Finite Sample Size Research Articles

Dirac fermion spectrum of the fractional quantum Hall states

Applying a unified approach, we study the integer quantum Hall effect (IQHE) and fractional quantum Hall effect (FQHE) in the Hofstadter model with short range interactions between fermions. An effective field, that takes into account the interaction between fermions, is determined by both amplitude and phase. Its amplitude is proportional to the interaction strength, the phase corresponds to the minimum energy. In fact, the problem is reduced to the Harper equation with two different scales: the first is a magnetic scale with the cell size corresponding to a unit quantum magnetic flux, the second scale determines the inhomogeneity of the effective field, forms the steady fine structure of the Hofstadter spectrum and leads to the realization of fractional quantum Hall states. In a sample of finite size with open boundary conditions, the fine structure of the Hofstadter spectrum consists of the Dirac branches of the fermion excitations and includes the fine structure of the edge chiral modes. The Chern numbers of the topological Hofstadter bands are conserved during the formation of their fine structure. The edge modes are formed into the Hofstadter bands. They connect the nearest-neighbor subbands and determine the conductance for the fractional filling.

Finitely Repeated Adversarial Quantum Hypothesis Testing

Quantum technologies have been considered as a method to enhance cyber-security due to the unique properties of quantum systems. However, quantum systems themselves could also suffer from strategic attacks. In this paper, we formulate the relationship between the strategic attacker and a passive quantum detector into a Stackelberg game. By deriving the optimal detection rule at equilibrium, we study the detection performance of the detector under the setting of finite sample size of corrupted observations. Under mild assumptions, we show that the miss. We illustrate the general decaying results of the miss rate numerically, depicting that the passive detector manages to achieve a miss rate and a false alarm rate both exponentially decaying to zero given infinitely many quantum states, although at a much slower rate than a quantum non-adversarial counterpart. Finally, we adopt our formulations upon a case study of detection with quantum radars.

On the Relation Between Discrete and Continuous-Time Refined Instrumental Variable Methods

The Refined Instrumental Variable method for discrete-time systems (RIV) and its variant for continuous-time systems (RIVC) are popular methods for the identification of linear systems in open-loop. The continuous-time equivalent of the transfer function estimate given by the RIV method is commonly used as an initialization point for the RIVC estimator. In this paper, we prove that these estimators share the same converging points for finite sample size when the continuous-time model has relative degree zero or one. This relation does not hold for higher relative degrees. Then, we propose a modification of the RIV method whose continuous-time equivalent is equal to the RIVC estimator for any non-negative relative degree. The implications of the theoretical results are illustrated via a simulation example.

Sample Complexity of the Sign-Perturbed Sums Identification Method: Scalar Case*

Sign-Perturbed Sum (SPS) is a powerful finite-sample system identification algorithm which can construct confidence regions for the true data generating system with exact coverage probabilities, for any finite sample size. SPS was developed in a series of papers and it has a wide range of applications, from general linear systems, even in a closed-loop setup, to nonlinear and nonparametric approaches. Although several theoretical properties of SPS were proven in the literature, the sample complexity of the method was not analysed so far. This paper aims to fill this gap and provides the first results on the sample complexity of SPS. Here, we focus on scalar linear regression problems, that is we study the behaviour of SPS confidence intervals. We provide high probability upper bounds, under three different sets of assumptions, showing that the sizes of SPS confidence intervals shrink at a geometric rate around the true parameter, if the observation noises are subgaussian. We also show that similar bounds hold for the previously proposed outer approximation of the confidence region. Finally, we present simulation experiments comparing the theoretical and the empirical convergence rates.

Goodness-of-Fit Test for the Bivariate Hermite Distribution

This paper studies the goodness of fit test for the bivariate Hermite distribution. Specifically, we propose and study a Cramér–von Mises-type test based on the empirical probability generation function. The bootstrap can be used to consistently estimate the null distribution of the test statistics. A simulation study investigates the goodness of the bootstrap approach for finite sample sizes.

Analysis of convolutional neural network image classifiers in a hierarchical max-pooling model with additional local pooling

In practical applications of image classification, methods based on convolutional neural networks are used as standard nowadays. However, the success of these methods is not explained sufficiently from a theoretical point of view. For this, a statistical model for image classification, namely a hierarchical max-pooling model with additional local pooling, is introduced. Here the additional local pooling enables the hierarchical model to combine parts of the image which have a variable relative distance towards each other. Various convolutional neural network image classifiers are introduced and analyzed in view of their rate of convergence of the misclassification risk of the estimates towards the optimal misclassification risk. This analysis provides a theoretical explanation why general convolutional neural network architectures that include some kind of local pooling layers are useful in some image classification situations and gives theoretical hints for choosing the right network architecture. Furthermore, the finite sample size performance of these network architectures is illustrated by applying them to simulated and real data.

Another unit Burr XII quantile regression model based on the different reparameterization applied to dropout in Brazilian undergraduate courses.

In many practical situations, there is an interest in modeling bounded random variables in the interval (0, 1), such as rates, proportions, and indexes. It is important to provide new continuous models to deal with the uncertainty involved by variables of this type. This paper proposes a new quantile regression model based on an alternative parameterization of the unit Burr XII (UBXII) distribution. For the UBXII distribution and its associated regression, we obtain score functions and observed information matrices. We use the maximum likelihood method to estimate the parameters of the regression model, and conduct a Monte Carlo study to evaluate the performance of its estimates in samples of finite size. Furthermore, we present general diagnostic analysis and model selection techniques for the regression model. We empirically show its importance and flexibility through an application to an actual data set, in which the dropout proportion of Brazilian undergraduate animal sciences courses is analyzed. We use a statistical learning method for comparing the proposed model with the beta, Kumaraswamy, and unit-Weibull regressions. The results show that the UBXII regression provides the best fit and the most accurate predictions. Therefore, it is a valuable alternative and competitive to the well-known regressions for modeling double-bounded variables in the unit interval.

Semiparametric inference of treatment effects on restricted mean survival time in two sample problems from length-biased samples

Estimating treatment effects on restricted mean survival time(RMST) is often of direct interest in clinical studies. We propose two approaches based on estimating equation to estimate treatment effects on RMST under the setting of length-biased sampling. The asymptotic properties for the proposed estimators are derived. Simulation studies are performed to evaluate the performance of the methods in finite sample sizes. In addition, we apply our procedures to Oscar Award data.

Using Behavioural Economics to Analyse and Enhance Contraception Usage Decisions

Existent literature helped narrow down variables influencing modern contraceptive adoption (usage), a behaviour carrying enormous positive externality. Using finite population sample size formula and probability proportional to size method of sample selection, primary data was collected from participants using inclusion and exclusion criterions. Binary logistic regression model was used to predict probability of occurrence of dependent variable ‘usage of modern method of contraception’ being treated at a dichotomous outcome level. Predictor variables after confirming association by cross tabulation were introduced stepwise to build model subject to elimination of those variables adding insignificantly to the overall predictability of the model. Variables such as gender, education level, spousal influence, extended family influence, financial well-being and contraceptive information were found to significantly predict the probability of occurrence of the dependent variable. Except for financial well-being with three sub-categories, other independent variables were treated at dichotomous level. Income level was found to be an important predictor although found statistically insignificant. Non-contributory factors such as age, occupation and years of marriage were dropped. Post-model construction, borrowing ‘nudges’ from behavioural economics (BE) domain, strategies to nurture the significant context specific influencing variables, were articulated. BE was particularly preferred for its openness to the paradigm of non-rational behavioural choices.

Robust Adaptive Beamforming via Modified Variable Loading with Subsampling Preprocessing

For robust adaptive beamforming (RAB), the variable loading (VL) technique can provide a better trade-off between robustness and adaptivity than diagonal loading (DL). Despite its importance, few research efforts have explored the loading factor for VL to ensure robustness in various environments. Moreover, the performance of VL is restricted by the sample covariance matrix in snapshot deficiency situations. This paper proposes a modified variable loading (VL) method for robust adaptive beamforming, considering imprecise steering vector effects and finite sample size impairments. First, a novel subsampling method is used to construct the calibrated covariance matrix to improve the robustness of the VL in sample-starving scenarios. Then, a parameter-free method for the VL factor is proposed to further enhance the insensitivity to the steering vector mismatches of the antenna array. Simulation results verify the effectiveness and robustness of the proposed method as compared to the traditional VL and other widely used robust techniques.

A generalized knockoff procedure for FDR control in structural change detection

Controlling false discovery rate (FDR) is crucial for variable selection, multiple testing, among other signal detection problems. In literature, there is certainly no shortage of FDR control strategies when selecting individual features, but the relevant works for structural change detection, such as profile analysis for piecewise constant coefficients and integration analysis with multiple data sources, are limited. In this paper, we propose a generalized knockoff procedure (GKnockoff) for FDR control under such problem settings. We prove that the GKnockoff possesses pairwise exchangeability, and is capable of controlling the exact FDR under finite sample sizes. We further explore GKnockoff under high dimensionality, by first introducing a new screening method to filter the high-dimensional potential structural changes. We adopt a data splitting technique to first reduce the dimensionality via screening and then conduct GKnockoff on the refined selection set. Furthermore, the powers of proposed methods are systematically studied. Numerical comparisons with other methods show the superior performance of GKnockoff, in terms of both FDR control and power. We also implement the proposed methods to analyze a macroeconomic dataset for detecting changes of driven effects of economic development on the secondary industry.

High-dimensional asymptotic expansion of the null distribution for [formula omitted] norm based MANOVA testing statistic under general distribution

This paper is concerned with the MANOVA test when the dimensionality of the observed vector may exceed the sample size. Zhou et al. (2017) proposed the L2 norm based testing statistic T and gave the limiting null distribution under the high-dimensional asymptotic framework that the minimum of sample sizes and the dimensionality go to infinity. In this paper we give a one-term asymptotic expansion of the null distribution for T under the asymptotic framework. We propose a correction of the critical point for Zhou et al. (2017)’s test based on this expansion. The finite sample size and the dimensionality performance for attained significance level is evaluated in a simulation study and the results are compared to those of Zhou et al. (2017)’s test.

Discussion of ‘Multi-scale Fisher’s independence test for multivariate dependence’

We discuss how MultiFIT, the Multiscale Fisher's Independence Test for Multivariate Dependence proposed by Gorsky and Ma (2022), compares to existing linear-time kernel tests based on the Hilbert-Schmidt independence criterion (HSIC). We highlight the fact that the levels of the kernel tests at any finite sample size can be controlled exactly, as it is the case with the level of MultiFIT. In our experiments, we observe some of the performance limitations of MultiFIT in terms of test power.

Accurate interval estimation for the risk difference in an incomplete correlated 2 × 2 table: Calf immunity analysis.

Interval estimation with accurate coverage for risk difference (RD) in a correlated 2 × 2 table with structural zero is a fundamental and important problem in biostatistics. The score test-based and Bayesian tail-based confidence intervals (CIs) have good coverage performance among the existing methods. However, as approximation approaches, they have coverage probabilities lower than the nominal confidence level for finite and moderate sample sizes. In this paper, we propose three new CIs for RD based on the fiducial, inferential model (IM) and modified IM (MIM) methods. The IM interval is proven to be valid. Moreover, simulation studies show that the CIs of fiducial and MIM methods can guarantee the preset coverage rate even for small sample sizes. More importantly, in terms of coverage probability and expected length, the MIM interval outperforms other intervals. Finally, a real example illustrates the application of the proposed methods.

On modified Anderson-Darling test statistics with asymptotic properties

This study proposes two modified Anderson-Darling test statistics that emphasize the upper or lower tails of distribution. Under the finite sample sizes, various models are used to estimate the upper tail percentage points of the proposed test statistic. The limiting distribution of the proposed test statistics is derived with a sum of infinite series. The polynomial approximation, the method of Durbin and Knott and the normal and non normal saddlepoint approximations are used to approximate the limiting distribution of the suggested test statistics. Simulations are used to investigate the accuracy of various approximations. The asymptotic power of the proposed statistics is obtained to compare with various statistics. The proposed test statistics are illustrated by the analysis of real data.

High-dimensional asymptotic expansion of the null distribution for Schott’s test statistic for complete independence of normal random variables

This article is concerned with the testing complete independence for the elements of observed vector. Schott proposed the testing statistic T and gave limiting null distribution under the high-dimensional asymptotic framework that the sample size n and the dimensionality p go to infinity together while p/n converges to a positive constant. In this article we give a one-term asymptotic expansion of the null distribution for T as min{n,p} tends toward infinity. We derive a correction of the critical point for Schott’s test based on this expansion. The finite sample size and dimensionality performance for attained significance level is evaluated in a simulation study and the results are compared to those of Schott’s test.

Empirical Correlations between the Spectral Input Energy and Spectral Acceleration

ABSTRACT Correlating different intensity measures (IMs) enables the joint consideration of multiple IMs in various performance-based earthquake engineering applications, such as ground motion selection. The correlations between the spectral input energy and spectral accelerations have not been previously examined. This study develops a correlation model quantifying the uncertainties due to GMPE selection and a finite sample size. The results show a strong correlation between the spectral amplitudes at the same period, and the correlation decreases as the period difference increases. With the proposed model, ground motion characteristics reflected by the input energy can be explicitly considered in performance-based earthquake engineering applications.

Forecasting of Daily Outpatient Visits Based on Genetic Programming.

The forecasting of daily outpatient visits has significant practical implications in outpatient clinic operation management, not only contributing to guiding long-term resource planning and scheduling but also making tactical resolutions for short-term adjustments on special days such as holidays. We here in propose an effective genetic programming (GP)-based forecasting model to predict daily outpatient visits (OV) in a primary hospital. In the GP-based model, the holiday-based distance outlier mining algorithm was used to determine the holiday effect. In addition, solar terms were applied as the smallest unit to more accurately determine the impact of a change in the climate on the outpatient volume. A segmental learning strategy also was used to predict the daily outpatient volume for the time series data. The GP-based prediction could more effectively extract depth information from a finite training sample size and achieve a better performance for predicting daily outpatient visits, with lower root mean square error (RMSE) and higher coefficient of determination (R2) values, than the seasonal autoregressive integrated moving average (SARIMA) model in the time range of holidays and the holiday effect. GP-based model can achieve better prediction performance by overcoming the shortcomings of the SARIMA model. The results can be applied to support decision-making and planning of outpatient clinic resources, to help managers implement periodic scheduling of available resources on the basis of periodic features, and to perform proactive scheduling of additional resources.

Permutation Tests at Nonparametric Rates

ABSTRACT Classical two-sample permutation tests for equality of distributions have exact size in finite samples, but they fail to control size for testing equality of parameters that summarize each distribution. This article proposes permutation tests for equality of parameters that are estimated at root-n or slower rates. Our general framework applies to both parametric and nonparametric models, with two samples or one sample split into two subsamples. Our tests have correct size asymptotically while preserving exact size in finite samples when distributions are equal. They have no loss in local asymptotic power compared to tests that use asymptotic critical values. We propose confidence sets with correct coverage in large samples that also have exact coverage in finite samples if distributions are equal up to a transformation. We apply our theory to four commonly-used hypothesis tests of nonparametric functions evaluated at a point. Lastly, simulations show good finite sample properties, and two empirical examples illustrate our tests in practice. Supplementary materials for this article are available online.

Potential of Hematologic Parameters in Predicting Mortality of Patients with Traumatic Brain Injury.

Traumatic brain injury (TBI) occurs frequently, and acute TBI requiring surgical treatment is closely related to patient survival. Models for predicting the prognosis of patients with TBI do not consider various factors of patient status; therefore, it is difficult to predict the prognosis more accurately. In this study, we created a model that can predict the survival of patients with TBI by adding hematologic parameters along with existing non-hematologic parameters. The best-fitting model was created using the Akaike information criterion (AIC), and hematologic factors including preoperative hematocrit, preoperative C-reactive protein (CRP), postoperative white blood cell (WBC) count, and postoperative hemoglobin were selected to predict the prognosis. Among several prediction models, the model that included age, Glasgow Coma Scale, Injury Severity Score, preoperative hematocrit, preoperative CRP, postoperative WBC count, postoperative hemoglobin, and postoperative CRP showed the highest area under the curve and the lowest corrected AIC for a finite sample size. Our study showed a new prediction model for mortality in patients with TBI using non-hematologic and hematologic parameters. This prediction model could be useful for the management of patients with TBI.