Bootstrap Confidence Regions Research Articles

Calibrating and Visualizing Some Bootstrap Confidence Regions

When the bootstrap sample size is moderate, bootstrap confidence regions tend to have undercoverage. Improving the coverage is known as calibrating the confidence region. Consider testing H0:θ=θ0 versus H1:θ≠θ0. We reject H0 only if θ0 is not contained in a large-sample 95% confidence region. If the confidence region has 3% undercoverage for the data set sample size, then the type I error is 8% instead of the nominal 5%. Hence, calibrating confidence regions is also useful for testing hypotheses. Several bootstrap confidence regions are also prediction regions for a future value of a bootstrap statistic. A new bootstrap confidence region uses a simple prediction region calibration technique to improve the coverage. The DD plot for visualizing prediction regions can also be used to visualize some bootstrap confidence regions.

Alternative bootstrap confidence regions for multiple correspondence analysis

A new approach for constructing elliptical bootstrap confidence regions for Multiple Correspondence Analysis is proposed, where the difference in method follows directly from the adoption of a different objective criterion to previous approaches. This includes a new correction for the well-known problems caused by the diagonal of the Burt matrix. Simulation experiments show that the method performs reasonably well in many cases, with exceptions being noted. Simulated data sets with known structure are also used to illustrate cases where the proposed method produces results in line with what would be expected, but some alternatives do not.

Bootstrapping ARMA time series models after model selection

Inference after model selection is a very important problem. This article derives the asymptotic distribution of some model selection estimators for autoregressive moving average time series models. Under strong regularity conditions, the model selection estimators are asymptotically normal, but generally the asymptotic distribution is a non normal mixture distribution. Hence bootstrap confidence regions that can handle this complicated distribution were used for hypothesis testing. A bootstrap technique to eliminate selection bias is to fit the model selection estimator β ̂ M S ∗ to a bootstrap sample to find a submodel, then draw another bootstrap sample, and fit the same submodel to get the bootstrap estimator β ̂ M I X ∗ .

Bootstrap Confidence Regions for Learned Feature Embeddings

Algorithmic feature learners provide high-dimensional vector representations for non-matrix structured data, like image or text collections. Low-dimensional projections derived from these representations, called embeddings, are often used to explore variation in these data. However, it is not clear how to assess the embedding uncertainty. We adapt methods developed for bootstrapping principal components analysis to the setting where features are algorithmically derived from nonmatrix data. We empirically compare the derived confidence areas in simulations, varying factors influencing feature learning and the bootstrap, like feature learning algorithm complexity and bootstrap sample size. We illustrate the proposed approaches on a spatial proteomics dataset, where we observe that embedding precision is not uniform across all tissue types. Code, data, and pretrained models are available as an R compendium in the supplementary materials. Supplementary files for this article are available online.

Wald type tests with the wrong dispersion matrix

A Wald type test with the wrong dispersion matrix is used when the dispersion matrix is not a consistent estimator of the asymptotic covariance matrix of the test statistic. One class of such tests occurs when there are p groups and it is assumed that the population covariance matrices from the p groups are equal, but the common covariance matrix assumption does not hold. The pooled t test, one-way ANOVA F test, and one-way MANOVA F test are examples of this class. Another class of such tests is used for weighted least squares. Two bootstrap confidence regions are modified to obtain large sample Wald type tests with the wrong dispersion matrix.

Directions Old and New: Palaeomagnetism and Fisher (1953) Meet Modern Statistics

SummaryMost modern articles in the palaeomagnetism literature are based on statistics developed by Fisher's 1953 paper ‘Dispersion on a sphere’, which assumes independent and identically distributed (iid) spherical data. However, palaeomagnetic sample designs are usually hierarchical, where specimens are collected within sites and the data are then combined across sites to calculate an overall mean direction for a geological formation. The specimens within sites are typically more similar than specimens between different sites, and so the iid assumptions fail. This article has three principal goals. The first is to review, contrast and compare both the statistics and geophysics literature on the topic of analysis methods for clustered data on spheres. The second is to present a new hierarchical parametric model, which avoids the unrealistic assumption of rotational symmetry in Fisher's 1953 paper ‘Dispersion on a sphere’ and may be broadly useful in the analysis of many palaeomagnetic datasets. To help develop the model, we use publicly available data as a case study collected from the Golan Heights volcanic plateau. The third goal is to explore different methods for constructing confidence regions for the overall mean direction based on clustered data. Two bootstrap confidence regions that we propose perform well and will be especially useful to geophysics practitioners.

Bootstrapping some GLM and survival regression variable selection estimators

Inference after variable selection is a very important problem. This paper derives the asymptotic distribution of many variable selection estimators, such as forward selection and backward elimination, when the number of predictors is fixed. Under strong regularity conditions, the variable selection estimators are asymptotically normal, but generally the asymptotic distribution is a nonnormal mixture distribution. The theory shows that the lasso variable selection and elastic net variable selection estimators are consistent estimators of when lasso and elastic net are consistent estimators of A bootstrap technique to eliminate selection bias is to fit the variable selection estimator to a bootstrap sample to find a submodel, then draw another bootstrap sample and fit the same submodel to get the bootstrap estimator Bootstrap confidence regions were used for hypothesis testing.

A Central Limit Theorem for extrinsic antimeans and estimation of Veronese–Whitney means and antimeans on planar Kendall shape spaces

This article is concerned with random objects in the complex projective space ℂPk−2. It is shown that the Veronese–Whitney (VW) antimean, which is the extrinsic antimean of a random point on ℂPk−2 relative to the VW-embedding, is given by the point on ℂPk−2 represented by the eigenvector corresponding to the smallest eigenvalue of the expected mean of the VW-embedding of the random point, provided this eigenvalue is simple. We also derive a CLT for extrinsic sample antimeans, and an asymptotic χ2-distribution of an appropriately studentized statistic, based on the extrinsic antimean, which in the particular case of a VW-embedding is then used to construct nonparametric bootstrap confidence regions for the VW-antimean planar Kendall shape. Simulations studies and an application to medical imaging are illustrating the proposed methodology.

Bootstrap confidence regions based on M-estimators under nonstandard conditions

Suppose that a confidence region is desired for a subvector $\theta $ of a multidimensional parameter $\xi =(\theta ,\psi )$, based on an M-estimator $\hat{\xi }_{n}=(\hat{\theta }_{n},\hat{\psi }_{n})$ calculated from a random sample of size $n$. Under nonstandard conditions $\hat{\xi }_{n}$ often converges at a nonregular rate $r_{n}$, in which case consistent estimation of the distribution of $r_{n}(\hat{\theta }_{n}-\theta )$, a pivot commonly chosen for confidence region construction, is most conveniently effected by the $m$ out of $n$ bootstrap. The above choice of pivot has three drawbacks: (i) the shape of the region is either subjectively prescribed or controlled by a computationally intensive depth function; (ii) the region is not transformation equivariant; (iii) $\hat{\xi }_{n}$ may not be uniquely defined. To resolve the above difficulties, we propose a one-dimensional pivot derived from the criterion function, and prove that its distribution can be consistently estimated by the $m$ out of $n$ bootstrap, or by a modified version of the perturbation bootstrap. This leads to a new method for constructing confidence regions which are transformation equivariant and have shapes driven solely by the criterion function. A subsampling procedure is proposed for selecting $m$ in practice. Empirical performance of the new method is illustrated with examples drawn from different nonstandard M-estimation settings. Extension of our theory to row-wise independent triangular arrays is also explored.

Density Level Sets: Asymptotics, Inference, and Visualization

ABSTRACTWe study the plug-in estimator for density level sets under Hausdorff loss. We derive asymptotic theory for this estimator, and based on this theory, we develop two bootstrap confidence regions for level sets. We introduce a new technique for visualizing density level sets, even in multidimensions, which is easy to interpret and efficient to compute. Supplementary materials for this article are available online.

Statistical inference for two exponential populations under joint progressive type-I censored scheme

ABSTRACTIn this article, we introduce a new scheme called joint progressive type-I (JPC-I) censored and as a special case, joint type-I censored scheme. Bayesian and non Bayesian estimators have been obtained for two exponential populations under both JPC-I censored scheme and joint type-I censored. The maximum likelihood estimators of the parameters, the asymptotic variance covariance matrix, have been obtained. Bayes estimators have been developed under squared error loss function using independent gamma prior distributions. Moreover, approximate confidence region based on the asymptotic normality of the maximum likelihood estimators and credible confidence region from a Bayesian viewpoint are also discussed and compared with two Bootstrap confidence regions. A numerical illustration for these new results is given.

On the Added Value of Bootstrap Analysis for K-Means Clustering

Because of its deterministic nature, K-means does not yield confidence information about centroids and estimated cluster memberships, although this could be useful for inferential purposes. In this paper we propose to arrive at such information by means of a non-parametric bootstrap procedure, the performance of which is tested in an extensive simulation study. Results show that the coverage of hyper-ellipsoid bootstrap confidence regions for the centroids is in general close to the nominal coverage probability. For the cluster memberships, we found that probabilistic membership information derived from the bootstrap analysis can be used to improve the cluster assignment of individual objects, albeit only in the case of a very large number of clusters. However, in the case of smaller numbers of clusters, the probabilistic membership information still appeared to be useful as it indicates for which objects the cluster assignment resulting from the analysis of the original data is likely to be correct; hence, this information can be used to construct a partial clustering in which the latter objects only are assigned to clusters.

Nonparametric confidence regions for the central orientation of random rotations

Three-dimensional orientation data, with observations as 3 × 3 rotation matrices, have applications in areas such as computer science, kinematics and materials sciences, where it is often of interest to estimate a central orientation parameter S represented by a 3 × 3 rotation matrix. A well-known estimator of this parameter is the projected arithmetic mean and, based on this statistic, two nonparametric methods for setting confidence regions for S exist. Both of these methods involve large-sample normal theory, with one approach based on a data-transformation of rotations to directions (four-dimensional unit vectors) prior to analysis. However, both of these nonparametric methods may result in poor coverage accuracy in small samples. As a remedy, we consider two bootstrap methods for approximating the sampling distribution of the projected mean statistic and calibrating nonparametric confidence regions for the central orientation parameter S . As with normal approximations, one bootstrap method is based on the rotation data directly while the other bootstrap approach involves a data-transformation of rotations into directions. Both bootstraps are shown to be valid for approximating sampling distributions and calibrating confidence regions based on the projected mean statistic. A simulation study compares the performance of the normal theory and proposed bootstrap confidence regions for S , based on common data-generating models for symmetric orientations. The bootstrap methods are shown to exhibit good coverage accuracies, thus providing an improvement over normal theory approximations especially for small sample sizes. The bootstrap methods are also illustrated with a real data example from materials science.

Bayesian and Non–Bayesian Estimation for Two Generalized Exponential Populations Under Joint Type II Censored Scheme

In this paper, Bayesian and non-Bayesian estimators have been obtained for two generalized exponential populations under joint type II censored scheme, which generalize results of Balakrishnan and Rasouli (2008) and Shafay et al. (2013) . The maximum likelihood estimators (MLEs) of the parameters and Bayes estimators have been developed under squared error loss function as well as under LINEX loss function. Moreover, approximate confidence region are also discussed and compared with two Bootstrap confidence regions. Also the MLE and three confidence intervals for the stress–strength parameter are explored. A numerical illustration for these new results is given.

Robust confidence regions for multinomial probabilities

Generally, confidence regions for the probabilities of a multinomial population are constructed based on the Pearson χ2 statistic. Morales et al. (Bootstrap confidence regions in multinomial sampling. Appl Math Comput. 2004;155:295–315) considered the bootstrap and asymptotic confidence regions based on a broader family of test statistics known as power-divergence test statistics. In this study, we extend their work and propose penalized power-divergence test statistics-based confidence regions. We only consider small sample sizes where asymptotic properties fail and alternative methods are needed. Both bootstrap and asymptotic confidence regions are constructed. We consider the percentile and the bias corrected and accelerated bootstrap confidence regions. The latter confidence region has not been studied previously for the power-divergence statistics much less for the penalized ones. Designed simulation studies are carried out to calculate average coverage probabilities. Mean absolute deviation between actual and nominal coverage probabilities is used to compare the proposed confidence regions.

Second-order accuracy of depth-based bootstrap confidence regions

We consider the problem of setting bootstrap confidence regions for multivariate parameters based on data depth functions. We prove, under mild regularity conditions, that depth-based bootstrap confidence regions are second-order accurate in the sense that their coverage error is of order n − 1 , given a random sample of size n . The results hold in general for depth functions of types A and D, which cover as special cases the Tukey depth, the majority depth, and the simplicial depth. A simulation study is also provided to investigate empirically the bootstrap confidence regions constructed using these three depth functions.

Conditional Maximum Likelihood and Interval Estimation for Two Weibull Populations under Joint Type-II Progressive Censoring

Comparative lifetime experiments are important when the object of a study is to determine the relative merits of two competing duration of life products. This study considers the interval estimation for two Weibull populations when joint Type-II progressive censoring is implemented. We obtain the conditional maximum likelihood estimators of the two Weibull parameters under this scheme. Moreover, simultaneous approximate confidence region based on the asymptotic normality of the maximum likelihood estimators are also discussed and compared with two Bootstrap confidence regions. We consider the behavior of probability of failure structure with different schemes. A simulation study is performed and an illustrative example is also given.

Bootstrap inference for mean reflection shape and size-and-shape with three-dimensional landmark data

Working within the framework of a multi-dimensional scaling approach to shape analysis, we develop bootstrap methods for inference about mean reflection shape and size-and-shape based on labelled landmark data. The approach is developed in general dimensions though we focus on the three-dimensional case. We consider two pivotal statistics which we use to construct bootstrap confidence regions for the mean reflection shape or size-and-shape, and present simulation results which show that these statistics perform well in a variety of examples. We also suggest regularized versions of the test statistics that are suitable for more challenging cases where sample size is not sufficiently large in relation to the number of landmarks and present numerical results confirming that regularization indeed leads to better performance. An algorithm for producing a graphical representation of the confidence region for the mean reflection shape is presented and applied in an example involving molecular dynamics simulation data.

Bootstrap confidence regions for the planar mean shape

We consider bootstrap methods for constructing confidence regions for the mean shape of objects specified by labelled landmarks in two dimensions. Two statistics are considered: a pivotal statistic, T, derived using matrix perturbation arguments; and a Hotelling-type statistic, H , based on partial Procrustes tangent projections of the observations. We give a rigorous proof, under weak conditions, that the null asymptotic distribution of T is χ 2 . Simulation results show that (i) the confidence region procedure obtained by bootstrapping each statistic is clearly superior to the corresponding ‘tabular’ procedure; and (ii) the pivotal T bootstrap confidence regions generally have smaller coverage error than the Hotelling bootstrap confidence regions, especially for distributions with low concentration.

Some nonasymptotic results on resampling in high dimension, II: Multiple tests

We study generalized bootstrap confidence regions for the mean of a random vector whose coordinates have an unknown dependency structure. The random vector is supposed to be either Gaussian or to have a symmetric and bounded distribution. The dimensionality of the vector can possibly be much larger than the number of observations and we focus on a nonasymptotic control of the confidence level, following ideas inspired by recent results in learning theory. We consider two approaches, the first based on a concentration principle (valid for a large class of resampling weights) and the second on a resampled quantile, specifically using Rademacher weights. Several intermediate results established in the approach based on concentration principles are of interest in their own right. We also discuss the question of accuracy when using Monte Carlo approximations of the resampled quantities.