Double Bootstrap Confidence Intervals in the Two-Stage DEA Approach

Contextual factors usually assume an important role in determining firms' productive efficiencies. Nevertheless, identifying them in a regression framework might be complicated. The problem arises from the efficiencies being correlated with each other when estimated by Data Envelopment Analysis, rendering standard inference methods invalid. Simar and Wilson (2007) suggest the use of bootstrap algorithms that allow for valid statistical inference in this context. This article extends their work by proposing a double bootstrap algorithm for obtaining confidence intervals with improved coverage probabilities. Moreover, acknowledging the computational burden associated with iterated bootstrap procedures, we provide an algorithm based on deterministic stopping rules, which is less computationally demanding. Monte Carlo evidence shows considerable improvement in the coverage probabilities after iterating the bootstrap procedure. The results also suggest that percentile confidence intervals perform better than their basic counterpart

Sequential Implementation of Monte Carlo Tests with Uniformly Bounded Resampling Risk

This paper introduces an open-ended sequential algorithm for computing the p-value of a test using Monte Carlo simulation. It guarantees that the resampling risk, the probability of a different decision than the one based on the theoretical p-value, is uniformly bounded by an arbitrarily small constant. Previously suggested sequential or non-sequential algorithms, using a bounded sample size, do not have this property. Although the algorithm is open-ended, the expected number of steps is finite, except when the p-value is on the threshold between rejecting and not rejecting. The algorithm is suitable as standard for implementing tests that require (re-)sampling. It can also be used in other situations: to check whether a test is conservative, iteratively to implement double bootstrap tests, and to determine the sample size required for a certain power.Comment: Major Revision 15 pages, 4 figure

On the Inversion of High Energy Proton

Inversion of the K-fold stochastic autoconvolution integral equation is an elementary nonlinear problem, yet there are no de facto methods to solve it with finite statistics. To fix this problem, we introduce a novel inverse algorithm based on a combination of minimization of relative entropy, the Fast Fourier Transform and a recursive version of Efron's bootstrap. This gives us power to obtain new perspectives on non-perturbative high energy QCD, such as probing the ab initio principles underlying the approximately negative binomial distributions of observed charged particle final state multiplicities, related to multiparton interactions, the fluctuating structure and profile of proton and diffraction. As a proof-of-concept, we apply the algorithm to ALICE proton-proton charged particle multiplicity measurements done at different center-of-mass energies and fiducial pseudorapidity intervals at the LHC, available on HEPData. A strong double peak structure emerges from the inversion, barely visible without it.Comment: 29 pages, 10 figures, v2: extended analysis (re-projection ratios, 2D

A subsampled double bootstrap for massive data

The bootstrap is a popular and powerful method for assessing precision of estimators and inferential methods. However, for massive datasets which are increasingly prevalent, the bootstrap becomes prohibitively costly in computation and its feasibility is questionable even with modern parallel computing platforms. Recently Kleiner, Talwalkar, Sarkar, and Jordan (2014) proposed a method called BLB (Bag of Little Bootstraps) for massive data which is more computationally scalable with little sacrifice of statistical accuracy. Building on BLB and the idea of fast double bootstrap, we propose a new resampling method, the subsampled double bootstrap, for both independent data and time series data. We establish consistency of the subsampled double bootstrap under mild conditions for both independent and dependent cases. Methodologically, the subsampled double bootstrap is superior to BLB in terms of running time, more sample coverage and automatic implementation with less tuning parameters for a given time budget. Its advantage relative to BLB and bootstrap is also demonstrated in numerical simulations and a data illustration