performance of the proposed tests. We also illustrate the test procedures using real data sets.

Cost-effectiveness analysis under multiple effectiveness outcomes: A probabilistic approach.

Probability based criteria are proposed for the assessment of cost-effectiveness of a new treatment compared to a standard treatment when there are multiple effectiveness measures. Depending on the preferences of a policy maker, there are several options to define such criteria. Two such metrics are investigated in detail. One metric is defined as the conditional probability that a new treatment is more effective with respect to the multiple effectiveness measures for patients having lower costs under the new treatment. A second metric is defined as the conditional probability that a new treatment is less costly for patients having greater health benefits under the new treatment. The metrics offer considerable flexibility to a policy maker as thresholds for cost and effectiveness can be incorporated into the metrics. Parametric confidence limits are developed using a percentile bootstrap approach assuming multivariate normality for the joint distribution of the log(cost) and effectiveness measures. A non-parametric estimation procedure is also developed using the theory of U-statistics. Numerical results indicate that the proposed confidence limits accurately maintain coverage probabilities. The methodologies are illustrated using a study on the treatment of type two diabetes. Code implementing the proposed methods are provided in the supporting information.

Two-sample nonparametric test for proportional reversed hazards

In the last few decades, several works have been undertaken in the context of proportional reversed hazard rates (PRHR) but any specific statistical methodology for the PRHR hypothesis is absent in the literature. This paper proposes a two-sample nonparametric test based on two independent samples to verify the PRHR assumption. Based on a consistent U-statistic three statistical methodologies have been developed exploiting U-statistics theory, jackknife empirical likelihood and adjusted jackknife empirical likelihood method. A simulation study has been performed to assess the merit of the proposed test procedure. Finally, the proposed procedure is applied to a data set in the context of brain injury-related biomarkers and a data set related to Duchenne muscular dystrophy.

Non-Parametric Conditional U-Processes for Locally Stationary Functional Random Fields under Stochastic Sampling Design

Stute presented the so-called conditional U-statistics generalizing the Nadaraya–Watson estimates of the regression function. Stute demonstrated their pointwise consistency and the asymptotic normality. In this paper, we extend the results to a more abstract setting. We develop an asymptotic theory of conditional U-statistics for locally stationary random fields {Xs,An:sinRn} observed at irregularly spaced locations in Rn=[0,An]d as a subset of Rd. We employ a stochastic sampling scheme that may create irregularly spaced sampling sites in a flexible manner and includes both pure and mixed increasing domain frameworks. We specifically examine the rate of the strong uniform convergence and the weak convergence of conditional U-processes when the explicative variable is functional. We examine the weak convergence where the class of functions is either bounded or unbounded and satisfies specific moment conditions. These results are achieved under somewhat general structural conditions pertaining to the classes of functions and the underlying models. The theoretical results developed in this paper are (or will be) essential building blocks for several future breakthroughs in functional data analysis.

U-statistic based on overlapping sample spacings

For testing goodness of fit, we consider a class of U-statistics of overlapping spacings of order two, and investigate their asymptotic properties. The standard U-statistic theory is not directly applicable here as the overlapping spacings form a dependent random sequence. The asymptotic distribution of the statistics under the null hypothesis and under a sequence of local alternatives are derived. In terms of the Pitman ARE, the U-statistic based on Gini’s mean square difference of overlapping spacings is found to be the asymptotically locally most powerful. Interestingly, this test has the same efficacy as the Greenwood test based on overlapping spacings.

Prognostic accuracy for predicting ordinal competing risk outcomes using ROC surfaces.

Many medical conditions are marked by a sequence of events in association with continuous changes in biomarkers. Few works have evaluated the overall accuracy of a biomarker in predicting disease progression. We thus extend the concept of receiver operating characteristic (ROC) surface and the volume under the surface (VUS) from multi-category outcomes to ordinal competing-risk outcomes that are also subject to noninformative censoring. Two VUS estimators are considered. One is based on the definition of the ROC surface and obtained by integrating the estimated ROC surface. The other is an inverse probability weighted U estimator that is built upon the equivalence of the VUS to the concordance probability between the marker and sequential outcomes. Both estimators have nice asymptotic results that can be derived using counting process techniques and U-statistics theory. We illustrate their good practical performances through simulations and applications to two studies of cognition and a transplant dataset.

The asymptotic distribution of the Net Benefit estimator in presence of right-censoring.

The benefit-risk balance is a critical information when evaluating a new treatment. The Net Benefit has been proposed as a metric for the benefit-risk assessment, and applied in oncology to simultaneously consider gains in survival and possible side effects of chemotherapies. With complete data, one can construct a U-statistic estimator for the Net Benefit and obtain its asymptotic distribution using standard results of the U-statistic theory. However, real data is often subject to right-censoring, e.g. patient drop-out in clinical trials. It is then possible to estimate the Net Benefit using a modified U-statistic, which involves the survival time. The latter can be seen as a nuisance parameter affecting the asymptotic distribution of the Net Benefit estimator. We present here how existing asymptotic results on U-statistics can be applied to estimate the distribution of the net benefit estimator, and assess their validity in finite samples. The methodology generalizes to other statistics obtained using generalized pairwise comparisons, such as the win ratio. It is implemented in the R package BuyseTest (version 2.3.0 and later) available on Comprehensive R Archive Network.

Confidence sets for dynamic poverty indexes

ABSTRACT In this study, we consider different poverty indexes in a dynamic framework where individuals change their rate of income randomly in time. The primary objective of this paper is to assess the accuracy of the approximation of the indexes that can be obtained by applying the strong law of large numbers to an economic system composed of an infinite number of agents. The main result is a multivariate central limit theorem for dynamic poverty measures, which is obtained applying the theory of U-statistics. We also show how to get the confidence sets for the considered dynamic indexes, which show the appropriateness of the model. An application to the Italian income data from 1998 to 2012 confirms the effectiveness of the considered approach and the possibility to determine the evolution of poverty and inequality in real economies.

Sample size formula for general win ratio analysis.

Originally proposed for the analysis of prioritized composite endpoints, the win ratio has now expanded into a broad class of methodology based on general pairwise comparisons. Complicated by the non-i.i.d. structure of the test statistic, however, sample size estimation for the win ratio has lagged behind. In this article, we develop general and easy-to-use formulas to calculate sample size for win ratio analysis of different outcome types. In a nonparametric setting, the null variance of the test statistic is derived using U-statistic theory in terms of a dispersion parameter called the standard rank deviation, an intrinsic characteristic of the null outcome distribution and the user-defined rule of comparison. The effect size can be hypothesized either on the original scale of the population win ratio, or on the scale of a "usual" effect size suited to the outcome type. The latter approach allows one to measure the effect size by, for example, odds/continuation ratio for totally/partially ordered outcomes and hazard ratios for composite time-to-event outcomes. Simulation studies show that the derived formulas provide accurate estimates for the required sample size across different settings. As illustration, real data from two clinical studies of hepatic and cardiovascular diseases are used as pilot data to calculate sample sizes for future trials.

An association test for functional data based on Kendall’s Tau

We consider the problem of testing for association between a functional variable belonging to a Hilbert space and a scalar variable. Particularly, we propose a distribution-free test statistic based on Kendall’s Tau, which is a popular method for determining the association between two random variables. The distribution of the test statistic under the null hypothesis of independence is established using the theory of U-statistics taking values in a Hilbert space. We also consider the case where the functional data is sparsely observed, a situation that arises in many applications. Simulations show that the proposed method outperforms the alternatives under multiple settings, demonstrating the effectiveness and robustness of our approach. We provide data applications that further showcase the utility of our method.

Patterns in Random Permutations

Every k entries in a permutation can have one of k! different relative orders, called patterns. How many times does each pattern occur in a large random permutation of size n?The distribution of this k!-dimensional vector of pattern densities was studied by Janson, Nakamura, and Zeilberger (2015). Their analysis showed that some component of this vector is asymptotically multi-normal of order \(1/\sqrt n \), while the orthogonal component is smaller.Using representations of the symmetric group, and the theory of U-statistics, we refine the analysis of this distribution. We show that it decomposes into k asymptotically uncorrelated components of different orders in n, that correspond to Sk-representations.Some combinations of pattern densities that arise in this decomposition have interpretations as practical nonparametric statistical tests.

Pairwise distance-based heteroscedasticity test for regressions

In this study, we propose nonparametric testing for heteroscedasticity in nonlinear regression models based on pairwise distances between points in a sample. The test statistic can beformulated such that U-statistic theory can be applied to it. Although the limiting null distribution of the statistic iscomplicated, we can derive a computationally feasible bootstrap approximation forsuch a distribution; the validity of the introduced bootstrap algorithm is proven.The test can detect any local alternatives that are different fromthe null at a nearly optimal rate in hypothesis testing. Theconvergence rate of this test statistic does not depend on thedimension of the covariates, which significantly alleviates the impact ofdimensionality. We provide three simulation studies anda real-data example to evaluate the performance of the test anddemonstrate its applications.

Abstract 3399: Uncovering hidden sources of transcriptional dysregulation arising from inter- and intra-tumor heterogeneity

Abstract Introduction: This study develops an innovative computational framework, Expression Variation Analysis (EVA), to model transcriptional dysregulation in cancer. Heterogeneity poses a major challenge in translational research. For example, inter-tumor heterogeneity limits the biomarker discovery and intra-tumor heterogeneity enables therapeutic resistance. Moreover, in some cancers driver mutations are insufficient to account for the widespread transcriptional variation responsible for these outcomes. Thus, new computational tools to model transcriptional variation are essential. Methods: EVA is a unified computational framework to model transcriptional variation in cancer. Briefly, EVA quantifies transcriptional heterogeneity for one set of samples or cells from one phenotype using the expected dissimilarity between pairs of expression profiles. U-statistics theory can then quantify the statistical significance of the difference in transcriptional heterogeneity between phenotypes. Results: We apply EVA to perform a comprehensive characterization of transcriptional variation in head and neck squamous cell carcinoma (HNSCC). At a pathway level, transcriptional variation in HNSCC tumors is higher than normal controls. Applying EVA to integrate ChIP-seq data with RNA-seq reveals that these pervasive transcriptional differences occur in enhancers. Similarly, applying EVA at a gene level to model splicing reveals more heterogeneity in transcript usage in tumor samples than normals. HPV- HNSCC tumors are unique in having mutations in genes that regulate the splicing machinery, and the HPV- tumors with these alterations have a greater number of dysregulated splice variants than those without. Nonetheless, the EVA analysis identifies a similar number of alternative splice variants in HPV+ as HPV- tumors suggesting an alternative mechanism of transcriptional heterogeneity in HPV+ disease. Adapting EVA to single cell data demonstrates that increased fibroblast composition is associated with greater variation in immune pathway activity in HNSCC. Moreover, we observe greater transcriptional heterogeneity in HNSCC primary tumors than lymph node metastasis consistent with a clonal outgrowth. Conclusions: We demonstrate that the statistical framework from EVA enables differential heterogeneity analysis in HNSCC ranging from pathway dysregulation, splice variation, epigenetic regulation, and single cell analysis. This algorithm provides a critical framework to model the hidden multi-molecular mechanisms underlying the complex patient outcomes that are pervasive in cancer. Citation Format: Bahman Afsari, Leslie Cope, Daria A. Gaykalova, Donald Geman, Sidharth Puram, Loyal A. Goff, Alexander Favorov, Elana Judith Fertig. Uncovering hidden sources of transcriptional dysregulation arising from inter- and intra-tumor heterogeneity [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2019; 2019 Mar 29-Apr 3; Atlanta, GA. Philadelphia (PA): AACR; Cancer Res 2019;79(13 Suppl):Abstract nr 3399.

Modeling functional data: a test procedure

The paper deals with a test procedure able to state the compatibility of observed data with a reference model, by using an estimate of the volumetric part in the small-ball probability factorization which plays the role of a real complexity index. As a preliminary by-product we state some asymptotics for a new estimator of the complexity index. A suitable test statistic is derived and, referring to the U-statistics theory, its asymptotic null distribution is obtained. A study of level and power of the test for finite sample sizes and a comparison with a competitor are carried out by Monte Carlo simulations. The test procedure is performed over a financial time series.

Statistical moments of Gaussian kernel correlation sum and weighted least square estimator of correlation dimension and noise level

In this paper, using non-central chi-squared distribution and polynomial chaos decomposition of a log-normal random variable, we derive the exact expressions for the covariance and variance of the Gaussian kernel correlation sum (Gkcs). The obtained results are combined with U-statistics theory and non-linear models theory to construct the exact confidence intervals for the correlation dimension and for the noise level in the case where deterministic time series are corrupted by additive Gaussian noise. The theoretical results are tested on two continuous chaotic dynamics corrupted by Gaussian noise for different values of signal to noise ratio (SNR). An application to a real data time series has been also conducted.