Dependent Bootstrap Research Articles

Mean convergence theorems for arrays of dependent random variables with applications to dependent bootstrap and non-homogeneous Markov chains

Mean convergence theorems for arrays of dependent random variables with applications to dependent bootstrap and non-homogeneous Markov chains

Bootstrap inference under cross‐sectional dependence

In this paper, we introduce a method of generating bootstrap samples with unknown patterns of cross‐ sectional/spatial dependence, which we call the spatial dependent wild bootstrap. This method is a spatial counterpart to the wild dependent bootstrap of Shao (2010) and generates data by multiplying a vector of independently and identically distributed external variables by the eigendecomposition of a bootstrap kernel. We prove the validity of our method for studentized and unstudentized statistics under a linear array representation of the data. Simulation experiments document the potential for improved inference with our approach. We illustrate our method in a firm‐level regression application investigating the relationship between firms' sales growth and the import activity in their local markets using unique firm‐level and imports data for Canada.

Comparing spontaneous and pellet-triggered ELMs via non-linear extended MHD simulations

Injecting frozen deuterium pellets into an ELMy H-mode plasma is a well established scheme for triggering edge localized modes (ELMs) before they naturally occur. This paper presents non-linear simulations of spontaneous type-I ELMs and pellet-triggered ELMs in ASDEX Upgrade performed with the extended MHD code JOREK. A thorough comparison of the non-linear dynamics of these events is provided. In particular, pellet-triggered ELMs are simulated by injecting deuterium pellets into different time points during the pedestal build-up described in A Cathey et al (2020 Nuclear Fusion 60 124007). Realistic ExB and diamagnetic background plasma flows as well as the time dependent bootstrap current evolution are included during the build-up to accurately capture the balance between stabilising and destabilising terms for the edge instabilities. Dependencies on the pellet size and injection times are studied. The spatio-temporal structures of the modes and the resulting divertor heat fluxes are compared in detail between spontaneous and triggered ELMs. We observe that the premature excitation of ELMs by means of pellet injection is caused by a helical perturbation described by a toroidal mode number of n = 1. In accordance with experimental observations, the pellet-triggered ELMs show reduced thermal energy losses and a narrower divertor wetted area with respect to spontaneous ELMs. The peak divertor energy fluence is seen to decrease when ELMs are triggered by pellets injected earlier during the pedestal build-up.

Bootstrapping factor models with cross sectional dependence

We consider bootstrap methods for factor-augmented regressions with cross sectional dependence among idiosyncratic errors. This is important to capture the bias of the OLS estimator derived recently by Gonçalves and Perron (2014). We first show that a common approach of resampling cross sectional vectors over time is invalid in this context because it induces a zero bias. We then propose the cross-sectional dependent (CSD) bootstrap where bootstrap samples are obtained by taking a random vector and multiplying it by the square root of a consistent estimator of the covariance matrix of the idiosyncratic errors. We show that if the covariance matrix estimator is consistent in the spectral norm, then the CSD bootstrap is consistent, and we verify this condition for the thresholding estimator of Bickel and Levina (2008). Finally, we apply our new bootstrap procedure to forecasting inflation using convenience yields as recently explored by Gospodinov and Ng (2013).

Standard error estimation by an automated blocking method

The sample mean $\overline{X}$ is probably the most popular estimator of the expected value in all sciences and $\mathrm{var}(\overline{X})$ measures the error (standard- and mean-square-errors). Here, an alternative approach to estimation of $\mathrm{var}(\overline{X})$ for time series data is presented. The method has an accuracy similar to dependent bootstrapping, but scales in $O(n)$ time, and applies to stationary time series, including stationary Markov chains. The computational complexity is bounded by $12n$ floating point operations, but this can be reduced to $n+O(1)$ in large computations. Convergence in relative error squared is faster than ${n}^{\ensuremath{-}1/2}$ and the method is insensitive to the probability distribution of the observations. It is proven that a small part of the correlation structure is relevant to the convergence rate of the method. From this, proof of the Blocking method [Flyvbjerg and Petersen, J. Chem. Phys. 91, 461 (1989)] follows as a corollary. The result is also used to propose a hypothesis test surveying the relevant part of the correlation structure. It yields a fully automatic method which is sufficiently robust to operate without supervision. An algorithm and sample code showing the implementation is available for Python, C++, and R [www.github.com/computative/block]. Method validation using autoregressive AR(1) and AR(2) processes and physics applications is included. Method self-evaluation is provided by bias and mean-square-error statistics. The method is easily adapted to multithread applications and data larger than computing cluster memory, such as ultralong time series or data streams. This way, the paper provides a stringent and modern treatment of the Blocking method using rigorous linear algebra, multivariate probability theory, real analysis, and Fisherian statistical inference.

Bootstrap Model Averaging Unit Root Inference

Classical unit root tests are known to suffer from potentially crippling size distortions, and a range of procedures have been proposed to attenuate this problem, including the use of bootstrap procedures. It is also known that the estimating equation’s functional form can affect the outcome of the test, and various model selection procedures have been proposed to overcome this limitation. In this paper, we adopt a model averaging procedure to deal with model uncertainty at the testing stage. In addition, we leverage an automatic model-free dependent bootstrap procedure where the null is imposed by simple differencing (the block length is automatically determined using recent developments for bootstrapping dependent processes). Monte Carlo simulations indicate that this approach exhibits the lowest size distortions among its peers in settings that confound existing approaches, while it has superior power relative to those peers whose size distortions do not preclude their general use. The proposed approach is fully automatic, and there are no nuisance parameters that have to be set by the user, which ought to appeal to practitioners.

Complete q-th moment convergence and its application in the dependent bootstrap

Let be an array of rowwise pairwise negatively quadrant dependent random variables with mean zero and as , where is a sequence of positive integers. The complete qth moment convergence results for and are established which are much stronger than the complete convergence results obtained in the literature. As applications of our main results, we obtain some complete moment theorems for the dependent bootstrap samples arising from a sequence of independent and identically distributed random variables.

Dependent bootstrapping for value-at-risk and expected shortfall

Estimation in extreme financial risk is often faced with challenges such as the need for adequate distributional assumptions, considerations for data dependencies, and the lack of tail information. Bootstrapping provides an alternative that overcomes some of these challenges. It does not assume a distributional form and asymptotically replicates the empirical density for resampled data. Moreover, advanced bootstrapping can cater for dependencies and stationarity in the data. In this paper, we evaluate the use of dependent bootstrapping, both for the original financial time series and for its GARCH innovations (under the Gaussian and Student t noise assumptions), in forecasting value-at-risk and expected shortfall. We also assess the effect of using different window sizes for these procedures. The two datasets used are daily returns of the S&P500 from NYSE and the ALSI from JSE.

Complete convergence theorems for normed row sums from an array of rowwise pairwise negative quadrant dependent random variables with application to the dependent bootstrap

Let {X n,j , 1 ⩽ j ⩽ m(n), n ⩾ 1} be an array of rowwise pairwise negative quadrant dependent mean 0 random variables and let 0 < b n → ∞. Conditions are given for $$\sum\nolimits_{j = 1}^{m(n)} {{X_{n,j}}/{b_n} \to 0} $$ completely and for $${\max _{1 \leqslant k \leqslant m(n)}}|\sum\nolimits_{j = 1}^k {{X_{n,j}}|/{b_n} \to 0} $$ completely. As an application of these results, we obtain a complete convergence theorem for the row sums $$\sum\nolimits_{j = 1}^{m(n)} {X_{n,j}^*} $$ of the dependent bootstrap samples $$\{ \{ X_{n,j}^*,1 \leqslant j \leqslant m(n)\} ,n \geqslant 1\} $$ arising from a sequence of i.i.d. random variables {X n , n ⩾ 1}.

Synergy Micro-Electronics Technology with a High Linearity BiCMOS Sample-and-Hold Circuit

A high accuracy BiCMOS sample and hold (S/H) circuit employed in the front end of a12bit 10 MS/s Pipeline ADC is presented. To reduce the nonlinearity error cause by the sampling switch, a signal dependent clock bootstrapping system is introduced. It is implemented using 0.6 um BiCMOS process. An 88.77 dB spurious-free dynamic range (SFDR), and a -105.20 dB total harmonic distortion (THD) are obtained.

Fixed-smoothing asymptotics for time series

In this paper, we derive higher order Edgeworth expansions for the finite sample distributions of the subsampling-based t-statistic and the Wald statistic in the Gaussian location model under the so-called fixed-smoothing paradigm. In particular, we show that the error of asymptotic approximation is at the order of the reciprocal of the sample size and obtain explicit forms for the leading error terms in the expansions. The results are used to justify the second-order correctness of a new bootstrap method, the Gaussian dependent bootstrap, in the context of Gaussian location model.

Bootstrapping Daily Returns

A daily log-return can be regarded as a test statistic - specifically the (unscaled) sample mean of a sequence of intraday random variables. We discuss sufficient conditions for a dependent bootstrap to consistently and non-parametrically estimate the entire distribution of this “test statistic”, up to, but not including, the location parameter. The method proposed is robust to market microstructure effects. There are many possible applications. In this paper, two are considered: 1) Estimating daily variance, and 2) Estimating Value-at-Risk (VaR). Of particular import: the VaR estimator is combined with the framework described in Patton & Li (2013) to produce forecast evaluation tests that are significantly more powerful than current popular techniques.

On the negative association property for the dependent bootstrap random variables

In this paper, we prove that the dependent bootstrap random variables possess the negative association property.

Correction to “Automatic Block-Length Selection for the Dependent Bootstrap” by D. Politis and H. White

A correction on the optimal block size algorithms of Politis and White (2004) is given following a correction of Lahiri's (Lahiri 1999) theoretical results by Nordman (2008).

Validity of dependent bootstrapping in finite populations

The traditional bootstrap resamples with replacement from the original sample observations to form arrays of row-wise independent and identically distributed bootstrap random variables. There are situations, for example, when sampling from finite populations, where resampling without replacement provides a more realistic bootstrap procedure and produces dependent bootstrap random variables. The desired properties of consistency and asymptotic validity are shown to hold for certain nonparametric dependent bootstrap estimators. In addition, it is shown that the smaller variation in dependent bootstrap estimators can be used to increase precision in some of the estimators.

Confidence estimation using dependent circular block bootstrapping: application to the statistical analysis of PIV measurements

In this paper the authors show an original application to time-resolved PIV of an existing method for confidence level and error determination, called dependent bootstrapping. Due to the high sampling frequencies the measured velocity samples are no longer uncorrelated making classical statistical procedures not applicable. Examples show that the various ways in calculating the number of independent samples based on the autocorrelation function question the reliability of the rarely, if ever, mentioned confidence levels in literature. Instead, the dependent bootstrapping technique reports consistent results of confidence estimates for both correlated and uncorrelated PIV velocity samples making this technique robust and general for further applications. The step-by-step description of the dependent circular block bootstrap implementation is given. The practical application and viability of the method are illustrated by two experimental cases.

Almost Sure lim sup Behavior of Dependent Bootstrap Means

In this article, the upper bound for the exact convergence rate (i.e., the law of the logarithm type result) is obtained for dependent bootstrap means.

Consistency of Dependent Bootstrap Estimators

Consistency of dependent bootstrap estimators is established in this paper. The major tool will be the concept of negative dependence which results from resampling without replacement from an enriched set of sample observations. In particular, strong limit theorems for arrays of rowwise negatively dependent random variables are obtained and used to obtain the consistency for the dependent bootstrap mean, variance and empirical distribution function.

Consistency equation for the Pomeron

A $t$-dependent bootstrap equation for the Pomeron is found by following a procedure for counting states very similar to that used in the Reggeon bootstrap equation. The relation ${\ensuremath{\alpha}}_{P}(0)\ensuremath{-}{\ensuremath{\alpha}}_{R}(0)={\ensuremath{\alpha}}_{R}(0)\ensuremath{-}{\ensuremath{\alpha}}_{c}(0)$, where ${\ensuremath{\alpha}}_{P}$, ${\ensuremath{\alpha}}_{R}$, and ${\ensuremath{\alpha}}_{c}$ are the Pomeron, Reggeon, and Reggeon-Reggeon cut trajectories, respectively, shown to hold approximately; therefore ${\ensuremath{\alpha}}_{P}(0)\ensuremath{\simeq}1$. Assuming short-range correlations among shadow particles, we find that the above relation holds on very general grounds because the cylinder has twice as much shadow as the Reggeon. Therefore, the cluster-cluster overlap configuration is argued to be dominant in the cylinder discontinuity in contrast to the usual way of constructing the Pomeron which includes only a cluster-gap or gap-gap overlap.