Exact Bootstrap Research Articles

Estimation of quantiles with the exact bootstrap method

A problem with the estimation of quantiles occurs when the sample comes from an unknown distribution. The estimation uses the bootstrap method in the version that the literature refers to as exact. Three bootstrap estimators were used: two of them based on one order statistic, and the third on a linear combination of two order statistics (for an integer). The distribution of the exact bootstrap estimator based on a single order statistic is known. It has been shown that there is no general form of the distribution of the exact bootstrap estimator based on two order statistics. However, it is possible to calculate such a distribution – the article presents the algorithm that performs such a task. The bootstrap confidence intervals were constructed using the exact percentile method. It has been shown that if the estimator is based on a single order statistic, it is known in advance which elements of the primary sample are the limits of the confidence intervals, so there is no need to resample. The intervals determined by the exact percentile method were compared with those constructed using other methods. It has been shown that the information on the direction of the asymmetry of the distribution that the sample comes from is worth considering when selecting the rank of the order statistic used as an estimator. Attention is paid to the influence of the quality of the pseudorandom number generators on the results of the Monte Carlo simulation.

Deep bootstrap for Bayesian inference.

For a Bayesian, the task to define the likelihood can be as perplexing as the task to define the prior. We focus on situations when the parameter of interest has been emancipated from the likelihood and is linked to data directly through a loss function. We survey existing work on both Bayesian parametric inference with Gibbs posteriors and Bayesian non-parametric inference. We then highlight recent bootstrap computational approaches to approximating loss-driven posteriors. In particular, we focus on implicit bootstrap distributions defined through an underlying push-forward mapping. We investigate independent, identically distributed (iid) samplers from approximate posteriors that pass random bootstrap weights through a trained generative network. After training the deep-learning mapping, the simulation cost of such iid samplers is negligible. We compare the performance of these deep bootstrap samplers with exact bootstrap as well as MCMC on several examples (including support vector machines or quantile regression). We also provide theoretical insights into bootstrap posteriors by drawing upon connections to model mis-specification. This article is part of the theme issue 'Bayesian inference: challenges, perspectives, and prospects'.

Cosmological bootstrap in slow motion

Speed matters. How the masses and spins of new particles active during inflation can be read off from the statistical properties of primordial density fluctuations is well understood. However, not when the propagation speeds of the new degrees of freedom and of the curvature perturbation differ, which is the generic situation in the effective field theory of inflationary fluctuations. Here we use bootstrap techniques to find exact analytical solutions for primordial 2-,3- and 4-point correlators in this context. We focus on the imprints of a heavy relativistic scalar coupled to the curvature perturbation that propagates with a reduced speed of sound cs, hence strongly breaking de Sitter boosts. We show that akin to the de Sitter invariant setup, primordial correlation functions can be deduced by acting with suitable weight-shifting operators on the four-point function of a conformally coupled field induced by the exchange of the massive scalar. However, this procedure requires the analytical continuation of this seed correlator beyond the physical domain implied by momentum conservation. We bootstrap this seed correlator in the extended domain from first principles, starting from the boundary equation that it satisfies due to locality. We further impose unitarity, reflected in cosmological cutting rules, and analyticity, by demanding regularity in the collinear limit of the four-point configuration, in order to find the unique solution. Equipped with this, we unveil that heavy particles that are lighter than H/cs leave smoking gun imprints in the bispectrum in the form of resonances in the squeezed limit, a phenomenon that we call the low speed collider. We characterise the overall shape of the signal as well as its unusual logarithmic mass dependence, both vividly distinct from previously identified signatures of heavy fields. Eventually, we demonstrate that these features can be understood in a simplified picture in which the heavy field is integrated out, albeit in a non-standard manner resulting in a single-field effective theory that is non-local in space. Nonetheless, the latter description misses the non-perturbative effects of spontaneous particle production, well visible in the ultra-squeezed limit in the form of the cosmological collider oscillations, and it breaks down for masses of order the Hubble scale, for which only our exact bootstrap results hold.

Evaluation of Inter-Observer Reliability of Animal Welfare Indicators: Which Is the Best Index to Use?

Simple SummaryIn order to be effective, on-farm welfare assessment protocols should always rely on reliable, as well as valid and feasible, indicators. Inter-observer reliability refers to the extent to which two or more observers are observing and recording data in the same way. The present study focuses on the problem of assessing inter-observer reliability in the case of dichotomous (e.g., yes/no) welfare indicators and the presence of two observers, in order to decide about the inclusion of indicators in welfare assessment protocols. We compared the performance of the most popular currently available agreement indexes. Some widely used indexes showed their inappropriateness to evaluate the inter-observer reliability when the agreement between observers was high. Other less used indexes, such as Bangdiwala’s or Gwet’s , were found to perform better and are therefore suggested to assess the inter-observer reliability of dichotomous indicators.This study focuses on the problem of assessing inter-observer reliability (IOR) in the case of dichotomous categorical animal-based welfare indicators and the presence of two observers. Based on observations obtained from Animal Welfare Indicators (AWIN) project surveys conducted on nine dairy goat farms, and using udder asymmetry as an indicator, we compared the performance of the most popular agreement indexes available in the literature: Scott’s , Cohen’s , , Holsti’s , Krippendorff’s , Hubert’s , Janson and Vegelius’ , Bangdiwala’s , Andrés and Marzo’s , and Gwet’s . Confidence intervals were calculated using closed formulas of variance estimates for , , , , , and , while the bootstrap and exact bootstrap methods were used for all the indexes. All the indexes and closed formulas of variance estimates were calculated using Microsoft Excel. The bootstrap method was performed with R software, while the exact bootstrap method was performed with SAS software. , and exhibited a paradoxical behavior, showing unacceptably low values even in the presence of very high concordance rates. and showed values very close to the concordance rate, independently of its value. Both bootstrap and exact bootstrap methods turned out to be simpler compared to the implementation of closed variance formulas and provided effective confidence intervals for all the considered indexes. The best approach for measuring IOR in these cases is the use of or , with bootstrap or exact bootstrap methods for confidence interval calculation.

WPŁYW ASYMETRII ROZKŁADU NA DOBÓR BOOTSTRAPOWEGO ESTYMATORA KWARTYLI

Badania dotyczą estymacji kwartyli (pierwszego, drugiego i trzeciego) w sytuacji gdy brak jest informacji o rozkładzie, z którego wylosowana została próba, zaś iloczyn np ma wartość całkowitą (gdzie n jest liczebnością próby, a p rzędem kwantyla). Jeśli np nie jest całkowite jako estymator kwantyla rzędu p wybierana jest zwykle statystyka pozycyjna rzędu [np]+1. Jeśli np nie jest całkowite rozwiązań jest znacznie więcej. W niniejszej pracy porównane zostały dwa dokładne bootstrapowe estymatory kwartyli w postaci pojedynczych statystyk pozycyjnych rzędu np i np +1. Do oceny wykorzystane zostały obciążenie i wariancja estymatora oraz szerokość przedziałów ufności i zliczeniowy poziom ufności. Przedziały ufności wyznaczone zostały dokładną metodą percentyli. Próby losowano z rozkładów o asymetrii prawo i lewostronnej oraz symetrycznych, co umożliwiło wybór estymatora najbardziej odpowiedniego w danej sytuacji.

PATH-22. COMPARISON OF SUPERVISED CLASSIFICATION METHODS FOR CENTRAL NERVOUS SYSTEM TUMORS BASED ON DNA-METHYLATION

Classification of brain tumors using methylation profiling is an important diagnostic advance, reducing subjectivity and improving interpretability of clinical outcome data. Despite the recognized value of methylation profiling in the clinical laboratory, the performance characteristics of different supervised classification models has not been directly compared. We developed 3 methods using methylation profiles to classify CNS tumors: an exact bootstrap k-nearest neighbor (kNN), a multi-layer perceptron neural net (NN), and a random forest classifier (RF). We trained these methods on the publicly available CNS tumor reference cohort (GSE90496) with 2,801 profiles and 91 classes. We evaluated the performance of these methods by leave-out-25% cross-validation. The relative performance of these methods were evaluated in terms of accuracy, precision, and recall for class or class family. The kNN, RF, and NN classifier had an estimate error rate of 10.74%, 4.01%, and 1.89%, respectively for class prediction and an error rate for family prediction of 5.97%, 0.90%, and 0.6%, respectively. At perfect recall for class assignment, the RF and kNN had a precision of 0.96 and 0.89 while the NN reached 0.98. For family assignment, the precision for the three classifiers was almost 1.0 with recall of nearly 0.8. At the recall rate of 1.0, the precision dropped to 0.94, 0.991 and 0.994 for kNN, RF, and NN, respectively. Overall, the NN showed improved performance metrics compared to the kNN and RF in CNS tumor classification for both class and class family assignment.

A semi-parametric bootstrap-based best linear unbiased estimator of location under symmetry

In this note we provide a novel semi-parametric best linear unbiased estimator (BLUE) of location and its corresponding variance estimator under the assumption the random variate is generated from a symmetric location-scale family of distributions. The approach follows in a two-stage fashion and is based on the exact bootstrap estimate of the covariance matrix of the order statistic. We generalize our approach to add a robustness component in order to derive a trimmed BLUE of location under a semi-parametric symmetry assumption.

Computation of Exact Bootstrap Confidence Intervals: Complexity and Deterministic Algorithms

The bootstrap method is one of the major developments in statistics in the 20th century for computing confidence intervals directly from data. However, the bootstrap method is traditionally approximated with a randomized algorithm, which can sometimes produce inaccurate confidence intervals. In “Computation of Exact Bootstrap Confidence Intervals: Complexity and Deterministic Algorithms,” Bertsimas and Sturt present a new perspective of the bootstrap method through the lens of counting integer points in a polyhedron. Through this perspective, the authors develop the first computational complexity results and efficient deterministic approximation algorithm (fully polynomial time approximation scheme) for bootstrap confidence intervals, which unlike traditional methods, has guaranteed bounds on its error. In experiments on real and synthetic data sets from clinical trials, the proposed deterministic algorithms quickly produce reliable confidence intervals, which are significantly more accurate than those from randomization.

Evaluation of inferential methods for the net benefit and win ratio statistics

ABSTRACT General Pairwise Comparison (GPC) statistics, such as the net benefit and the win ratio, have been applied in clinical trial data analysis and design. In the literature, inferential methods based on re-sampling, asymptotic or exact methods have been proposed for these GPC statistics, but they have not been compared to each other. In this paper, the small sample bias of the variance estimation, Type I error control and 95% confidence interval coverage of the GPC inferential methods are evaluated using simulations. The exact permutation and bootstrap tests perform best in all evaluated aspects for the net benefit, while the exact bootstrap test performs best for the win ratio.

Efficiency balanced designs for bootstrap simulations

This paper considers the balanced resampling design approach of generating bootstrap samples to handle the simulation error problem in bootstrap estimation of mean and variance of a sample statistic of interest. A class of balanced n-arry block design, slightly different from the definition of Tocher (1952), is defined and its necessary conditions for the existence of these designs are derived. In particular, an infinite series of balanced n-arry block designs are found to be efficiency balanced, i.e., every contrast of element (observation) totals has the same loss of information. As an application in bootstrap method, the eccentric class of efficiency balanced n-arry block design (termed Efficient Bootstrap Sample Design (EBSD(n)) can be used to generate resamples of sample size n to achieve an exact bootstrap estimation of mean and variance of statistic “sample average”, i.e., a collection of bootstrap data samples and corresponding bootstrap sample averages where mean and variance of this empirical collection of sample averages match the first two theoretical central moments i.e., moments about the sample average. The EBSD(n) solely depends on the existence of balanced incomplete block design B[ 4n-2, 2n-1, n, n-1] for n ≥ 2. Thus the implementation of computer algorithm for spawning bootstrap samples using EBSD(n) is simple and straightforward for recapitulating sampling characteristics such as standard error of predefined computable symmetric and non symmetric sample statistic of interest.

HW_TEST, a program for comprehensive HARDY-WEINBERG equilibrium testing.

This article deals with a Windows (© Microsoft Inc.) executable, user-friendly program that tests the hypothesis of Hardy-Weinberg (HW) proportions from autosomal multiallelic data using different methods that include parametric, nonparametric and exact bootstrap tests, the latter obtained through computer simulations. The program can be obtained free of charge directly from the internet repository https://github.com/Lemes-RenanB/HardyWeinbergTesting.

Bayesian and Classical Estimation of the Inverse Pareto Distribution and Its Application to Strength-Stress Models

SYNOPTIC ABSTRACTThis article deals with the estimation of R = P(Y < X) when X and Y are two independent random variables following inverse Pareto distributions with different parameters. The maximum-likelihood estimator of R and its asymptotic sampling distribution are proposed. The asymptotic sampling distribution is used to construct an asymptotic confidence interval for R. The exact confidence interval and bootstrap confidence interval for R are also presented. Bayes estimate and credible interval are studied using Gibbs sampling technique. Monte Carlo simulations are performed to compare the performance of different proposed estimation methods. Analysis of a real data set is presented for illustrative purposes.

Exact bootstrap confidence intervals for regression coefficients in small samples

ABSTRACTRegression analysis is one of the important tools in statistics to investigate the relationships among variables. When the sample size is small, however, the assumptions for regression analysis can be violated. This research focuses on using the exact bootstrap to construct confidence intervals for regression parameters in small samples. The comparison of the exact bootstrap method with the basic bootstrap method was carried out by a simulation study. It was found that on a very small sample (n ≈ 5) under Laplace distribution with the independent variable treated as random, the exact bootstrap was more effective than the standard bootstrap confidence interval.

Bootstrapping data arrays of arbitrary order

In this paper we study a bootstrap strategy for estimating the variance of a mean taken over large multifactor crossed random effects data sets. We apply bootstrap reweighting independently to the levels of each factor, giving each observation the product of independently sampled factor weights. No exact bootstrap exists for this problem [McCullagh (2000) Bernoulli 6 285-301]. We show that the proposed bootstrap is mildly conservative, meaning biased toward overestimating the variance, under sufficient conditions that allow very unbalanced and heteroscedastic inputs. Earlier results for a resampling bootstrap only apply to two factors and use multinomial weights that are poorly suited to online computation. The proposed reweighting approach can be implemented in parallel and online settings. The results for this method apply to any number of factors. The method is illustrated using a 3 factor data set of comment lengths from Facebook.

The exact bootstrap method shown on the example of the mean and variance estimation

The bootstrap method is based on resampling of the original randomsample drawn from a population with an unknown distribution. In the article it was shown that because of the progress in computer technology resampling is actually unnecessary if the sample size is not too large. It is possible to automatically generate all possible resamples and calculate all realizations of the required statistic. The obtained distribution can be used in point or interval estimation of population parameters or in testing hypotheses. We should stress that in the exact bootstrap method the entire space of resamples is used and therefore there is no additional bias which results from resampling. The method was used to estimate mean and variance. The comparison of the obtained distributions with the limit distributions confirmed the accuracy of the exact bootstrap method. In order to compare the exact bootstrap method with the basic method (with random sampling) probability that 1,000 resamples would allow for estimating a parameter with a given accuracy was calculated. There is little chance of obtaining the desired accuracy, which is an argument supporting the use of the exact method. Random sampling may be interpreted as discretization of a continuous variable.

Capital Allocation Using the Bootstrap

This paper investigates the use of the bootstrap in capital allocation. In particular, for the distortion risk measure (DRM) class, we show that the exact bootstrap estimate is available in analytic form for the allocated capital. We then theoretically justify the bootstrap bias correction for the allocated capital induced from the concave DRM when the conditional mean function is strictly monotone. A numerical example shows a tradeoff exists between the bias reduction and variance increase in bootstrapping the allocated capital. However, unlike the aggregate capital case, the variance increase of the bias-corrected allocated capital estimate substantially outweighs the benefit of bias correction, making the bootstrap bias correction at the allocated capital level not as useful. Overall, the exact bootstrap without bias correction offers an efficient method for determining allocation over the ordinary resampling bootstrap estimate and the empirical counterpart.

An exact bootstrap for variance of finite-population L-statistic

We give an exact formula of a finite-population bootstrap variance estimator for a general class of L-statistic. It is aimed to reduce the computational burden and to eliminate the approximation error, typically present in resampling approximations based on simulation. In the case of the classical nonparametric Efron bootstrap, a similar formula was shown by Hutson and Ernst [A.D. Hutson and M.D. Ernst, The exact bootstrap mean and variance of an L-estimator, J. R. Stat. Soc., Ser. B, 62:89–94, 2000].

Dokładna metoda bootstrapowa i jej zastosowanie do estymacji wariancji

The article presents the exact bootstrap method, which can be used to estimate the parameters of the estimators of random variables with unknown distribution. The method allows to determine an estimate of any parameter, the error of estimation, the distribution of the estimator and confidence intervals. Traditionally this task is carried out using the bootstrap method, which consists of resampling of the original sample. Random sampling is necessary if examining the entire population data is impossible or too costly. Note that the fundamental sample property is of finite size and we know its distribution – it is the empirical distribution. Rather than driving a resample, we can generate automatically the entire resample space and calculate the values of a statistic which is looking for a parameter estimator. This article describes a method for performing exact algorithm for bootstrapping, which correctness was verified on an example of the unbiased estimator of variance. It is shown that the expected value of the estimator calculated with exact bootstrap is exactly equal the variance of the sample. The method does not introduce bias of the resampling, as it may be for the classic bootstrap. The distribution of the estimator determined by the exact bootstrap compared with the limit distribution for estimator of variance, which does not require assumptions of normality of the original sample. Research has shown that if we estimate variance we cannot ignore the issue of normality of the original sample.

Bootstrap, jackknife and Edgeworth approximations for finite population L-statistics

In this paper we give exact bootstrap estimators for the parameters defining one-term Edgeworth expansion of distribution function of finite population L-statistic and compare these estimators with corresponding jackknife estimators. We also compare `````` true’ distribution of L-statistic with its normal approximation, Edgeworth expansion, empirical Edgeworth expansion and bootstrap approximation.

An exact bootstrap approach towards modification of the Harrell–Davis quantile function estimator for censored data

A new kernel quantile estimator is proposed for right-censored data, which takes the form of , where w j(u, c) is based on a beta kernel with bandwidth parameter c. The advantage of this estimator is that exact bootstrap methods may be employed to estimate the mean and variance of [Qcirc](u; c). It follows that a novel solution for finding the optimal bandwidth may be obtained through minimization of the exact bootstrap mean squared error (MSE) estimate of [Qcirc](u; c). We prove the large sample consistency of [Qcirc](u; c) for fixed values of c. A Monte Carlo simulation study shows that our estimator is significantly better than the product-limit quantile estimator [Qcirc] KM(u)=inf{t:[Fcirc] n (t)≥u}, with respect to various MSE criteria. For general simplicity, setting c=1 leads to an extension of classical Harrell–Davis estimator for censored data and performs well in simulations. The procedure is illustrated by an application to lung cancer survival data.