Edgeworth Expansion Research Articles

Non-uniform Bounds and Edgeworth Expansions in Self-normalized Limit Theorems

We study Edgeworth expansions in limit theorems for self-normalized sums. Non-uniform bounds for expansions in the central limit theorem are established while imposing only minimal moment conditions. Within this result, we address the case of non-integer moments leading to a reduced remainder. Furthermore, we provide non-uniform bounds for expansions in local limit theorems. The enhanced tail accuracy of our non-uniform bounds allows for deriving an Edgeworth-type expansion in the entropic central limit theorem as well as a central limit theorem in total variation distance for self-normalized sums.

Comparing Confidence Intervals for the Mean of Symmetric and Skewed Distributions

In context-aware decision analysis, mean can be an important measure, even when the distribution is skewed. Previous comparative studies showed that it is a real challenge to construct a confidence interval that performs well for highly skewed data. In this study, we propose new confidence intervals for the population mean based on Edgeworth expansion that include both skewness and kurtosis corrections. We compared existing and newly proposed confidence intervals for a range of samples from symmetric and skewed distributions of varying levels of kurtosis. Using Monte Carlo simulations, we evaluated the performance of these intervals based on the coverage probability, mean length, and standard deviation of the length. The proposed bootstrap Edgeworth-based confidence interval outperformed other confidence intervals in terms of coverage probability for both symmetric and skewed distributions and can be recommended for general use in practice.

Distribution of angular momenta ML and MS in non-relativistic configurations: statistical analysis using cumulants and Gram-Charlier series

Abstract The distributions P(ML,MS) of the total magnetic quantum numbers ML and MS for N electrons of angular momentum l, as well as the enumeration of LS spectroscopic terms and spectral lines, are crucial for the calculation of atomic structure and spectra, in particular for the modeling of emission or absorption properties of hot plasmas. However, no explicit formula for P(ML,MS) is known yet. In the present work, we show that the generating function for the cumulants, which characterize the distribution, obeys a recurrence relation, similar to the Newton-Girard identities relating elementary symmetric polynomials to power sums. This enables us to provide an explicit formula for the generating function. We also analyze the possibility of representing the P(ML,MS) distribution by a bi-variate Gram-Charlier series, which coefficients are obtained from the knowledge of the exact moments of P(ML,MS). It is shown that a simple approximation is obtained by truncating this series to the first few terms, though it is not convergent.

Higher-order coverage errors of batching methods via Edgeworth expansions on t-statistics

Higher-order coverage errors of batching methods via Edgeworth expansions on t-statistics

Increasing measurement accuracy by nonparametric data reconciliation

This paper discusses two approaches to the reconciliation of joint measurements with non-Gaussian errors’ distributions based on considering the functional relationships between measurands: semi-nonparametric based on polynomial approximation or Gram-Charlier series expansions and completely nonparametric based on kernel density estimator. The article presents mathematical expressions for all proposed methods in detail and the results of diverse statistical modelling for determining the limits of their possible applicability. We compared the effectiveness of the mentioned methods in a series of Monte Carlo experiments and showed that the nonparametric approach based on the kernel density estimation is less demanding on sample size than Gram-Charlier approximation for samples of size n > 30. However, in the case of asymmetric densities (skewness greater than 2.0), the reconciling algorithm with Gram-Charlier approximation is preferable for samples of medium size. The paper shows that the recommended measurement amount should be n > 150 for each measured quantity if we use not more than four terms in Gram-Charlier expansion.

Valid Edgeworth Expansion of the Bootstrap t-statistic of the Whittle MLE for Linear Regression Models with Long-Memory Residuals

Valid Edgeworth Expansion of the Bootstrap t-statistic of the Whittle MLE for Linear Regression Models with Long-Memory Residuals

Comparing annual extreme winds in Iran predicted by numerical weather forecasting and Gram-Charlier statistical model with meteorological observation data

Iran has been experiencing severe dust storms due to strong wind speeds that cause sand erosion. This erosion is particularly pronounced in the semi-arid and arid zones. Wind speed predictions using numerical weather forecasting (NWF) are indispensable; however, the applicability of NWF for predicting extreme annual wind speeds is not well known. To validate the NWF model, this study compared NWF-model-predicted 3-hour averaged wind speeds over one year and those observed at 390 weather stations. In addition, a statistical model was employed to estimate the probability densities of wind speeds for both the observational data and the NWF output. As an NWF model, the mesoscale Weather Research and Forecasting (WRF) model was utilized to simulate wind speeds. The results indicate that the observation data are consistent with the modeled datasets regarding the relationships among the mean, standard deviation, and skewness, whereas the WRF model tends to overestimate the mean wind speeds. In addition, the predictability of annual extreme wind speeds was determined using the peak factor. Moreover, skewness has emerged as an influential parameter for predicting extreme winds. Finally, the Gram-Charlier series model was utilized to estimate probability density functions of the wind speeds, demonstrating its effectiveness in capturing positively skewed distributions. The present analyses broaden the use of both NWF outputs and statistical methods to predict extreme wind speeds in Iran.

5th-Order Multivariate Edgeworth Expansions for Parametric Estimates

The only cases where exact distributions of estimates are known is for samples from exponential families, and then only for special functions of the parameters. So statistical inference was traditionally based on the asymptotic normality of estimates. To improve on this we need the Edgeworth expansion for the distribution of the standardised estimate. This is an expansion in n−1/2 about the normal distribution, where n is typically the sample size. The first few terms of this expansion were originally given for the special case of a sample mean. In earlier work we derived it for any standard estimate, hugely expanding its application. We define an estimate w^ of an unknown vector w in Rp, as a standard estimate, if Ew^→w as n→∞, and for r≥1 the rth-order cumulants of w^ have magnitude n1−r and can be expanded in n−1. Here we present a significant extension. We give the expansion of the distribution of any smooth function of w^, say t(w^) in Rq, giving its distribution to n−5/2. We do this by showing that t(w^), is a standard estimate of t(w). This provides far more accurate approximations for the distribution of t(w^) than its asymptotic normality.

A method for assessing the characteristics of the virtual machine migration process, taking into account the type of migrations and applications

Dynamic resource allocation in cloud computing is a critical aspect. It is necessary for computing resources scaling to maintain application performance, reduce costs and ensure the reliability of IT systems. Dynamic resource allocation is possible using open virtual machines (VM), but live migration is a relatively expensive and resource-intensive operation. A number of VM migration algorithms have been proposed, each with different performance characteristics depending on the state of the host system, network, and the VM workload. Recently, a large number of works have emerged that have developed models for assessing the effectiveness of of VM migration, but many of them do not provide a standard of satisfactory prediction accuracy and are built for a single migration algorithm, which limits the use of data models. This work uses a statistical approach to estimate the total migration time and VM downtime based on the approximation of probability densities by Gram-Charlier and Laguerre Series. The proposed method, in comparison with the known ones, allows one to answer the question of what is the probability of total migration time and VM downtime for a given migration type and VM application. The analysis of total migration time and VM downtime is based on the Virtual Machine Live Migration Dataset includes more than 40 000 virtual machine migration records with five different live migration algorithms: post-copy migration, pre-copy migration and its modifications: CPU throttling delta compression, and data compression. The dataset includes approximately 8000 migration records of each migration algorithm with 9 types of workloads. Analysis of the results allows us to conclude that the type of application significantly influences both the shape of the empirical distribution (histograms) of total migration time and its characteristics, therefore it is advisable to obtain an analytical expression for the distribution law of the total migration time and VM downtime taking into account these circumstances. The results of the experiments shows that this method does not depend on the migration algorithm and the type of working application. The use of the Laguerre series can be recommended as giving more reliable results compared to Gram-Charlier series. A method for estimating total migration time and VM downtime can be integrated into a management system to select the best monitoring window for servers and evaluate service-level agreement terms.

Saddlepoint approximations for the P-values and probability mass functions of some bivariate sign tests

Bivariate analysis is essential in several applied fields, such as medicine, engineering, biology, and econometrics. Many parametric and non-parametric procedures can be used for bivariate analysis. Non-parametric procedures require fewer conditions than their parametric counterparts. Therefore, this article proposes an accurate approximation of the p-values and the probability mass functions of some non-parametric bivariate tests, including bivariate sign tests and medians. The proposed approximation, saddlepoint approximation, is compared to the traditional approximation method, asymptotic normal approximation method, and Edgeworth expansion through three real data examples and a simulation study. The numerical results show that the proposed approximation method is more accurate than the asymptotic method and Edgeworth expansion. Furthermore, it is computationally less demanding than the simulation method, which is permutation-based and time-consuming.

Fatigue life and fatigue reliability mechanism of ball bearings

Abstract As the main supporting parts of rotating machinery, the fatigue life and fatigue reliability of ball bearings directly affect the accuracy, stability and reliability of equipment. Based on the quasi-static model of bearing and the fatigue life theory, this paper analyzes the effect of the external loads on the mechanical properties and fatigue life of bearing, and studies the action mechanism of bearing working conditions and other parameters. In addition, based on the stress–strength model and the fatigue life model of bearing, the fatigue reliability and sensitivity of bearings are analyzed by means of perturbation method, Edgeworth series and fourth moment technique, and the correctness is verified by the Monte Carlo simulation. Results have important theoretical and practical value, and can provide theoretical support for the research of reliability optimization design, fault mechanism and fault diagnosis method of ball bearings.

Neural network field theories: non-Gaussianity, actions, and locality

Both the path integral measure in field theory (FT) and ensembles of neural networks (NN) describe distributions over functions. When the central limit theorem can be applied in the infinite-width (infinite-N) limit, the ensemble of networks corresponds to a free FT. Although an expansion in 1/N corresponds to interactions in the FT, others, such as in a small breaking of the statistical independence of network parameters, can also lead to interacting theories. These other expansions can be advantageous over the 1/N -expansion, for example by improved behavior with respect to the universal approximation theorem. Given the connected correlators of a FT, one can systematically reconstruct the action order-by-order in the expansion parameter, using a new Feynman diagram prescription whose vertices are the connected correlators. This method is motivated by the Edgeworth expansion and allows one to derive actions for NN FT. Conversely, the correspondence allows one to engineer architectures realizing a given FT by representing action deformations as deformations of NN parameter densities. As an example, φ 4 theory is realized as an infinite-N NN FT.

Machine Learning-Based Probabilistic Forecasting of Wind Power Generation: A Combined Bootstrap and Cumulant Method

Probabilistic forecasting provides complete probability information of renewable generation and load, which assists the diverse decision-making tasks in power systems under uncertainties. Conventional machine learning-based probabilistic forecasting methods usually consider the predictive uncertainty following prior distributional assumptions. This paper develops a novel combined bootstrap and cumulant (CBC) method to generate nonparametric predictive distribution using higher order statistics for probabilistic forecasting. The CBC method successfully integrates machine learning with conditional moments and cumulants to describe the overall predictive uncertainty. A bootstrap-based conditional moment estimation method is proposed to quantify both the epistemic and aleatory uncertainties involved in machine learning. Higher order cumulants are utilized for overall uncertainty quantification based on the estimated conditional moments with its unique additivity. Three types of series expansions including Gram-Charlier, Edgeworth, and Cornish-Fisher expansions are adopted to improve the overall performance and the generalization ability. Comprehensive numerical studies using the actual wind power data validate the effectiveness of the proposed CBC method.

Local Edgeworth Expansions

Local Edgeworth Expansions

Correlations among high-order statistics and low-occurrence wind speeds within a simplified urban canopy based on particle image velocimetry datasets

Predicting infrequent and extreme wind speeds in a built environment is essential for ensuring comfortable and safe pedestrian spaces. Recent studies have employed statistical methods that assume a distribution function for wind speeds at pedestrian levels. Fundamental studies on the relationship between canopy flow and statistics are required to further develop statistical models. Therefore, this study aimed to understand the characteristics of strong and weak wind events within a simplified urban canopy and to scrutinize the relationship between high-order moments and extreme wind events. Particle image velocimetry (PIV) was employed to capture the velocities within a canopy consisting of cubes arranged in a staggered layout with a packing density of 25 %. The probability density functions (PDFs) of the velocity components classified by the mean flow patterns revealed that the PDF shapes were altered by the reverse and spanwise flows. In addition, strong correlations were verified between the gust or peak factor (PF) and high-order moments such as skewness, kurtosis, fifth-order, and sixth-order moments. Accordingly, the PF of the velocity components and wind speed were compared with the predictions by statistical methods based on the Weibull and Gram–Charlier series (GCS). These observations validate the previous statistical methods based on Weibull or GCS distributions. Although the physical interpretation of these statistics is ambiguous, the present analyses indicate that PF can be predicted by high-order moments, especially in particular, by skewness and kurtosis.

Explicit Non-Asymptotic Bounds for the Distance to the First-Order Edgeworth Expansion

Explicit Non-Asymptotic Bounds for the Distance to the First-Order Edgeworth Expansion

Bayesian inference with finitely wide neural networks.

The analytic inference, e.g., predictive distribution being in closed form, may be an appealing benefit for machine learning practitioners when they treat wide neural networks as Gaussian process in a Bayesian setting. The realistic widths, however, are finite and cause weak deviation from the Gaussianity under which partial marginalization of random variables in a model is straightforward. On the basis of multivariate Edgeworth expansion, we propose a non-Gaussian distribution in differential form to model a finite set of outputs from a random neural network, and derive the corresponding marginal and conditional properties. Thus, we are able to derive the non-Gaussian posterior distribution in Bayesian regression task. In addition, in the bottlenecked deep neural networks, a weight space representation of a deep Gaussian process, the non-Gaussianity is investigated through the marginal kernel and the accompanying small parameters.

Distribution of squared sum of products of independent Nakagami-m random variables

The need for the distribution of combination of random variables (RVs) arises in many areas of sciences and engineering. In this paper, closed-form approximations for the distribution of squared sum of products of independent Nakagami-m RVs are derived. Three different approaches (central limit theorem, Edgeworth expansion, and a one-term gamma approximation) are considered. The Kolmogorov-Smirnov, Anderson-Darling, and Cramer-von Mises statistics are used as a quantitative metrics to compare the derived distribution forms with the empirical distribution obtained from a simulation study. Furthermore, an application for the derived distributions in wireless communication field is presented. As a result, it is shown that the most accurate and simplest closed-form expression is the one obtained by a one-term gamma approximation.

Weak Edgeworth expansion for the mean-field Bose gas

We consider the ground state and the low-energy excited states of a system of N identical bosons with interactions in the mean-field scaling regime. For the ground state, we derive a weak Edgeworth expansion for the fluctuations of bounded one-body operators, which yields corrections to a central limit theorem to any order in 1/sqrt{N}. For suitable excited states, we show that the limiting distribution is a polynomial times a normal distribution, and that higher-order corrections are given by an Edgeworth-type expansion.

Investigation of the Static Performance of Hydrostatic Thrust Bearings Considering Non-Gaussian Surface Topography

The dynamic and static characteristics of hydrostatic thrust bearings are significantly affected by the bearing surface topography. Previous studies on hydrostatic thrust bearings have focused on Gaussian distribution models of bearing surface topography. However, based on actual measurements, the non-Gaussianity of the distribution characteristics of bearing surface topography is clear. To accurately characterize the non-Gaussian distribution of bearing surface topography, the traditional probability density function of Gaussian distribution was modified by introducing Edgeworth expansion. The non-Gaussian surface was then reflected by two parameters: kurtosis and skewness. This had an effect on the static characteristics of hydrostatic thrust bearings with both circumferential and radial surface topographies. The comparison between the Gaussian distribution results and those of the non-Gaussian model showed that errors between the two models could reach more than 10%. Therefore, it is important to take into account the non-Gaussianity of bearing surface when discussing static characteristics of hydrostatic thrust bearings considering the surface topography.