Finite Sample Size Research Articles

Application of an improved version of McDiarmid inequality in finite-key-length decoy-state quantum key distribution

In practical decoy-state quantum key distribution, the raw key length is finite. Thus, deviation of the estimated single photon yield and single photon error rate from their respective true values due to finite sample size can seriously lower the provably secure key rate R. Current method to obtain a lower bound of R follows an indirect path by first bounding the yields and error rates both conditioned on the type of decoy used. These bounds are then used to deduce the single photon yield and error rate, which in turn are used to calculate a lower bound of the key rate R. Here we report an improved version of McDiarmid inequality in statistics and show how use it to directly compute a lower bound of R via the so-called centering sequence. A novelty in this work is the optimization of the bound through the freedom of choosing possible centering sequences. The provably secure key rate of realistic 100 km long quantum channel obtained by our method is at least twice that of the state-of-the-art procedure when the raw key length ℓraw is ≈105–106. In fact, our method can improve the key rate significantly over a wide range of raw key length from about 105 to 1011. More importantly, it is achieved by pure theoretical analysis without altering the experimental setup or the post-processing method. In a boarder context, this work introduces powerful concentration inequality techniques in statistics to tackle physics problem beyond straightforward statistical data analysis especially when the data are correlated so that tools like the central limit theorem are not applicable.

Quantum tomography by regularized linear regressions

In this paper, we study extended linear regression approaches for quantum state tomography based on regularization techniques. For unknown quantum states represented by density matrices, performing measurements under certain basis yields random outcomes, from which a classical linear regression model can be established. First of all, for complete or over-complete measurement bases, we show that the empirical data can be utilized for the construction of a weighted least squares estimate (LSE) for quantum tomography. Taking into consideration the trace-one condition, a constrained weighted LSE can be explicitly computed, being the optimal unbiased estimation among all linear estimators. Next, for general measurement bases, we show that ℓ2-regularization with proper regularization gain provides even a lower mean-square error under a cost in bias. The optimal regularization parameter is defined in terms of a risk characterization for any finite sample size and a resulting implementable estimator is proposed. Finally, a concise and unified formula is established for the regularization parameter with complete measurement basis under an equivalent regression model, which proves that the proposed implementable tuning estimator is asymptotically optimal as the number of copies grows to infinity. Additionally, several numerical examples are provided to validate the established results.

Modeling Volatility in the Stock Market for Accuracy in Forecasting

In this paper, the best GARCH type model was selected and compared with the machine learning models, such as Extreme Learning Machine (ELM) and Multilayer Perceptron Neural Network (MLP-NN) models in modeling and forecasting monthly return of the financial market data. The objective of the study was to compare the best model in forecasting New York and Shanghai Stock Composite indices, for the period 01.01.1996 to 01.09.2019. The GJR-GARCH model outperformed other GARCH type models based on the Schwarz Bayesian Information Criterion (SSBIC). The Monte Carlo simulation carried at 1000, 2000, 3000, 4000 and 5000 finite sample (window) sizes to test the consistency of the GJR-GARCH model parameters has shown perfect results between true and the simulated coefficients. Finally, the GJR-GARCH model was compared with the MLP-NN and ELM machine learning models. The monthly return forecasting of two years (24 months) was done starting from period 01.09.2019 to 01.09.2021. The study found the MLP-NN model as the best in the modeling and forecasting monthly returns of the two composite stock indices for the two years by considering the Root Mean Square Error (RMSE).The study recommends that further research should focus on the formulation of the hybrid model that combines machine learning and the GJR-GARCH models in forecasting stock market volatility.

Frequency domain estimation of cointegrating vectors with mixed frequency and mixed sample data

This paper proposes a simple method for exploiting the information contained in mixed frequency and mixed sample data in the estimation of cointegrating vectors. The asymptotic properties of easy-to-compute spectral regression estimators of the cointegrating vectors are derived and these estimators are shown to belong to the class of optimal cointegration estimators. Furthermore, Wald statistics based on these estimators have asymptotic chi-square distributions which enable inferences to be made straightforwardly. Simulation experiments suggest that the spectral regression estimators considered perform well in finite samples and are at least as good as time domain fully modified estimators. The finite sample size and power properties of the spectral regression-based Wald statistic are also found to be good.

Bootstrap Model-Based Constrained Optimization Tests of Indirect Effects.

In mediation analysis, conditions necessary for commonly recommended tests, including the confidence interval (CI)-based tests, to produce an accurate Type I error, do not generally hold for finite sample sizes and non-normally distributed model residuals. This is typically the case because of the complexity of testing a null hypothesis about indirect effects. To remedy these issues, we propose two extensions of the recently developed asymptotic Model-based Constrained Optimization (MBCO) likelihood ratio test (LRT), a promising new model comparison method for testing a general function of indirect effects. The proposed tests, semi-parametric and parametric bootstrap MBCO LRT are shown to yield a more accurate Type I error rate in smaller sample sizes and under various degrees of non-normality of the model residuals compared to the asymptotic MBCO LRT and the CI-based methods. We provide R script in the Supplemental Materials to perform all three MBCO LRTs.

The gradient test and its finite sample size properties in a conditional maximum likelihood and psychometric modeling context

In asymptotic theory, the gradient test proposed by Terrell is a recent likelihood-based hypothesis testing approach which can be considered as an alternative to the well-established trinity of likelihood ratio, Rao score, and Wald tests. The gradient test has not yet entered into the mainstream of applied statistics. This is particularly true for the psychometric context. This research discusses a novel application of the gradient test within the conditional maximum likelihood and the Rasch modeling framework. It also investigates some of its finite sample size properties and compares it with the classical trinity of chi square tests by conducting an extensive Monte Carlo study. The results confirm that the gradient test has its pros and cons.

Adaptive, Rate-Optimal Testing in Instrumental Variables Models

This paper proposes simple, data-driven, optimal rate-adaptive inferences on a structural function in semi-nonparametric conditional moment restrictions. We consider two types of hypothesis tests based on leave-one-out sieve estimators. A structure-space test (ST) uses a quadratic distance between the structural functions of endogenous variables; while an image-space test (IT) uses a quadratic distance of the conditional moment from zero. For both tests, we analyze their respective classes of nonparametric alternative models that are separated from the null hypothesis by the minimax rate of testing. That is, the sum of the type I and the type II errors of the test, uniformly over the class of nonparametric alternative models, cannot be improved by any other test. Our new minimax rate of ST differs from the known minimax rate of estimation in nonparametric instrumental variables (NPIV) models. We propose computationally simple and novel exponential scan data-driven choices of sieve regularization parameters and adjusted chi-squared critical values. The resulting tests attain the minimax rate of testing, and hence optimally adapt to the unknown smoothness of functions and are robust to the unknown degree of ill-posedness (endogeneity). Data-driven confidence sets are easily obtained by inverting the adaptive ST. Monte Carlo studies demonstrate that our adaptive ST has good size and power properties in finite samples for testing monotonicity or equality restrictions in NPIV models. Empirical applications to nonparametric multi-product demands with endogenous prices are presented.

Tail Granger Causalities and Where to Find Them: Extreme Risk Spillovers vs Spurious Linkages

Identifying risk spillovers in financial markets is of great importance for assessing systemic risk and portfolio management. Granger causality in tail (or in risk) tests whether past extreme events of a time series help predicting future extreme events of another time series. The topology and connectedness of networks built with Granger causality in tail can be used to measure systemic risk and to identify risk transmitters. Here we introduce a novel test of Granger causality in tail which adopts the likelihood ratio statistic and is based on the multivariate generalization of a discrete autoregressive process for binary time series describing the sequence of extreme events of the underlying price dynamics. The proposed test has very good size and power in finite samples, especially for large sample size, allows inferring the correct time scale at which the causal interaction takes place, and it is flexible enough for multivariate extension when more than two time series are considered in order to decrease false detections as spurious effect of neglected variables. An extensive simulation study shows the performances of the proposed method with a large variety of data generating processes and it introduces also the comparison with the test of Granger causality in tail by [Hong et al., 2009]. We report both advantages and drawbacks of the different approaches, pointing out some crucial aspects related to the false detections of Granger causality for tail events. An empirical application to high frequency data of a portfolio of US stocks highlights the merits of our novel approach.

Thickness measurement of metallic plates with finite planar dimension using eddy current method

Until now, the Dodd–Deeds model has been used to evaluate the inductance change in an air-cored sensor in the presence of an infinitely large conducting plate. In practice, it can be applied to the sample relatively larger than the radius of the sensor coil (normally >3–5 times larger), which can be regarded as a plate sample with infinite planar size. However, in various practical applications, the size of the tested sample may not satisfy this condition. In this article, a modified analytical solution based on the Dodd–Deeds model is proposed, which has introduced a new initial integration point instead of 0 for the analytical inductance of the finite-size metallic plate. Theoretical derivation has been presented for the derivation of the initial integration point. Moreover, an inversely proportional relation can be observed between the initial integration point and the radius of the test sample. The simulation and experimental measurements for the thickness of several finite-size metallic samples have been carried out for the verification of the proposed method.

A New Approach for Estimating Rock Discontinuity Trace Intensity Based on Rectangular Sampling Windows

Trace intensity is defined as mean total trace length of discontinuities per unit area, which is an important geometric parameter to describe fracture networks. The probability of each trace appearing in the sampling surface is different since discontinuity orientation has a scatter and is probabilistically distributed, so this factor should be taken into account in trace intensity estimation. This paper presents an approach to estimate the two‐dimensional trace intensity by considering unequal appearing probability for discontinuities sampled by rectangular windows. The estimation method requires the number of discontinuities intersecting the window, the appearing probability of discontinuities with both ends observed, one end observed, and both ends censored, and the mean trace length of discontinuities intersecting the window. The new estimator is validated by using discontinuity data from an outcrop in Wenchuan area in China. Similarly, circular windows are used along with Mauldon’s equation to calculate trace intensity using discontinuity trace data of the same outcrop as a contrast. Results indicate that the proposed new method based on rectangular windows shows close accuracy and less variability than that of the method based on circular windows due to the influence of finite sample size and the variability of location of the window and has advantage in application to sampling surfaces longer in one direction than in the other such as tunnel cross sections and curved sampling surfaces such as outcrops that show some curvature.

Spectral estimation for non-linear long range dependent discrete time trawl processes

Discrete time trawl processes constitute a large class of time series parameterized by a trawl sequence (a j) j∈N and defined though a sequence of independent and identically distributed (i.i.d.) copies of a continuous time process (γ(t)) t∈R called the seed process. They provide a general framework for modeling linear or non-linear long range dependent time series. We investigate the spectral estimation, either pointwise or broadband, of long range dependent discrete-time trawl processes. The difficulty arising from the variety of seed processes and of trawl sequences is twofold. First, the spectral density may take different forms, often including smooth additive correction terms. Second, trawl processes with similar spectral densities may exhibit very different statistical behaviors. We prove the consistency of our estimators under very general conditions and we show that a wide class of trawl processes satisfy them. This is done in particular by introducing a weighted weak dependence index that can be of independent interest. The broadband spectral estimator includes an estimator of the long memory parameter. We complete this work with numerical experiments to evaluate the finite sample size performance of this estimator for various integer valued discrete time trawl processes.

A Nonparametric Measure of Heteroskedasticity

We introduce a nonparametric measure to quantify the degree of heteroskedasticity at a fixed quantile of the conditional distribution of a random variable. Our measure of heteroskedasticity is based on nonparametric quantile regressions and is expressed in terms of unrestricted and restricted expectations of quantile loss functions. It can be consistently estimated by replacing the unknown expectations by their nonparametric estimates. We derive a Bahadur-type representation for the nonparametric estimator of the measure. We provide the asymptotic distribution of this estimator, which one can use to build tests for the statistical significance of the measure. Thereafter, we establish the validity of a fixed regressor bootstrap that one can use in finite-sample settings to perform tests. A Monte Carlo simulation study reveals that the bootstrap-based test has a good finite sample size and power for a variety of data generating processes and different sample sizes. Finally, two empirical applications are provided to illustrate the importance of the proposed measure.

A spatial concordance correlation coefficient with an application to image analysis

In this work we define a spatial concordance coefficient for second-order stationary processes. This problem has been widely addressed in a non-spatial context, but here we consider a coefficient that for a fixed spatial lag allows one to compare two spatial sequences along a 45°line. The proposed coefficient was explored for the bivariate Matérn and Wendland covariance functions. The asymptotic normality of a sample version of the spatial concordance coefficient for an increasing domain sampling framework was established for the Wendland covariance function. To work with large digital images, we developed a local approach for estimating the concordance that uses local spatial models on non-overlapping windows. Monte Carlo simulations were used to gain additional insights into the asymptotic properties for finite sample sizes. As an illustrative example, we applied this methodology to two similar images of a deciduous forest canopy. The images were recorded with different cameras but similar fields-of-view and within minutes of each other. Our analysis showed that the local approach helped to explain a percentage of the non-spatial concordance and provided additional information about its decay as a function of the spatial lag.

Model-free two-sample test for network-valued data

In the framework of Object Oriented Data Analysis, a permutation approach to the two-sample testing problem for network-valued data is proposed. In detail, the present framework proceeds in four steps: (i) matrix representation of the networks, (ii) computation of the matrix of pairwise (inter-point) distances, (iii) computation of test statistics based on inter-point distances and (iv) embedding of the test statistics within a permutation test. The proposed testing procedures are proven to be exact for every finite sample size and consistent. Two new test statistics based on inter-point distances (i.e., IP-Student and IP-Fisher) are defined and a method to combine them to get a further inferential tool (i.e., IP-StudentFisher) is introduced. Simulated data shows that tests with our statistic exhibit a statistical power that is either the best or second-best but very close to the best on a variety of possible alternatives hypotheses and other statistics. A second simulation study that aims at better understanding which features are captured by specific combinations of matrix representations and distances is presented. Finally, a case study on mobility networks in the city of Milan is carried out. The proposed framework is fully implemented in the R package nevada (NEtwork-VAlued Data Analysis).

Coulomb Drag in Mesoscopic Hopping Insulators

As it is known, the role of Coulomb correlations in hopping transport is still not understood in detail and different works contain controversial conclusions in this concern. In what follows, we are going to consider an effect which is completely controlled by the Coulomb correlations. Namely, we mean the Coulomb drag between two-dimensional hopping insulators. Assuming that the principal role is played by the critical resistors in both the active layer and the passive one, we have analyzed the “ratchet” mechanism of the drag. The effect consists of the modulation of an activation energy in the “passive” resistor by the non-equilibrium electric dipole formed on the “active” resistor in the presence of electric field. The dipole originates due to a hop of a single electron and has a transient character. Namely, the effect of this non-equilibrium dipole on the “passive” resistor takes place at the timescale shorter than typical relaxation times for an establishment of the equilibrium in the “passive” plane. The latter fact is of principal importance. The effect vanishes in a sample with infinite size in the plane provided that the distance between the layers is large enough. For finite-size samples, both the effect magnitude and sign depend on the sample’s size. We have analyzed the dependencies in relevant limiting cases. It is important to note that the effect drastically depends on the distance between the two planes. Thus, we hope that studies of this effect can, in particular, throw light on the role of Coulomb correlations in hopping transport. We have compared the results with those of previously published papers.

Optimal design of overall yield-based variable repetitive sampling plans for processes with multiple characteristics

In this study, we developed two repetitive types of sampling plans for processes with multiple quality characteristics based on the overall yield index SpkT. These plans can be implemented for mutually independent and normally distributed characteristics. The plans are optimally designed based on the asymptotic sampling distribution of SpkT using an efficient nonlinear optimization algorithm. During the solution of optimization problems, the average sample number required for inspection and the contract requirements are treated as an objective function and constraints, respectively. The optimal parameters were determined for use in industrial environments with various combinations of requirements in tables. A simulation study was also conducted to show that the tabulated parameters based on the results obtained by large sample theory can guarantee the specified risks for finite sample sizes. Moreover, the limitations of the proposed plans with respect to the sample size were analyzed based on extensive simulations. Numerical calculations and graphical illustrations are presented to demonstrate the properties of the proposed plans. In addition, the advantages of the schemes are discussed compared with existing plans. Finally, the efficient plan is applied to two real industrial problems.

Competing effects of strain and vacancy defect on thermal conductivity of silicene: A computational study

We theoretically investigate the thermal conductivity of freestanding silicene under isotropic tensile strain in a wide range of temperature and in the presence of important scatterings. Based on the phonon Boltzmann transport equation within the relaxation time approximation and using the strain-dependent force constants obtained from the first-principle calculations, we calculate the variations of thermal conductivity of strained infinite and finite-size silicene with the single-vacancy defects and boundary scatterings and compare them with those in the case of unstrained silicene. Particularly, we are interested in exploring the competition between the two opposing effects; strain induces enhancement and vacancy defects cause reduction in thermal conductivity. We show that the presence of vacancy defects has a more remarkable and much stronger effect in strained silicene and is able to remove or shift the peak created by strain in the thermal conductivity of infinite silicene. Interestingly, we find that the thermal conductivity suppression due to the vacancy defects varies with strain. Furthermore, presented results indicate that by increasing the temperature, the thermal conductivity becomes less sensitive to the strain and the difference between infinite and finite size samples gradually disappears. Finally, our calculations show that the effect of specularity parameter of boundary scattering is more pronounced at intermediate strain values.

On the variance parameter estimator in general linear models

In the present note we consider general linear models where the covariates may be both random and non-random, and where the only restrictions on the error terms are that they are independent and have finite fourth moments. For this class of models we analyse the variance parameter estimator. In particular we obtain finite sample size bounds for the variance of the variance parameter estimator which are independent of covariate information regardless of whether the covariates are random or not. For the case with random covariates this immediately yields bounds on the unconditional variance of the variance estimator—a situation which in general is analytically intractable. The situation with random covariates is illustrated in an example where a certain vector autoregressive model which appears naturally within the area of insurance mathematics is analysed. Further, the obtained bounds are sharp in the sense that both the lower and upper bound will converge to the same asymptotic limit when scaled with the sample size. By using the derived bounds it is simple to show convergence in mean square of the variance parameter estimator for both random and non-random covariates. Moreover, the derivation of the bounds for the above general linear model is based on a lemma which applies in greater generality. This is illustrated by applying the used techniques to a class of mixed effects models.

Influence of Langmuir adsorption and viscous fingering on transport of finite size samples in porous media

We examine the transport in a homogeneous porous medium of a finite slice of a solute which adsorbs on the porous matrix following a Langmuir adsorption isotherm and can influence the dynamic viscosity of the solution. In the absence of any viscosity variation, the Langmuir adsorption induces the formation of a shock layer wave at the frontal interface and of a rarefaction wave at the rear interface of the sample. For a finite width sample, these waves interact after a given time that varies nonlinearly with the adsorption properties to give a triangle-like concentration profile in which the mixing efficiency of the solute is larger in comparison to the linear or no-adsorption cases. In the presence of a viscosity contrast such that a less viscous carrier fluid displaces the more viscous finite slice, viscous fingers are formed at the rear rarefaction interface. The fingers propagate through the finite sample to preempt the shock layer at the viscously stable front. In the reverse case i.e. when the shock layer front features viscous fingering, the fingers are unable to intrude through the rarefaction zone and the qualitative properties of the expanding rear wave are preserved. A non-monotonic dependence with respect to the Langmuir adsorption parameter $b$ is observed in the onset time of interaction between the nonlinear waves and viscous fingering. The coupled effect of viscous fingering at the rear interface and of Langmuir adsorption provides a powerful mechanism to enhance the mixing efficiency of the adsorbed solute.

The quantum Hall effect at a weak magnetic field

Using a weak limit for the hopping integral in one direction in the Hofstadter model, we show that the fermion states in the gaps of the spectrum are determined within the Kitaev chain. The proposed approach allows us to study the behavior of Chern insulators (CI) in different classes of symmetry. We consider the Hofstadter model on the square and honeycomb lattices in the case of rational and irrational magnetic fluxes ϕ, and discuss the behavior of the Hall conductance at a weak magnetic field in a sample of finite size. We show that in the semiclassical limit at the center of the fermion spectrum, the Bloch states of fermions turn into chiral Majorana fermion liquid when the magnetic scale 1ϕ is equal to the sample size N. We are talking about the dielectric–metal phase transition, which is determined by the behavior of the Landau levels in 2D fermion systems in a transverse magnetic field. When a magnetic scale, which determines the wave function of fermions, exceeds the size of the sample, a jump in the longitudinal conductance occurs. The wave function describes non-localized states of fermions, the sample becomes a conductor, the system changes from the dielectric state to the metallic one. It is shown, that at 1∕ϕ>N the quantum Hall effect and the Landau levels are not realized, which makes possibility to study the behavior of CI in irrational magnetic fluxes.