Gaussian Random Variables Research Articles

A Improved , ) Effective df Approximation

This study introduces a correction to the approximation of effective df as proposed by Satterthwaite, specifically addressing scenarios where component df are small. The correction is grounded in analytical results concerning the moments of standard normal random variables. This modification is applicable to complex variance estimates that involve both small and large df, offering an enhanced approximation of the higher moments required by Satterthwaite’s framework. Additionally, this correction extends and partially validates the empirically derived adjustment by Johnson and Rust, as it is based on theoretical foundations rather than simulations used to derive empirical transformation constants. Finally, the proposed adjustment also provides a correction to the estimate of the total variance in cases missing data have been replaced by multiple imputations such as in the case of plausible values in national and international large scale assessments.

Optimal guaranteed estimation methods for the Cox–Ingersoll–Ross models

In this paper, we study parameter estimation problems for the Cox–Ingersoll–Ross (CIR) processes. For the first time for such models, sequential estimation procedures are proposed. In the non-asymptotic setting, the proposed sequential procedures provide the estimation with non-asymptotic fixed mean square accuracy. For the scalar parameter estimation problems non-asymptotic normality properties for the proposed estimators are established even in the cases when the classical non-sequential maximum likelihood estimators cannot be calculated. Moreover, the Laplace transformations for the mean observation durations are obtained. In the asymptotic setting, the limit forms for the mean observation durations are found and it is shown that the constructed sequential estimators uniformly converge in distribution to normal random variables. Then using the Local Asymptotic Normality (LAN) property, it is obtained asymptotic sharp lower bound for the minimax risks in the class of all sequential procedures with the same mean observation duration and as a consequence, it is established that the proposed sequential procedures are optimal in the minimax sense in this class.

The distribution of the ratio of normal random variables and the ellipticity of the Earth

In this paper we consider inference regarding the ratio of two normal and the ratio of two t-distributed random variables, using both the popular Fieller interval, as well as, exact distributions. We apply these methods to a historical dataset regarding the shape of the Earth, and estimate the Earth's flatness coefficient as a ratio of regression coefficients. We demonstrate the equivalence of the inference using the exact density of this ratio with that using the Fieller interval.

Asymptotic approximations for the distribution of the product of correlated normal random variables

Asymptotic approximations for the distribution of the product of correlated normal random variables

Improved Variance Estimation From Trimmed Samples

ABSTRACTSeveral methods for the robust estimation of the variance of a normal random variable are based on trimming. For example, the popular estimator is based on trimming the pairs of samples with the largest absolute distance. Another example is least trimmed squares regression, where all unusually large residuals are removed before the estimation of the variance. Here, in this work, we propose two new classes of estimators that use an optimal linear combination of the trimmed samples to achieve a lower mean squared error (MSE). The first class guarantees the smallest variance among all unbiased estimators that are based on linear combinations, while the second class of estimators guarantees the smallest MSE at the cost of some small bias. Through simulations, we demonstrate that our proposed estimators can have considerably smaller MSE than other robust estimators both in the presence and absence of outliers.

Non-Hermitian diluted banded random matrices: Scaling of eigenfunction and spectral properties.

Here we introduce the non-Hermitian diluted banded random matrix (nHdBRM) ensemble as the set of N×N real nonsymmetric matrices whose entries are independent Gaussian random variables with zero mean and variance one if |i-j|<b and zero otherwise, moreover off-diagonal matrix elements within the bandwidth b are randomly set to zero such that the sparsity α is defined as the fraction of the N(b-1)/2 independent nonvanishing off-diagonal matrix elements. By means of a detailed numerical study we demonstrate that the eigenfunction and spectral properties of the nHdBRM ensemble scale with the parameter x=γ[(bα)^{2}/N]^{δ}, where γ,δ∼1. Moreover, the normalized localization length β of the eigenfunctions follows a simple scaling law: β=x/(1+x). For comparison purposes, we also report eigenfunction and spectral properties of the Hermitian diluted banded random matrix ensemble.

Stochastic stability of random stacking of blocks

Stochastic stability of random stacking of blocks

Context-Based Meta-Reinforcement Learning With Bayesian Nonparametric Models.

Deep reinforcement learning agents usually need to collect a large number of interactions to solve a single task. In contrast, meta-reinforcement learning (meta-RL) aims to quickly adapt to new tasks using a small amount of experience by leveraging the knowledge from training on a set of similar tasks. State-of-the-art context-based meta-RL algorithms use the context to encode the task information and train a policy conditioned on the inferred latent task encoding. However, most recent works are limited to parametric tasks, where a handful of variables control the full variation in the task distribution, and also failed to work in non-stationary environments due to the few-shot adaptation setting. To address those limitations, we propose MEta-reinforcement Learning with Task Self-discovery (MELTS), which adaptively learns qualitatively different nonparametric tasks and adapts to new tasks in a zero-shot manner. We introduce a novel deep clustering framework (DPMM-VAE) based on an infinite mixture of Gaussians, which combines the Dirichlet process mixture model (DPMM) and the variational autoencoder (VAE), to simultaneously learn task representations and cluster the tasks in a self-adaptive way. Integrating DPMM-VAE into MELTS enables it to adaptively discover the multi-modal structure of the nonparametric task distribution, which previous methods using isotropic Gaussian random variables cannot model. In addition, we propose a zero-shot adaptation mechanism and a recurrence-based context encoding strategy to improve the data efficiency and make our algorithm applicable in non-stationary environments. On various continuous control tasks with both parametric and nonparametric variations, our algorithm produces a more structured and self-adaptive task latent space and also achieves superior sample efficiency and asymptotic performance compared with state-of-the-art meta-RL algorithms.

Statistical distribution of surface elevation by using quadratic forms of normal random variables

This paper is concerned with the statistical properties of surface elevation, which is crucial for the design and operation of marine and coastal structures and is helpful to the prediction of rogue waves. A semi-analytic quadratic model is proposed to describe the statistical distribution of surface elevation by using the theory of quadratic forms of normal random variables, in which the characteristic function and cumulants are given in analytical form, and the probability density is obtained by the numerical inversion of the characteristic function using the fast Fourier transform algorithm. Compared with Monte Carlo simulations, the quadratic model presents excellent performance in describing the probability density function of surface elevation, especially at the tails, for varying degrees of steepness, bandwidth, and directional spreading in both finite and infinite water depth. The effect of these parameters that characterize sea states on the statistical properties of surface elevation is also investigated. Through a comparative study, the quadratic model is superior to previously existing theoretical models.

A Stein characterisation of the distribution of the product of correlated normal random variables

A Stein characterisation of the distribution of the product of correlated normal random variables

Capacity analysis over fractional order Rayleigh fading channel under additive white generalized Gaussian noise

AbstractThis study presents an innovative fractional order Rayleigh fading model that can be used for channel capacity estimation in the presence of additive white generalized Gaussian noise. The proposed model assumes that the real and imaginary parts of channel gains are generalized Gaussian random variables, which makes it possible to consider the traditional Rayleigh fading model as a special case of fractional order Rayleigh fading. Compared to the Rayleigh model, the fractional order Rayleigh fading model offers a more precise representation of new real‐world communication, such as integrating terrestrial and underwater networks in sixth‐generation communications channels. The probability density function of the channel gain with additive white generalized Gaussian noise is analyzed here. Furthermore, the ergodic and outage capacities of the channel are determined, taking into account the assumption that the channel state information is only available at the receiver. The ergodic capacity is calculated using Meijer's G‐functions, resulting in a closed‐form expression. Numerical simulations demonstrate the superiority of the fractional order Rayleigh fading model over the Rayleigh channel. Moreover, the impact of ergodic and outage capacities under diverse channel characteristics is assessed.

Absolute moments of the variance-gamma distribution

Absolute moments of the variance-gamma distribution

Chance-constrained programming for drinking water quality management under fuzziness and randomness

ABSTRACT In this paper, an inexact fuzzy random chance-constrained programming (IFRCCP) model was proposed to deal with the randomness and fuzziness in the optimization of booster cost for the water distribution system (WDS). The IFRCCP model was applied to a WDS to analyze the effects of randomness and fuzziness on the booster costs. The booster cost intervals were obtained under various violation levels and confidence levels. The results indicated that the lower and upper booster costs increased with the confidence levels of the lower limits and decreased with the violation levels of the lower limits for both normal and log-normal distribution fuzzy random variables (FRVs). Moreover, the increased magnitude of lower and upper booster costs under normal distribution FRV is a little less than that under log-normal distribution FRV, while the decreased magnitude of lower and upper booster costs under normal distribution FRV is larger than that under log-normal distribution FRV. The same regulation can be found for the FVR with both triangular and trapezoidal distribution FRVs. With the increase of booster number, the increased magnitude of the lower and upper booster costs decreased for both normal and log-normal distribution FRVs.

Importance sampling for counting statistics in one-dimensional systems.

In this paper, we consider the problem of numerical investigation of the counting statistics for a class of one-dimensional systems. Importance sampling, the cornerstone technique usually implemented for such problems, critically hinges on selecting an appropriate biased distribution. While an exponential tilt in the observable stands as the conventional choice for various problems, its efficiency in the context of counting statistics may be significantly hindered by the genuine discreteness of the observable. To address this challenge, we propose an alternative strategy, which we call importance sampling with the local tilt. We demonstrate the efficiency of the proposed approach through the analysis of three prototypical examples: a set of independent Gaussian random variables, Dyson gas, and symmetric simple exclusion process with a steplike initial condition.

Combination of dependent and partially reliable Gaussian random fuzzy numbers

Combination of dependent and partially reliable Gaussian random fuzzy numbers

Causalized Convergent Cross Mapping and Its Implementation in Causality Analysis.

Rooted in dynamic systems theory, convergent cross mapping (CCM) has attracted increased attention recently due to its capability in detecting linear and nonlinear causal coupling in both random and deterministic settings. One limitation with CCM is that it uses both past and future values to predict the current value, which is inconsistent with the widely accepted definition of causality, where it is assumed that the future values of one process cannot influence the past of another. To overcome this obstacle, in our previous research, we introduced the concept of causalized convergent cross mapping (cCCM), where future values are no longer used to predict the current value. In this paper, we focus on the implementation of cCCM in causality analysis. More specifically, we demonstrate the effectiveness of cCCM in identifying both linear and nonlinear causal coupling in various settings through a large number of examples, including Gaussian random variables with additive noise, sinusoidal waveforms, autoregressive models, stochastic processes with a dominant spectral component embedded in noise, deterministic chaotic maps, and systems with memory, as well as experimental fMRI data. In particular, we analyze the impact of shadow manifold construction on the performance of cCCM and provide detailed guidelines on how to configure the key parameters of cCCM in different applications. Overall, our analysis indicates that cCCM is a promising and easy-to-implement tool for causality analysis in a wide spectrum of applications.

The Measurement Errors and Their Effects on the Cumulative Sum Schemes for Monitoring the Ratio of Two Correlated Normal Variables

Monitoring the ratio of two correlated normal random variables is often used in many industrial manufacturing processes. At the same time, measurement errors inevitably exist in most processes, which have different effects on the performance of various charting schemes. This paper comprehensively analyses the impacts of measurement errors on the detection ability of the cumulative sum (CUSUM) charting schemes for the ratio of two correlated normal variables. A thorough numerical assessment is performed using the Monte Carlo simulation, and the results indicate that the measurement errors negatively impact the performance of the CUSUM scheme for the ratio of two correlated normal variables. Increasing the number of measurements per set is not a lucrative approach for minimizing the negative impact of measurement errors on the performance of the CUSUM charting scheme when monitoring the ratio of two correlated normal variables. We consider a food formulation as an example that illustrates the quality control problems involving the ratio of two correlated normal variables in an industry with a measurement error. The results are presented, along with some suggestions for further study.

Gauge Field Marginal of an Abelian Higgs Model

We study the gauge field marginal of an Abelian Higgs model with Villain action defined on a 2D lattice in finite volume. Our first main result, which holds for gauge theories on arbitrary finite graphs and does not assume that the structure group is Abelian, is a loop expansion of the Radon–Nikodym derivative of the law of the gauge field marginal with respect to that of the pure gauge theory. This expansion is similar to the one of Seiler (Gauge theories as a problem of constructive quantum field theory and statistical mechanics, volume 159 of lecture notes in physics, Springer, Berlin, p v+192. https://doi.org/10.1007/3-540-11559-5, 1982) but holds in greater generality and uses a different graph theoretic approach. Furthermore, we show ultraviolet stability for the gauge field marginal of the model in a fixed gauge. More specifically, we show that moments of the Hölder–Besov-type norms introduced in Chevyrev (Commun Math Phys 372(3):1027–1058. https://doi.org/10.1007/s00220-019-03567-5, 2019) are bounded uniformly in the lattice spacing. This latter result relies on a quantitative diamagnetic inequality that in turn follows from the loop expansion and elementary properties of Gaussian random variables.

A fast algorithm for computing product moments of multivariate normal random variables

We provide a simple identity that decomposes a product moment of multivariate normal random variables as a sum of various products of univariate moments of one of the random variables and multivariate moments of the other random variables. The new identity allows for much faster computation of the product moments of multivariate normal random variables than existing methods.

The distribution of the product of independent variance-gamma random variables

The distribution of the product of independent variance-gamma random variables