Rank-based Inference Research Articles

Efficient Fully Distribution-Free Center-Outward Rank Tests for Multiple-Output Regression and MANOVA

Extending rank-based inference to a multivariate setting such as multiple-output regression or MANOVA with unspecified d-dimensional error density has remained an open problem for more than half a century. None of the many solutions proposed so far is enjoying the combination of distribution-freeness and efficiency that makes rank-based inference a successful tool in the univariate setting. A concept of center-outward multivariate ranks and signs based on measure transportation ideas has been introduced recently. Center-outward ranks and signs are not only distribution-free but achieve in dimension d > 1 the (essential) maximal ancillarity property of traditional univariate ranks. In the present case, we show that fully distribution-free testing procedures based on center-outward ranks can achieve parametric efficiency. We establish the Hájek representation and asymptotic normality results required in the construction of such tests in multiple-output regression and MANOVA models. Simulations and an empirical study demonstrate the excellent performance of the proposed procedures.

Fully Nonparametric Methods for Multivariate Data in Factorial Designs. Asymptotics, Finite Sample Approximations, and Implementation in R

Abstract This article methodologically complements the R package nparMD, and it presents the follow-up to a previous article that had built the foundation for the procedure implemented initially within the package. We consider non-parametric rank-based inference methods for multivariate samples in completely randomized factorial designs. Assumption on the data are minimal: the variables need to be at least ordinal (including binary), and the multivariate observation vectors from different experimental units are assumed independent although there may be dependencies within the vectors. Different endpoints may be measured on different scales. That is, multivariate observation vectors with a count variable, a continuous metric response, and ordinal endpoints, are allowed for. Two different asymptotic settings are considered: (1) sample sizes are large while the number of factor levels is small and bounded, (2) small samples while the number of factor levels is large. Special attention is paid to deriving inference methods for the interaction effect, as here the variance-covariance matrix does not simplify under the null. In addition, we discuss some finite sample approximations and demonstrate application of the methods using real data from a psychological study on Seasonal Affective Disorder (SAD), also commonly referred to as Winter depression

Hypothesis Testing With Rank Conditions in Phylogenetics.

A phylogenetic model of sequence evolution for a set of n taxa is a collection of probability distributions on the 4n possible site patterns that may be observed in their aligned DNA sequences. For a four-taxon model, one can arrange the entries of these probability distributions into three flattening matrices that correspond to the three different unrooted leaf-labeled four-leaf trees, or quartet trees. The flattening matrix corresponding to the tree parameter of the model is known to satisfy certain rank conditions. Methods such as ErikSVD and SVDQuartets take advantage of this observation by applying singular value decomposition to flattening matrices consisting of empirical data. Each possible quartet is assigned an “SVD score” based on how close the flattening is to the set of matrices of the predicted rank. When choosing among possible quartets, the one with the lowest score is inferred to be the phylogeny of the four taxa under consideration. Since an n-leaf phylogenetic tree is determined by its quartets, this approach can be generalized to infer larger phylogenies. In this article, we explore using the SVD score as a test statistic to test whether phylogenetic data were generated by a particular quartet tree. To do so, we use several results to approximate the distribution of the SVD score and to give upper bounds on the p-value of the associated hypothesis tests. We also apply these hypothesis tests to simulated phylogenetic data and discuss the implications for interpreting SVD scores in rank-based inference methods.

High-dimensional rank-based inference

ABSTRACTExisting high-dimensional inferential methods for comparing multiple groups test hypotheses are formulated in terms of mean vectors or location parameters. These methods are applicable mainly for metric data. Furthermore, the mean-based methods assume that moments exist and the nonparametric (location-based) ones assume elliptical-contoured distributions for the populations. In this paper, a fully nonparametric (rank-based) method is proposed. The method is applicable for metric as well as non-metric data and, hence, is applicable for ordered categorical as well as skewed and heavy tailed data. To develop the theory, we prove a novel result for studying asymptotic behaviour of quadratic forms in ranks. Simulation study shows that the developed rank-based method performs comparably well with mean-based methods when the assumptions of those methods are satisfied. However, it has significantly superior power for heavy tailed distributions with the possibility of outliers. The rank method is applied to an EEG data with the objective of examining associations between alcohol use and change in brain function.

Rank-based inference tools for copula regression, with property and casualty insurance applications

Rank-based procedures are commonly used for inference in copula models for continuous responses whose behavior does not depend on covariates. This paper describes how these procedures can be adapted to the broader framework in which (possibly non-linear) regression models for the marginal responses are linked by a copula that does not depend on covariates. The validity of many of these techniques can be derived from the asymptotic equivalence between the classical empirical copula process and its analog based on suitable residuals from the marginal models. Moment-based parameter estimation and copula goodness-of-fit tests are shown to remain valid under weak conditions on the marginal error term distributions, even when the residual-based empirical copula process fails to converge weakly. The performance of these procedures is evaluated through simulation in the context of two general insurance applications: micro-level multivariate insurance claims, and dependent loss triangles.

A privacy-preserving framework for ranked retrieval model

In this paper, we address privacy issues related to ranked retrieval model in web databases, each of which takes private attributes as part of input in the ranking function. Many web databases keep private attributes invisible to public and believe that the adversary is unable to reveal the private attribute values from query results. However, prior research (Rahman et al. in Proc VLDB Endow 8:1106–17, 2015) studied the problem of rank-based inference of private attributes over web databases. They found that one can infer the value of private attributes of a victim tuple by issuing well-designed queries through a top-k query interface. To address the privacy issue, in this paper, we propose a novel privacy-preserving framework. Our framework protects private attributes’ privacy not only under inference attacks but also under arbitrary attack methods. In particular, we classify adversaries into two widely existing categories: domain-ignorant and domain-expert adversaries. Then, we develop equivalent set with virtual tuples (ESVT) for domain-ignorant adversaries and equivalent set with true tuples (ESTT) for domain-expert adversaries. The ESVT and the ESTT are the primary parts of our privacy-preserving framework. To evaluate the performance, we define a measurement of privacy guarantee for private attributes and measurements for utility loss. We prove that both ESVT and ESTT achieve the privacy guarantee. We also develop heuristic algorithms for ESVT and ESTT, respectively, under the consideration of minimizing utility loss. We demonstrate the effectiveness of our techniques through theoretical analysis and extensive experiments over real-world dataset.

Rank-based inference for the accelerated failure time model in the presence of interval censored data

Semiparametric analysis and rank-based inference for the accelerated failure time model are complicated in the presence of interval censored data. The main difficulty with the existing rank-based methods is that they involve estimating functions with the possibility of multiple roots. In this paper a class of asymptotically normal rank estimators is developed which can be acquired via linear programming for estimating the parameters of the model, and a two-step iterative algorithm is introduced for solving the estimating equations. The proposed inference procedures are assessed through a real example. The results of applying the proposed methodology on the breast cancer data show that the algorithm converges after three iterations, and the estimations of model parameter based on Log-rank and Gehan weight functions are fairly close with small standard errors.

A new nonparametric test for high-dimensional regression coefficients

ABSTRACTSeveral tests for high-dimensional (large p, small n) regression coefficients have been proposed in the recent literature, but they are adversely affected by outlying observations and heavy-tailed distributions. In order to attack these challenges, a novel nonparametric testing procedure is developed under the framework of rank-based inference and is robust with respect to both the responses and the covariates. Besides, the proposed test statistic is invariant under the group of scalar transformations, which implies that our test statistic can integrate all the individual information in a relatively fair way. The newly defined test has many desirable general asymptotic properties, such as normality and consistency when We assess the finite-sample performance of the proposed test by examining its size and power via Monte Carlo simulation, which demonstrates an improvement over the previous literature.

Rank-based ridge estimation in multiple linear regression

Multicollinearity and model misspecification are frequently encountered problems in practice that produce undesirable effects on classical ordinary least squares (OLS) regression estimator. The ridge regression estimator is an important tool to reduce the effects of multicollinearity, but it is still sensitive to a model misspecification of error distribution. Although rank-based statistical inference has desirable robustness properties compared to the OLS procedures, it can be unstable in the presence of multicollinearity. This paper introduces a rank regression estimator for regression parameters and develops tests for general linear hypotheses in a multiple linear regression model. The proposed estimator and the tests have desirable robustness features against the multicollinearity and model misspecification of error distribution. Asymptotic behaviours of the proposed estimator and the test statistics are investigated. Real and simulated data sets are used to demonstrate the feasibility and the performance of the estimator and the tests.

Fast accelerated failure time modeling for case-cohort data

Semiparametric accelerated failure time (AFT) models directly relate the expected failure times to covariates and are a useful alternative to models that work on the hazard function or the survival function. For case-cohort data, much less development has been done with AFT models. In addition to the missing covariates outside of the sub-cohort in controls, challenges from AFT model inferences with full cohort are retained. The regression parameter estimator is hard to compute because the most widely used rank-based estimating equations are not smooth. Further, its variance depends on the unspecified error distribution, and most methods rely on computationally intensive bootstrap to estimate it. We propose fast rank-based inference procedures for AFT models, applying recent methodological advances to the context of case-cohort data. Parameters are estimated with an induced smoothing approach that smooths the estimating functions and facilitates the numerical solution. Variance estimators are obtained through efficient resampling methods for nonsmooth estimating functions that avoids full blown bootstrap. Simulation studies suggest that the recommended procedure provides fast and valid inferences among several competing procedures. Application to a tumor study demonstrates the utility of the proposed method in routine data analysis.

Rank-Based Analysis of Unbalanced Repeated Measures Data

Normal 0 false false false EN-US X-NONE X-NONE /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman","serif";} In this article, we have developed a rank (intra-subject) based analysis of clinical trials with unbalanced repeated measures data. We assume that the errors within each patient are exchangeable and continuous random variables. This rank-based inference is valid when the unbalanced data are missing either completely at random or by design. A drop in dispersion test is developed for general linear hypotheses. A numerical example is given to illustrate the procedure.

Efficient Estimation for Rank‐Based Regression with Clustered Data

Rank-based inference is widely used because of its robustness. This article provides optimal rank-based estimating functions in analysis of clustered data with random cluster effects. The extensive simulation studies carried out to evaluate the performance of the proposed method demonstrate that it is robust to outliers and is highly efficient given the existence of strong cluster correlations. The performance of the proposed method is satisfactory even when the correlation structure is misspecified, or when heteroscedasticity in variance is present. Finally, a real dataset is analyzed for illustration.

Rank-based inference for the single-index model

This paper is concerned with estimating the coefficients in single-index models. We develop a robust estimator, which combines the ideas of rank-based regression inference and outer product of gradients. Both asymptotic and numerical results show that the proposed procedure has better performance than the least-squares-based method when the errors deviate from normal.

Semiparametrically efficient inference based on signed ranks in symmetric independent component models

We consider semiparametric location-scatter models for which the $p$-variate observation is obtained as $X=\Lambda Z+\mu$, where $\mu$ is a $p$-vector, $\Lambda$ is a full-rank $p\times p$ matrix and the (unobserved) random $p$-vector $Z$ has marginals that are centered and mutually independent but are otherwise unspecified. As in blind source separation and independent component analysis (ICA), the parameter of interest throughout the paper is $\Lambda$. On the basis of $n$ i.i.d. copies of $X$, we develop, under a symmetry assumption on $Z$, signed-rank one-sample testing and estimation procedures for $\Lambda$. We exploit the uniform local and asymptotic normality (ULAN) of the model to define signed-rank procedures that are semiparametrically efficient under correctly specified densities. Yet, as is usual in rank-based inference, the proposed procedures remain valid (correct asymptotic size under the null, for hypothesis testing, and root-$n$ consistency, for point estimation) under a very broad range of densities. We derive the asymptotic properties of the proposed procedures and investigate their finite-sample behavior through simulations.

Discussion: Statistical models and methods for dependence in insurance data

The paper by Haug et al. presents an overview of copulas and their use for dependence modeling in multivariate risk analysis. Various aspects of rank-based inference are covered, from pointwise and interval estimation to goodness-of-fit testing. The exposition is clearly not comprehensive and many details are glossed over but this is offset by the inclusion of newmaterial, most notably empirical likelihood methods. Readers interested in further details can refer, e.g., to Genest and Favre (2007), where additional graphical tools and inferential procedures are illustrated on a small ‘‘learning data set’’. A distinctive feature of the paper is its emphasis on extreme-value and tail copulas. These concepts are clearly relevant for risk management, as failure to account for dependence between extreme events can have disastrous financial, economic or environmental consequences; see, e.g., Chavez-Demoulin and Embrechts (2010). There is a pressing need for inference techniques tailored to extreme dependence models and the area is currently undergoing rapid development. To attempt a survey at this early stage was clearly a challenge, and the authors lived up to it. Relevant additional contributions of recent vintage include new goodness-of-fit tests for extreme-value copulas by Kojadinovic and Yan (2010) and Quessy (in press), and alternative estimators of the Pickands dependence function due to Bucher, Dette, and Volgushev (2010), Guillotte and Perron (2008), and Guillotte, Perron, and Segers (in press). In this discussion, wewould like to complement thework of Haug et al. by highlighting some of the difficulties associated with the occurrence of ties in rank-based inference for copulamodels. As illustrated in Section 2, this phenomenon is present in the Danish fire insurance data set and cannot be ignored. Unfortunately, the validity of rank-based procedures is often compromised when the margins are discontinuous (Genest & Neslehova, 2007). The Danish data are used in Sections 3 and 4 to illustrate the impact of ties on testing for extremeness and making inference for extreme-value copulas, respectively. While the problem is easily recognized, solutions are hard to come by. Some of the methodological challenges at hand are sketched in Section 5, and concluding remarks are made in Section 6.

Rank-Based Inference in Linear Models with Stable Errors

Linear models with stable error densities are considered. The local asymptotic normality of the resulting model is established. We use this result, combined with Le~Cam's third lemma, to obtain local powers of various classical rank tests (Wilcoxon's and van der Waerden's test, the median test, and their counterparts for regression and analysis of variance) under stable laws. The same results are used to construct new rank tests achieving parametric optimality at specified stable densities. A Monte-Carlo study is conducted to compare their relative performances.

Testing and interval estimation for two-sample survival comparisons with small sample sizes and unequal censoring

While the commonly used log-rank test for survival times between 2 groups enjoys many desirable properties, sometimes the log-rank test and its related linear rank tests perform poorly when sample sizes are small. Similar concerns apply to interval estimates for treatment differences in this setting, though their properties are less well known. Standard permutation tests are one option, but these are not in general valid when the underlying censoring distributions in the comparison groups are unequal. We develop 2 methods for testing and interval estimation, for use with small samples and possibly unequal censoring, based on first imputing survival and censoring times and then applying permutation methods. One provides a heuristic justification for the approach proposed recently by Heinze and others (2003, Exact log-rank tests for unequal follow-up. Biometrics 59, 1151-1157). Simulation studies show that the proposed methods have good Type I error and power properties. For accelerated failure time models, compared to the asymptotic methods of Jin and others (2003, Rank-based inference for the accelerated failure time model. Biometrika 90, 341-353), the proposed methods yield confidence intervals with better coverage probabilities in small-sample settings and similar efficiency when sample sizes are large. The proposed methods are illustrated with data from a cancer study and an AIDS clinical trial.

Multivariate quantiles and multiple-output regression quantiles: From L1 optimization to halfspace depth

A new multivariate concept of quantile, based on a directional version of Koenker and Bassett's traditional regression quantiles, is introduced for multivariate location and multiple-output regression problems. In their empirical version, those quantiles can be computed efficiently via linear programming techniques. Consistency, Bahadur representation and asymptotic normality results are established. Most importantly, the contours generated by those quantiles are shown to coincide with the classical halfspace depth contours associated with the name of Tukey. This relation does not only allow for efficient depth contour computations by means of parametric linear programming, but also for transferring from the quantile to the depth universe such asymptotic results as Bahadur representations. Finally, linear programming duality opens the way to promising developments in depth-related multivariate rank-based inference.

Rank-based Inference for Multivariate Nonlinear and Long-memory Time Series Models

The portfolio of the Japanese Government Pension Investment Fund (GPIF) consists of a linear combination of five benchmarks of financial assets. Some of these exhibit long-memory and nonlinear behavior. Their analysis therefore requires multivariate nonlinear and long-memory time series models. Moreover, the assumption that the innovation densities underlying those models are known seems quite unrealistic. If those densities remain unspecified, the model becomes a semiparametric one, and rank-based inference methods naturally come into the picture. Rank-based inference methods under very general conditions are known to achieve the semiparametric efficiency bounds. Defining ranks in the context of multivariate time series models, however, is not obvious. We propose two distinct definitions. The first one relies on the assumption that the innovation density is some unspecified elliptical density. The second one relies on the assumption that the innovation process is described by some unspecified independent component analysis model. Applications to portfolio management problems are discussed.

Rank-based inference for bivariate extreme-value copulas

Consider a continuous random pair (X, Y) whose dependence is characterized by an extreme-value copula with Pickands dependence function A. When the marginal distributions of X and Y are known, several consistent estimators of A are available. Most of them are variants of the estimators due to Pickands [Bull. Inst. Internat. Statist. 49 (1981) 859–878] and Capéraà, Fougères and Genest [Biometrika 84 (1997) 567–577]. In this paper, rank-based versions of these estimators are proposed for the more common case where the margins of X and Y are unknown. Results on the limit behavior of a class of weighted bivariate empirical processes are used to show the consistency and asymptotic normality of these rank-based estimators. Their finite- and large-sample performance is then compared to that of their known-margin analogues, as well as with endpoint-corrected versions thereof. Explicit formulas and consistent estimates for their asymptotic variances are also given.