Showing papers on "Resampling published in 2003"

Resampling Methods for Dependent Data

Abstract: 1 Scope of Resampling Methods for Dependent Data.- 2 Bootstrap Methods.- 3 Properties of Block Bootstrap Methods for the Sample Mean.- 4 Extensions and Examples.- 5 Comparison of Block Bootstrap Methods.- 6 Second-Order Properties.- 7 Empirical Choice of the Block Size.- 8 Model-Based Bootstrap.- 9 Frequency Domain Bootstrap.- 10 Long-Range Dependence.- 11 Bootstrapping Heavy-Tailed Data and Extremes.- 12 Resampling Methods for Spatial Data.- A.- B.- References.- Author Index.

Generalized discriminant analysis based on distances

Abstract: Summary This paper describes a method of generalized discriminant analysis based on a dissimilarity matrix to test for differences in a priori groups of multivariate observations. Use of classical multidimensional scaling produces a low-dimensional representation of the data for which Euclidean distances approximate the original dissimilarities. The resulting scores are then analysed using discriminant analysis, giving tests based on the canonical correlations. The asymptotic distributions of these statistics under permutations of the observations are shown to be invariant to changes in the distributions of the original variables, unlike the distributions of the multi-response permutation test statistics which have been considered by other workers for testing differences among groups. This canonical method is applied to multivariate fish assemblage data, with Monte Carlo simulations to make power comparisons and to compare theoretical results and empirical distributions. The paper proposes classification based on distances. Error rates are estimated using cross-validation.

Comparison of Bayesian and Maximum Likelihood Bootstrap Measures of Phylogenetic Reliability

Abstract: Owing to the exponential growth of genome databases, phylogenetic trees are now widely used to test a variety of evolutionary hypotheses. Nevertheless, computation time burden limits the application of methods such as maximum likelihood nonparametric bootstrap to assess reliability of evolutionary trees. As an alternative, the much faster Bayesian inference of phylogeny, which expresses branch support as posterior probabilities, has been introduced. However, marked discrepancies exist between nonparametric bootstrap proportions and Bayesian posterior probabilities, leading to difficulties in the interpretation of sometimes strongly conflicting results. As an attempt to reconcile these two indices of node reliability, we apply the nonparametric bootstrap resampling procedure to the Bayesian approach. The correlation between posterior probabilities, bootstrap maximum likelihood percentages, and bootstrapped posterior probabilities was studied for eight highly diverse empirical data sets and were also investigated using experimental simulation. Our results show that the relation between posterior probabilities and bootstrapped maximum likelihood percentages is highly variable but that very strong correlations always exist when Bayesian node support is estimated on bootstrapped character matrices. Moreover, simulations corroborate empirical observations in suggesting that, being more conservative, the bootstrap approach might be less prone to strongly supporting a false phylogenetic hypothesis. Thus, apparent conflicts in topology recovered by the Bayesian approach were reduced after bootstrapping. Both posterior probabilities and bootstrap supports are of great interest to phylogeny as potential upper and lower bounds of node reliability, but they are surely not interchangeable and cannot be directly compared.

Resampling-based multiple testing for microarray data analysis

Abstract: The burgeoning field of genomics has revived interest in multiple testing procedures by raising new methodological and computational challenges. For example, microarray experiments generate large multiplicity problems in which thousands of hypotheses are tested simultaneously. Westfall and Young (1993) propose resampling-based p-value adjustment procedures which are highly relevant to microarray experiments. This article discusses dierent criteria for error control in resampling-based multiple testing, including (a) the family wise error rate of Westfall and Young (1993) and (b) the false discovery rate developed by Benjamini and Hochberg (1995), both from a frequentist viewpoint; and (c) the positive false discovery rate of Storey (2002a), which has a Bayesian motivation. We also introduce our recently developed fast algorithm for implementing the minP adjustment to control family-wise error rate. Adjusted p-values for dierent approaches are applied to gene expression data from two recently published microarray studies. The properties of these procedures for multiple testing are compared.

Consistent Testing for Stochastic Dominance under General Sampling Schemes

Abstract: We propose a procedure for estimating the critical values of the extended Kolmogorov- Smirnov tests of First and Second Order Stochastic Dominance in the general K-prospect case. We allow for the observations to be serially dependent and, for the first time, we can accommodate general dependence amongst the prospects which are to be ranked. Also, the prospects may be the residuals from certain conditional models, opening the way for conditional ranking. We also propose a test of Prospect Stochastic Dominance. Our method is subsampling; we show that the resulting tests are consistent and powerful against some N|1/2 local alternatives even when computed with a data-based subsample size. We also propose some heuristic methods for selecting subsample size and demonstrate in simulations that they perform reasonably. We show that our test is asymptotically similar on the entire boundary of the null hypothesis, and is unbiased. In comparison, any method based on resampling or simulating from the least favorable distribution does not have these properties and consequently will have less power against some alternatives.

Rank-based inference for the accelerated failure time model

Abstract: SUMMARY A broad class of rank-based monotone estimating functions is developed for the semiparametric accelerated failure time model with censored observations. The corresponding estimators can be obtained via linear programming, and are shown to be consistent and asymptotically normal. The limiting covariance matrices can be estimated by a resampling technique, which does not involve nonparametric density estimation or numerical derivatives. The new estimators represent consistent roots of the non-monotone estimating equations based on the familiar weighted log-rank statistics. Simulation studies demonstrate that the proposed methods perform well in practical settings. Two real examples are provided.

Giving meaningful interpretation to ordination axes: assessing loading significance in principal component analysis

Abstract: Principal component analysis (PCA) is one of the most commonly used tools in the analysis of ecological data. This method reduces the effective dimensionality of a multivariate data set by producing linear combinations of the original variables (i.e., com- ponents) that summarize the predominant patterns in the data. In order to provide meaningful interpretations for principal components, it is important to determine which variables are associated with particular components. Some data analysts incorrectly test the statistical significance of the correlation between original variables and multivariate scores using standard statistical tables. Others interpret eigenvector coefficients larger than an arbitrary absolute value (e.g., 0.50). Resampling, randomization techniques, and parallel analysis have been applied in a few cases. In this study, we compared the performance of a variety of approaches for assessing the significance of eigenvector coefficients in terms of type I error rates and power. Two novel approaches based on the broken-stick model were also evaluated. We used a variety of simulated scenarios to examine the influence of the number of real dimensions in the data; unique versus complex variables; the magnitude of eigen- vector coefficients; and the number of variables associated with a particular dimension. Our results revealed that bootstrap confidence intervals and a modified bootstrap confidence interval for the broken-stick model proved to be the most reliable techniques.

Permutation Tests for Classification: Towards Statistical Significance in Image-Based Studies

Abstract: Estimating statistical significance of detected differences between two groups of medical scans is a challenging problem due to the high dimensionality of the data and the relatively small number of training examples. In this paper, we demonstrate a non-parametric technique for estimation of statistical significance in the context of discriminative analysis (i.e., training a classifier function to label new examples into one of two groups). Our approach adopts permutation tests, first developed in classical statistics for hypothesis testing, to estimate how likely we are to obtain the observed classification performance, as measured by testing on a hold-out set or cross-validation, by chance. We demonstrate the method on examples of both structural and functional neuroimaging studies.

Bootstrap Methods For Time Series

Abstract: The chapter gives a review of the literature on bootstrap methods for time series data. It describes various possibilities on how the bootstrap method, initially introduced for independent random variables, can be extended to a wide range of dependent variables in discrete time, including parametric or nonparametric time series models, autoregressive and Markov processes, long range dependent time series and nonlinear time series, among others. Relevant bootstrap approaches, namely the intuitive residual bootstrap and Markovian bootstrap methods, the prominent block bootstrap methods as well as frequency domain resampling procedures, are described. Further, conditions for consistent approximations of distributions of parameters of interest by these methods are presented. The presentation is deliberately kept non-technical in order to allow for an easy understanding of the topic, indicating which bootstrap scheme is advantageous under a specific dependence situation and for a given class of parameters of interest. Moreover, the chapter contains an extensive list of relevant references for bootstrap methods for time series.

The Impact of Bootstrap Methods on Time Series Analysis

Abstract: Sparked by Efron's seminal paper, the decade of the 1980s was a period of active research on bootstrap methods for independent data--mainly i.i.d. or regression set-ups. By contrast, in the 1990s much research was directed towards resampling dependent data, for example, time series and random fields. Consequently, the availability of valid nonparametric inference procedures based on resampling and/or subsampling has freed practitioners from the necessity of resorting to simplifying assumptions such as normality or linearity that may be misleading.

Introduction to Modern Nonparametric Statistics

Abstract: 1. ONE-SAMPLE METHODS. Preliminaries. A Nonparametric Test and Confidence Interval for the Median. Estimating the Population CDF and Quantiles. A Comparison of Statistical Tests. 2. TWO-SAMPLE METHODS. A Two-Sample Permutation Test. Permutation Tests Based on the Median and Trimmed Means. Random Sampling the Permutations. Wilcoxon Rank-Sum Test. Wilcoxon Rank-Sum Test Adjusted for Ties. Mann-Whitney Test and a Confidence Interval. Scoring Systems. Test for Equality of Scale Parameters and an Omnibus Test. Selecting Among Two-Sample Tests. Large Sample Approximations. Exercises. 3. K-SAMPLE METHODS. K-Sample Permutation Tests. The Kruskal-Wallis Test. Multiple Comparisons. Ordered Alternatives. Exercises. 4. PAIRED COMPARISONS AND BLOCKED DESIGNS. Paired Comparison Permutation Test. Signed-Rank Test. Other Paired-Comparison Tests. A Permutation Test for a Randomized Complete Block Design. Friedman"s Test for a Randomized Complete Block Design. Ordered Alternatives for a Randomized Complete Block Design. Exercises. 5. TESTS FOR TRENDS AND ASSOCIATION. A Permutation Test for Correlation and Slope. Spearman Rank Correlation. Kendall"s Tau. Permutation Tests for Contingency Tables. Fisher"s Exact Test for a 2 "e 2 Contingency Table. Contingency Tables With Ordered Categories. Mantel-Haenszel Test. Exercises. 6. MULTIVARIATE TESTS. Two-Sample Multivariate Permutation Tests. Two-Sample Multivariate Rank Tests. Multivariate Paired Comparisons. Multivariate Rank Tests for Paired Comparisons. Multi-response Categorical Data. Exercises. 7. ANALYSIS OF CENSORED DATA. Estimating the Survival Function. Permutation Tests for Two-Sample Censored Data. Gehan"s Generalization of the Mann-Whitney-Wilcoxon Test. Scoring Systems for Censored Data. Tests Using Scoring Systems for Censored Data. Exercises. 8. NONPARAMETRIC BOOTSTRAP METHODS. The Basic Bootstrap Method. Bootstrap Intervals for Location-Scale Models. BCA and Other Bootstrap Intervals. Correlation and Regression. Two-Sample Inference. Bootstrap Sampling from Several Populations. Bootstrap Sampling for Multiple Regression. Multivariate Bootstrap Sampling. Exercises. 9. MULTIFACTOR EXPERIMENTS. Analysis of Variance Models. Aligned Rank Transform. Testing for Lattice-Ordered Alternatives. Exercises. 10. SMOOTHING METHODS AND ROBUST MODEL FITTING. Estimating the Probability Density Function. Nonparametric Curve Smoothing. Robust and Rank-Based Regression. Exercises. TABLES. REFERENCES.

Accurate tests of statistical significance for r(WG) and average deviation interrater agreement indexes.

Abstract: The authors demonstrated that the most common statistical significance test used with r(WG)-type interrater agreement indexes in applied psychology, based on the chi-square distribution, is flawed and inaccurate. The chi-square test is shown to be extremely conservative even for modest, standard significance levels (e.g., .05). The authors present an alternative statistical significance test, based on Monte Carlo procedures, that produces the equivalent of an approximate randomization test for the null hypothesis that the actual distribution of responding is rectangular and demonstrate its superiority to the chi-square test. Finally, the authors provide tables of critical values and offer downloadable software to implement the approximate randomization test for r(WG)-type and for average deviation (AD)-type interrater agreement indexes. The implications of these results for studying a broad range of interrater agreement problems in applied psychology are discussed.

Exact Statistical Methods for Data Analysis

Abstract: 1 Preliminary Notions.- 1.1 Introduction.- 1.2 Sufficiency.- 1.3 Complete Sufficient Statistics.- 1.4 Exponential Families of Distributions.- 1.5 Invariance.- 1.6 Maximum Likelihood Estimation.- 1.7 Unbiased Estimation.- 1.8 Least Squares Estimation.- 1.9 Interval Estimation.- Exercises.- 2 Notions in significance testing of hypotheses.- 2.1 Introduction.- 2.2 Test Statistics and Test Variables.- 2.3 Definition of p-Value.- 2.4 Generalized Likelihood Ratio Method.- 2.5 Invariance in Significance Testing.- 2.6 Unbiasedness and Similarity.- 2.7 Interval Estimation and Fixed-Level Testing.- Exercises.- 3 Review of Special Distributions.- 3.1 Poisson and Binomial Distributions.- 3.2 Point Estimation and Interval Estimation.- 3.3 Significance Testing of Parameters.- 3.4 Bayesian Inference.- 3.5 The Normal Distribution.- 3.6 Inferences About the Mean.- 3.7 Inferences About the Variance.- 3.8 Quantiles of a Normal Distribution.- 3.9 Conjugate Prior and Posterior Distributions.- 3.10 Bayesian Inference About the Mean and the Variance.- Exercises.- 4 Exact Nonparametric Methods.- 4.1 Introduction.- 4.2 The Sign Test.- 4.3 The Signed Rank Test and the Permutation Test.- 4.4 The Rank Sum Test and Allied Tests.- 4.5 Comparing k Populations.- 4.6 Contingency Tables.- 4.7 Testing the Independence of Criteria of Classification.- 4.8 Testing the Homogeneity of Populations.- Exercises.- 5 Generalized p-Values.- 5.1 Introduction.- 5.2 Generalized Test Variables.- 5.3 Definition of Generalized p-Values.- 5.4 Frequency Interpretations and Generalized Fixed-Level Tests.- 5.5 Invariance.- 5.6 Comparing the Means of Two Exponential Distributions.- 5.7 Unbiasedness and Similarity.- 5.7 Comparing the Means of an Exponential Distribution and a Normal Distribution.- Exercises.- 6 Generalized Confidence Intervals.- 6.1 Introduction.- 6.2 Generalized Definitions.- 6.3 Frequency Interpretations and Repeated Sampling Properties.- 6.4 Invariance in Interval Estimation.- 6.5 Interval Estimation of the Difference Between Two Exponential Means.- 6.6 Similarity in Interval Estimation.- 6.7 Generalized Confidence Intervals Based on p-Values.- 6.8 Resolving an Undesirable Feature of Confidence Intervals.- 6.9 Bayesian and Conditional Confidence Intervals.- Exercises.- 7 Comparing Two Normal Populations.- 7.1 Introduction.- 7.2 Comparing the Means when the Variances are Equal.- 7.3 Solving the Behrens-Fisher Problem.- 7.4 Inferences About the Ratio of Two Variances.- 7.5 Inferences About the Difference in Two Variances.- 7.6 Bayesian Inference.- 7.7 Inferences About the Reliability Parameter.- 7.8 The Case of Known Stress Distribution.- Exercises.- 8 Analysis of Variance.- 8.1 Introduction.- 8.2 One-way Layout.- 8.3 Testing the Equality of Means.- 8.4 ANOVA with Unequal Error Variances.- 8.5 Multiple Comparisons.- 8.6 Testing the Equality of Variances.- 8.7 Two-way ANOVA without Replications.- 8.8 ANOVA in a Balanced Two-way Layout with Replications.- 8.9 Two-way ANOVA under Heteroscedasticity.- Exercises.- 9 Mixed Models.- 9.1 Introduction.- 9.2 One-way Layout.- 9.3 Testing Variance Components.- 9.4 Confidence Intervals.- 9.5 Two-way Layout.- 9.6 Comparing Variance Components.- Exercises.- 10 Regression.- 10.1 Introduction.- 10.2 Simple Linear Regression Model.- 10.3. Inferences about Parameters of the Simple Regression Model.- 10.3 Multiple Linear Regression.- 10.4 Distributions of Estimators and Significance Tests.- 10.5 Comparing Two Regressions with Equal Variances.- 10.6 Comparing Regressions without Common Parameters.- 10.7 Comparison of Two General Models.- Exercises.- Appendix A.- Elements of Bayesian Inference.- A.1 Introduction.- A.2 The Prior Distribution.- A.3 The Posterior Distribution.- A.4 Bayes Estimators.- A.5 Bayesian Interval Estimation.- A.6 Bayesian Hypothesis Testing.- Appendix B Technical Arguments.- References.

Particle-systems implementation of the PHD multitarget tracking filter

Abstract: We report here on the implementation of a particle systems approximation to the probability hypothesis density (PHD). The PHD of the multitarget posterior density has the property that, given any volume of state space, the integral of the PHD over that volume yields the expected number of targets present in the volume. As in the single target setting, upon receipt of an observation, the particle weights are updated, taking into account the sensor likelihood function, and then propagated forward in time by sampling from a Markov transition density. We also incorporate resampling and regularization into our implementation, introducing the new concept of cluster resampling.

Choosing between two learning algorithms based on calibrated tests

Abstract: Designing a hypothesis test to determine the best of two machine learning algorithms with only a small data set available is not a simple task. Many popular tests suffer from low power (5×2 cv [2]), or high Type I error (Weka's 10×10 cross validation [11]). Furthermore, many tests show a low level of replicability, so that tests performed by different scientists with the same pair of algorithms, the same data sets and the same hypothesis test still may present different results. We show that 5×2 cv, resampling and 10 fold cv suffer from low replicability. The main complication is due to the need to use the data multiple times. As a consequence, independence assumptions for most hypothesis tests are violated. In this paper, we pose the case that reuse of the same data causes the effective degrees of freedom to be much lower than theoretically expected. We show how to calibrate the effective degrees of freedom empirically for various tests. Some tests are not calibratable, indicating another flaw in the design. However the ones that are calibratable all show very similar behavior. Moreover, the Type I error of those tests is on the mark for a wide range of circumstances, while they show a power and replicability that is a considerably higher than currently popular hypothesis tests.

How do bootstrap and permutation tests work

Abstract: Resampling methods are frequently used in practice to adjust critical values of nonparametric tests In the present paper a comprehensive and unified approach for the conditional and unconditional analysis of linear resampling statistics is presented Under fairly mild assumptions we prove tightness and an asymptotic series representation for their weak accumulation points From this series it becomes clear which part of the resampling statistic is responsible for asymptotic normality The results leads to a discussion of the asymptotic correctness of resampling methods as well as their applications in testing hypotheses They are conditionally correct iff a central limit theorem holds for the original test statistic We prove unconditional correctness iff the central limit theorem holds or when symmetric random variables are resampled by a scheme of asymptotically random signs Special cases are the m (n) out of k (n) bootstrap, the weighted bootstrap, the wild bootstrap and all kinds of permutation statistics The program is carried out for convergent partial sums of rowwise independent infinitesimal triangular arrays in detail These results are used to compare power functions of conditional resampling tests and their unconditional counterparts The proof uses the method of random scores for permutation type statistics

A Lack-of-Fit Test for Quantile Regression

Abstract: We propose an omnibus lack-of-fit test for linear or nonlinear quantile regression based on a cusum process of the gradient vector. The test does not involve nonparametric smoothing but is consistent for all nonparametric alternatives without any moment conditions on the regression error. In addition, the test is suitable for detecting the local alternatives of any order arbitrarily close to n−1/2 from the null hypothesis. The limiting distribution of the proposed test statistic is non-Gaussian but can be characterized by a Gaussian process. We propose a simple sequential resampling scheme to carry out the test whose nominal levels are well approximated in our empirical study for

Residual‐Based Block Bootstrap for Unit Root Testing

Abstract: A nonparametric, residual-based block bootstrap procedure is proposed in the context of testing for integrated (unit root) time series. The resampling procedure is based on weak assumptions on the dependence structure of the stationary process driving the random walk and successfully generates unit root integrated pseudo-series retaining the important characteristics of the data. It is more general than previous bootstrap approaches to the unit root problem in that it allows for a very wide class of weakly dependent processes and it is not based on any parametric assumption on the process generating the data. As a consequence the procedure can accurately capture the distribution of many unit root test statistics proposed in the literature. Large sample theory is developed and the asymptotic validity of the block bootstrap-based unit root testing is shown via a bootstrap functional limit theorem. Applications to some particular test statistics of the unit root hypothesis, i.e., least squares and Dickey-Fuller type statistics are given. The power properties of our procedure are investigated and compared to those of alternative bootstrap approaches to carry out the unit root test. Some simulations examine the finite sample performance of our procedure.

Introduction to the Bootstrap World

Abstract: The bootstrap has made a fundamental impact on how we carry out statistical inference in problems without analytic solutions. This fact is illustrated with examples and comments that emphasize the parametric bootstrap and hypothesis testing.

Bootstrap Methods for Markov Processes

Abstract: The block bootstrap is the best known bootstrap method for time-series data when the analyst does not have a parametric model that reduces the data generation process to simple random sampling. However, the errors made by the block bootstrap converge to zero only slightly faster than those made by first-order asymptotic approximations. This paper describes a bootstrap procedure for data that are generated by a Markov process or a process that can be approximated by a Markov process with sufficient accuracy. The procedure is based on estimating the Markov transition density nonparametrically. Bootstrap samples are obtained by sampling the process implied by the estimated transition density. Conditions are given under which the errors made by the Markov bootstrap converge to zero more rapidly than those made by the block bootstrap.

From the help desk: Bootstrapped standard errors

Abstract: Bootstrapping is a nonparametric approach for evaluating the dis-tribution of a statistic based on random resampling This article illustrates the bootstrap as an alternative method for estimating

New resampling algorithms for particle filters

Abstract: Resampling is a critically important operation in the implementation of particle filtering. In parallel hardware implementations, resampling becomes a bottleneck due to its sequential nature and the increased complexity it imposes on the traffic of the designed interconnection network. To circumvent some of these difficulties, we propose two new resampling algorithms. The first one, called residual-systematic resampling, combines the merits of both systematic and residual resampling and is suitable for pipelined implementation. It also guarantees the fixed duration of the resampling procedure irrespective of the weight distribution of the particles. The second algorithm, referred to as partial resampling, has low complexity and reduces traffic load through the hardware network. These two algorithms should also be considered as resampling methods in simulations on standard computers.

Partitioned likelihood support and the evaluation of data set conflict

Abstract: In simultaneous analyses of multiple data partitions, the trees relevant when measuring support for a clade are the optimal tree, and the best tree lacking the clade (i.e., the most reasonable alternative). The parsimony-based method of partitioned branch support (PBS) "forces" each data set to arbitrate between the two relevant trees. This value is the amount each data set contributes to clade support in the combined analysis, and can be very different to support apparent in separate analyses. The approach used in PBS can also be employed in likelihood: a simultaneous analysis of all data retrieves the maximum likelihood tree, and the best tree without the clade of interest is also found. Each data set is fitted to the two trees and the log-likelihood difference calculated, giving "partitioned likelihood support" (PLS) for each data set. These calculations can be performed regardless of the complexity of the ML model adopted. The significance of PLS can be evaluated using a variety of resampling methods, such as the Kishino-Hasegawa test, the Shimodiara-Hasegawa test, or likelihood weights, although the appropriateness and assumptions of these tests remains debated.

Resampling approach for anomaly detection in multispectral images

Abstract: We propose a novel approach for identifying the 'most unusual' samples in a data set, based on a resampling of data attributes. The resampling produces a 'background class' and then binary classification is used to distinguish the original training set from the background. Those in the training set that are most like the background (i e, most unlike the rest of the training set) are considered anomalous. Although by their nature, anomalies do not permit a positive definition (if I knew what they were, I wouldn't call them anomalies), one can make 'negative definitions' (I can say what does not qualify as an interesting anomaly). By choosing different resampling schemes, one can identify different kinds of anomalies. For multispectral images, anomalous pixels correspond to locations on the ground with unusual spectral signatures or, depending on how feature sets are constructed, unusual spatial textures.

A new test for the multivariate two-sample problem based on the concept of minimum energy

Abstract: We introduce a new statistical quantity the energy to test whether two samples originate from the same distributions. The energy is a simple logarithmic function of the distances of the observations in the variate space. The distribution of the test statistic is determined by a resampling method. The power of the energy test in one dimension was studied for a variety of different test samples and compared to several nonparametric tests. In two and four dimensions a comparison was performed with the Friedman-Rafsky and nearest neighbor tests. The two-sample energy test was shown to be especially powerful in multidimensional applications.

Resampling-based Multiple Testing: Asymptotic Control of Type I Error and Applications to Gene Expression Data

Abstract: We define a general statistical framework for multiple hypothesis testing and show that the correct null distribution for the test statistics is obtained by projecting the true distribution of the test statistics onto the space of mean zero distributions. For common choices of test statistics (based on an asymptotically linear parameter estimator), this distribution is asymptotically multivariate normal with mean zero and the covariance of the vector influence curve for the parameter estimator. This test statistic null distribution can be estimated by applying the non-parametric or parametric bootstrap to correctly centered test statistics. We prove that this bootstrap estimated null distribution provides asymptotic control of most type I error rates. We show that obtaining a test statistic null distribution from a data null distribution, e.g. projecting the data generating distribution onto the space of all distributions satisfying the complete null), only provides the correct test statistic null distribution if the covariance of the vector influence curve is the same under the data null distribution as under the true data distribution. This condition is a weak version of the subset pivotality condition. We show that our multiple testing methodology controlling type I error is equivalent to constructing an error-specific confidence region for the true parameter and checking if it contains the hypothesized value. We also study the two sample problem and show that the permutation distribution produces an asymptotically correct null distribution if (i) the sample sizes are equal or (ii) the populations have the same covariance structure. We include a discussion of the application of multiple testing to gene expression data, where the dimension typically far exceeds the sample size. An analysis of a cancer gene expression data set illustrates the methodology.

Recent Advances and Trends in Nonparametric Statistics

Abstract: Preface. 1. Algorithmic Approaches to Statistics An introduction to support vector machines (B. Scholkopf). Bagging, subagging and bragging for improving some prediction algorithms (P. Buhlmann). Data compression by geometric quantization (Nkem-Amin Khumbah , E. J. Wegman). 2. Functional Data Analysis Functional data analysis in evolutionary biology (N. E. Heckman). Functional nonparametric statistics: a double infinite dimensional framework (F. Ferraty, P. Vieu). 3. Nonparametric Model Building Nonparametric models for ANOVA and ANCOVA: a review (M. G. Akritas, E. Brunner). Isotonic additive interaction models (I. Gluhovsky). A nonparametric alternative to analysis of covariance (A. Bathke, E. Brunner). 4. Goodness Of Fit Assessing structural relationships between distributions - a quantile process approach based on Mallows distance (G. Freitag, A. Munk, M. Vogt). Almost sure representations in survival analysis under censoring and truncation: applications to goodness-of-fit tests (R. Cao, W. Gonzalez Manteiga, C. Iglesias Perez) 5. High-Dimensional Data And Visualization Data depth: center-outward ordering of multivariate data and nonparametric multivariate statistics (R. Y. Liu). Visual exploration of data through their graph representations (G. Michailidis). 6. Nonparametric Regression Inference for nonsmooth regression curves and surfaces using kernel-based methods (I. Gijbels). Nonparametric smoothing methods for a class of non-standard curve estimation problems (O. Linton, E. Mammen). Weighted local linear approach to censored nonparametric regression (Z. Cai). 7. Topics In Nonparametrics Adaptive quantile regression (S. van de Geer). Set estimation: an overview and some recent developments (A. Cuevas, A. Rodriguez-Casal). Nonparametric methods for heavy tailed vector data: a survey with applications from finance and hydrology (M. M. Meerschaert, Hans-Peter Scheffler). 8. Nonparametrics in Finance Nonparametric methods in continuous-time finance: a selective review (Z. Cai, Y. Hong). Nonparametric estimation in a stochastic volatility model (J. Franke, W. Hardle, Jens-Peter Kreiss). Dynamic nonparametric filtering with application to volatility estimation (Ming-Yen Cheng, J. Fan, V. Spokoiny). A normalizing and variance-stabilizing transformation for financial time series (D. N. Politis). 9. Bioinformatics and Biostatistics Biostochastics and nonparametrics: oranges and apples? (P. K. Sen). Some issues concerning length-biased sampling in survival analysis (M. Asgharian, D. B. Wolfson). Covariate centering and scaling in varying-coefficient regression with application to longitudinal growth studies (C. O. Wu, K. F. Yu, V. W.S. Yuan). Directed peeling and covering of patient rules (M. LeBlanc, J. Moon, J. Crowley). 10. Resampling and Subsampling Statistical analysis of survival models with Bayesian bootstrap (J. Lee, Y. Kim). On optimal variance estimation under different spatial subsampling schemes (D. J. Nordman, S. N. Lahiri). Locally stationary processes and the local block bootstrap (A. Dowla, E. Paparoditis, D. N. Politis). 11. Time Series and Stochastic Processes Spectral analysis and a class of nonstationary processes (M. Rosenblatt). Curve estimation for locally stationary time series models (R. Dahlhaus). Assessing spatial isotropy (M. Sherman, Y. Guan, J. A. Calvin). 12. Wavelet and Multiresolution Methods Automatic landmark registration of 1D curves (J. Bigot). Stochastic multiresolution models for turbulence (B. Whitcher, J.B. Weiss, D.W. Nychka, T.J. Hoar). List of Contributors.

Improved sampling-importance resampling and reduced bias importance sampling

Abstract: . The sampling-importance resampling (SIR) algorithm aims at drawing a random sample from a target distribution π. First, a sample is drawn from a proposal distribution q, and then from this a smaller sample is drawn with sample probabilities proportional to the importance ratios π/q. We propose here a simple adjustment of the sample probabilities and show that this gives faster convergence. The results indicate that our version converges better also for small sample sizes. The SIR algorithms are compared with the Metropolis–Hastings (MH) algorithm with independent proposals. Although MH converges asymptotically faster, the results indicate that our improved SIR version is better than MH for small sample sizes. We also establish a connection between the SIR algorithms and importance sampling with normalized weights. We show that the use of adjusted SIR sample probabilities as importance weights reduces the bias of the importance sampling estimate.

Exact Log-Rank Tests for Unequal Follow-Up

Abstract: The asymptotic log-rank and generalized Wilcoxon tests are the standard procedures for comparing samples of possibly censored survival times. For comparison of samples of very different sizes, an exact test is available that is based on a complete permutation of log-rank or Wilcoxon scores. While the asymptotic tests do not keep their nominal sizes if sample sizes differ substantially, the exact complete permutation test requires equal follow-up of the samples. Therefore, we have developed and present two new exact tests also suitable for unequal follow-up. The first of these is an exact analogue of the asymptotic log-rank test and conditions on observed risk sets, whereas the second approach permutes survival times while conditioning on the realized follow-up in each group. In an empirical study, we compare the new procedures with the asymptotic log-rank test, the exact complete permutation test, and an earlier proposed approach that equalizes the follow-up distributions using artificial censoring. Results confirm highly satisfactory performance of the exact procedure conditioning on realized follow-up, particularly in case of unequal follow-up. The advantage of this test over other options of analysis is finally exemplified in the analysis of a breast cancer study.