Symmetric Probability Distributions Research Articles

A New Modification of the Weibull Distribution: Model, Theory, and Analyzing Engineering Data Sets

Symmetrical as well as asymmetrical statistical models play a prominent role in describing and predicting the real-world phenomena of nature. Among other fields, these models are very useful for modeling data in the sector of civil engineering. Due to the applicability of the statistical models in civil engineering and other related sectors, this paper offers a statistical methodology to improve the distributional flexibility of traditional models. The suggested method/approach is called the extended-X family of distributions. The proposed method has the ability to generate symmetrical and asymmetrical probability distributions. Based on the extended-X family approach, an updated version of the Weibull model, namely, the extended Weibull model, is studied. The proposed model is very flexible and has the ability to capture the symmetrical and asymmetrical shapes of its density function. For the extended-X method, the estimation of parameters, a simulation study, and some mathematical properties are derived. Finally, the practical illustration/usefulness of the suggested model is shown by analyzing two data sets taken from the field of engineering. Both data sets represent the fracture toughness of alumina (Al2O3).

Cooperative search model to detect a lost target based on its probabilistic distribution in a bounded zone

ABSTRACT This study uses two unit-speed searchers to solve the challenge of finding a randomly located target in a bounded region, such as searching for bodies found under the rubble resulting from the erosion of residential units due to horrific natural disasters. The search procedure begins at ( 0 , 0 ) . The region is subdivided into a limited even number of cells. The target’s position has a symmetric probability distribution. It is preferable to search in an optimal method in order to reduce the predicted time for discovering the target. An algorithm is used to generate the optimal search strategy. We discuss an example when the target has a symmetric truncated normal distribution. We provide an illustrated case to demonstrate this technique’s applicability.

A new probabilistic approach for modeling the confirmation time of transactions on blockchain technology

Probability distributions are frequently used for statistical modeling of real-life phenomena in every field of life. Most often, real-life data sets are skewed in nature, and therefore, asymmetrical probability distributions are very competent for such scenarios. In the recent gains in distribution theory, researchers are focusing on developing new symmetrical and asymmetrical trigonometric-based probability distributions for data modeling in various fields. In the literature, however, there is no published work on statistical modeling of the transaction confirmation times on blockchain technology using trigonometric-originated probability distributions. For the first time covering the aforesaid research gap, this paper introduces a new trigonometric-based probabilistic approach for modeling the transaction confirmation times on blockchain technology. The new approach is called a new sine-G family of distributions. Using the proposed method, a new probability distribution called a new sine-Weibull distribution is studied. The proposed model is very flexible and obeys the symmetrical and asymmetrical shapes of its density function. The estimators of the new sine-Weibull distribution are derived, and their evaluation is tested through a simulation study. The applicability of the new sine-Weibull distribution is demonstrated by analyzing the average waiting time until the Bitcoin transaction is confirmed on blockchain technology.

On Expected Value Strong Controllability

The Probabilistic Simple Temporal Network with Uncertainty (PSTNU) is a variant of the Simple Temporal Network with Uncertainty (STNU) in which known probability distributions govern the timing of uncontrollable timepoints. Previous approaches to solving PSTNUs focus mininizing risk, that is, the probability of violating constraints. These approaches are not applicable in over-constrained controllability problems, when it is certain that all constraints can’t be satisfied. We introduce the Weighted Probabilistic Simple Temporal Network with Uncertainty (WPSTNU), which extends the PSTNU by attaching a fixed value to the satisfaction of temporal constraints, and allows the schedule to violate some constraints in order to maximize the expected value of satisfying others. We study the problem of Expected Value Strong Controllability (EvSC) of WPSTNUs, which seeks a fixed-time schedule maximizing the expected value of satisfied constraints. We solve the EvSC problem using a mixed integer linear program (MILP) that bounds below the probability of satisfying constraints involving uncontrollable timepoints. While solving MILPs generally takes exponential time, we demonstrate our formulation’s effective performance using scheduling problems derived from the HEATlab and MIT ROVERS data sets. We then show how to use this MILP to reschedule during execution, after time has passed and uncertainty is reduced. We describe different fixed-period rescheduling approaches, including time-based and event-based, and report on the most successful strategies compared to the expected value of the fixed schedule produced by the MILP. All of our methods are evaluated on problems with both symmetric and asymmetric (skewed) probability distributions. We show that periodically rescheduling improves the expected value when compared to the fixed schedule, and describe how the benchmark and skewness impact the schedule value improvement. The resulting analysis shows that solving EvSC problems on WPSTNUs is a viable alternative to solving over-constrained controllability problems.

Novel transformation of unimodal (a)symmetric possibility distributions into probability distributions

The paper introduces a new transformation of possibility distributions into continuous symmetric and asymmetric probability distributions. This transformation is called Inverse Two-Sided Normalized and could be possibly the inverse transformation of Two-Sided Normalized [1] (Foulloy et al., 2021) and the inverse construction presented in [2] (Souliotis et al., 2022). In addition, it is shown that the proposed transformation is the extension of the optimal transformation [3] (Dubois et al., 1992) and of its inverse only in specific cases. Those cases are the symmetric distributions or specific distributions such as the two-piece skewed distribution [1] (Foulloy et al., 2021). It is also shown that the asymmetry preservation of the probability distribution has 1-1 correspondence with the asymmetry preservation of the possibility distribution about the mode.

A New Sampling Scheme for an Improved Monitoring of the Process Mean

ABSTRACT This article introduces an innovative sampling scheme, the median sampling (MS), utilizing individual observations over time to efficiently estimate the mean of a process characterized by a symmetric (non-uniform) probability distribution. The mean estimator based on MS is not only unbiased but also boasts enhanced precision compared to its simple random sampling-based counterpart. Moreover, a new EWMA chart based on the mean estimator within the MS scheme is proposed. The performance of the EWMA charts, derived from both simple random sampling (SRS) and MS schemes, is evaluated using the metrics of steady-state average run-length and average number of items-to-signal. The findings underscore the superiority of the EWMA-MS chart over the EWMA-SRS chart. Additionally, as the magnitude of ranking errors escalates, the behavior of the EWMA-MS chart converges toward that of the EWMA-SRS chart. The practical implementation of the newly introduced EWMA chart is demonstrated through an illustrative example.

Statistical learning algorithms for dendritic neuron model artificial neural network based on sine cosine algorithm

Training of dendritic neuron model artificial neural networks is generally achieved by using nonlinear least square methods. The distribution of random error terms is ignored in training algorithms although error terms are random variables. Maximum likelihood estimators can be obtained for dendritic neuron model artificial neural networks by using some indefinite symmetric probability distributions. In this study, statistical learning algorithms are proposed for dendritic neuron model artificial neural networks. Maximum likelihood estimators for dendritic neuron model artificial neural networks are obtained by using Normal, Cauchy, Logistic, Gumbel and Laplace distributions. The Sine cosine algorithm is used for maximization of the likelihood function under error terms that have Normal, Cauchy, Logistic, Gumbel and Laplace distributions. The proposed learning algorithms are applied to Istanbul Stock Exchange time series data sets. At the end of the analysis of application results, the performance of the proposed method is statistically better than well-known deep and shallow artificial neural networks.

Experimentally implementing the step-dependent discrete-time quantum walk on quantum computers

The discrete-time quantum walk (DTQW) with step-dependent (SD) scattering operator is implemented on quantum computer. The probabilities of different states, with their respective fidelities, are calculated. This is done by generalizing the coin with a rotation gate using the quantum gate model. The CNOT gates in the shift operator are replaced with the alternative to CNOT gates Rx( π). They are applied on a quantum device and a quantum simulator (QS). The fidelities varied around 50% and the probability distribution of SD-DTQW for the angle π/4 spread symmetrically, while the step-independent DTQW (SI-DTQW) tended to peak at one side. The symmetric probability distribution of SD-DTQW can help in better control of the walk on QS. In the case of angle π/2, the SI-DTQW spread equally across the states with four peaks, while the SD-DTQW spread with two peaks to one side.

Symmetric and Asymmetric Distributions: Theoretical Developments and Applications III

A summary of the eleven papers published in this special issue is presented here. This volume was the last in a series of special issues dealing with symmetric and non-symmetric continuous probability distributions. The works presented in this issue propose new probabilistic models and extend the properties of other existing models in the statistical literature.

On Wendel’s equality for intersections of balls

We study the analogue of Wendel’s equality in random polytope models in which the hull of the random points is formed by intersections of congruent balls, called the spindle (or hyper-) convex hull. According to the classical identity of Wendel the probability that the origin is contained in the (linear) convex hull of n i.i.d. random points distributed according to an origin symmetric probability distribution in the d-dimensional Euclidean space mathbb {R}^{d} that assigns measure zero to hyperplanes is a constant depending only on n and d. While in the classical convex case one gets nonzero probabilities only for nge d+1 points in mathbb {R}^{d}, for the spindle convex hull this happens for all nge 2. We study this question for the uniform and normally distributed random models.

Generalized Chain Regression-cum-Chain Ratio Estimator for Population Mean under Stratified Extreme-cum-Median Ranked Set Sampling

Estimation of population mean of study variable Y suffers loss of precision in the presence of high variation in the data set. The use of auxiliary information incorporated in construction of an estimator under ranked set sampling scheme results in efficient estimation of population mean. In this paper, we propose an efficient generalized chain regression-cum-chain ratio type estimator to estimate finite population mean of study variable under stratified extreme-cum-median ranked set sampling utilizing information on two auxiliary variables. Mean square error (MSE) of the proposed generalized estimator is derived up to first order of approximation. The applications of the proposed estimator under symmetrical and asymmetrical probability distributions are discussed using simulation study and real-life data set for comparisons of efficiency. It is concluded that the proposed generalized estimator performs efficiently as compared to some existing estimators. It is also observed that the efficiency of the proposed estimator is directly proportional to the correlations between the study variable and its auxiliary variables.

Efficient Sampling of Bernoulli-Gaussian-Mixtures for Sparse Signal Restoration

This paper introduces a new family of prior models called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Bernoulli-Gaussian-Mixtures</i> (BGM), with a view to efficiently address sparse linear inverse problems or sparse linear regression, in the Bayesian framework. The BGM family is based on continuous Location and Scale Mixtures of Gaussians (LSMG), which includes a wide range of symmetric and asymmetric heavy-tailed probability distributions. Particular attention is paid to the decomposition of probability laws as Gaussian mixtures, from which we derive a Partially Collapsed Gibbs Sampler (PCGS) for the BGM, in a systematic way. PCGS is shown to be more efficient than the standard Gibbs sampler, both in terms of number of iterations and CPU time. Moreover, special attention is paid to BGM involving a density defined over a real half-line. An asymptotically exact LSMG approximation is introduced, which allows us to expand the applicability of PCGS to cases such as BGM models with a non-negative support.

On Calculation of Constants in the Arak Inequalities for Concentration Functions of Convolutions of Probability Distributions

The aim of the present work is evaluation of absolute constants in the Arak inequalities for the concentration functions of convolutions of probability distributions. This result allows us to calculate the constant in the inequality for the uniform distance between n and (n + 1)-fold convolutions of one-dimensional symmetric probability distributions with a characteristic function separated from −1, as well as a number of other estimates, in particular, the accuracy of approximation of samples of rare events by the Poisson point process.

Behavior of the Fréchet mean and Central Limit Theorems on spheres

We compute higher derivatives of the Fréchet function on spheres with an absolutely continuous and rotationally symmetric probability distribution. Consequences include (i) a practical condition to test if the mode of the symmetric distribution is a local Fréchet mean; (ii) a central limit theorem on spheres with practical assumptions and an explicit limiting distribution; and (iii) an answer to the question of whether the smeary effect can occur on spheres with absolutely continuous and rotationally symmetric distributions: with the method presented here, it can in dimension at least 4.

On The Estimation Of The Shape Parameter Of A Symmetric Distribution

The shape parameter of a symmetric probability distribution is often more difficult to estimate accurately than the location and scale parameters. In this paper, we suggest an intuitive but innovative matching quantile estimation method for this parameter. The proposed shape parameter estimate is obtained by setting its value to a level such that the central 1/���� portion of the distribution will just cover all n observations, while the location and scale parameters are estimated using existing methods such as maximum likelihood (ML). This hybrid estimator is proved to be consistent and is illustrated by two distributions, namely Student-t and Exponential Power. Simulation studies show that the hybrid method provides reasonably accurate estimates. In the presence of extreme observations, this method provides thicker tails than the full ML method and protect inference on the location and scale parameters. This feature offered by the hybrid method is also demonstrated in the empirical study using two real data sets.

Modeling Liver Cancer and Leukemia Data Using Arcsine-Gaussian Distribution

The main objective of this paper is to discuss a general family of distributions generated from the symmetrical arcsine distribution. The considered family includes various asymmetrical and symmetrical probability distributions as special cases. A particular case of a symmetrical probability distribution from this family is the Arcsine–Gaussian distribution. Key statistical properties of this distribution including quantile, mean residual life, order statistics and moments are derived. The Arcsine–Gaussian parameters are estimated using two classical estimation methods called moments and maximum likelihood methods. A simulation study which provides asymptotic distribution of all considered point estimators, 90% and 95% asymptotic confidence intervals are performed to examine the estimation efficiency of the considered methods numerically. The simulation results show that both biases and variances of the estimators tend to zero as the sample size increases, i.e., the estimators are asymptotically consistent. Also, when the sample size increases the coverage probabilities of the confidence intervals increase to the nominal levels, while the corresponding length decrease and approach zero. Two real data sets from the medicine filed are used to illustrate the flexibility of the Arcsine–Gaussian distribution as compared with the normal, logistic, and Cauchy models. The proposed distribution is very versatile to fit real applications and can be used as a good alternative to the traditional gaussian distribution.

Statistics of the number of records for random walks and Lévy flights on a 1D lattice

We study the statistics of the number of records Rn for a symmetric, n-step, discrete jump process on a 1D lattice. At a given step, the walker can jump by arbitrary lattice units drawn from a given symmetric probability distribution. This process includes, as a special case, the standard nearest neighbor lattice random walk. We derive explicitly the generating function of the distribution P(Rn) of the number of records, valid for arbitrary discrete jump distributions. As a byproduct, we provide a relatively simple proof of the generalized Sparre Andersen theorem for the survival probability of a random walk on a line, with discrete or continuous jump distributions. For the discrete jump process, we then derive the asymptotic large n behavior of P(Rn) as well as of the average number of records E(Rn). We show that unlike the case of random walks with symmetric and continuous jump distributions where the record statistics is strongly universal (i.e. independent of the jump distribution for all n), the record statistics for lattice walks depends on the jump distribution for any fixed n. However, in the large n limit, we show that the distribution of the scaled record number Rn/E(Rn) approaches a universal, half-Gaussian form for any discrete jump process. The dependence on the jump distribution enters only through the scale factor E(Rn), which we also compute in the large n limit for arbitrary jump distributions. We present explicit results for a few examples and provide numerical checks of our analytical predictions.

Dying ReLU and Initialization: Theory and Numerical Examples

The dying ReLU refers to the problem when ReLU neurons become inactive and only output 0 for any input. There are many empirical and heuristic explanations of why ReLU neurons die. However, little is known about its theoretical analysis. In this paper, we rigorously prove that a deep ReLU network will eventually die in probability as the depth goes to infinite. Several methods have been proposed to alleviate the dying ReLU. Perhaps, one of the simplest treatments is to modify the initialization procedure. One common way of initializing weights and biases uses symmetric probability distributions, which suffers from the dying ReLU. We thus propose a new initialization procedure, namely, a randomized asymmetric initialization. We prove that the new initialization can effectively prevent the dying ReLU. All parameters required for the new initialization are theoretically designed. Numerical examples are provided to demonstrate the effectiveness of the new initialization procedure.

Numerical Investigation of High-Speed Turbulent Boundary Layers of Dense Gases

High-speed turbulent boundary layers of a dense gas (PP11) and a perfect gas (air) over flat plates are investigated by means of direct numerical simulations and large eddy simulations. The thermodynamic conditions of the incoming flow are chosen to highlight dense gas effects, and laminar-to-turbulent transition is triggered by suction and blowing. In the paper, the behavior of the fully developed turbulent flow region is investigated. Due to the low characteristic Eckert number of dense gas flows ( $$\hbox {Ec}=U_\infty ^2/c_{p,\infty }T_\infty$$ ), the mean velocity profiles are largely insensitive to the Mach number and very close to the incompressible case even at high speeds. Second-order velocity statistics are also weakly affected by the flow Mach number and the velocity spectra are characterized by a secondary peak in the outer region of the boundary layer because of the higher local friction Reynolds number. Despite the incompressible-like velocity and Reynolds-stress profiles, the strongly non-ideal thermodynamic and transport-property behavior of the dense gas results in unconventional distributions of the fluctuating thermo-physical quantities. Specifically, density and viscosity fluctuations reach a peak close to the wall, instead of vanishing as in perfect gas flows. Additionally, dense gas boundary layers exhibit higher values of the fluctuating Mach number and velocity divergence and a larger dilatational-to-solenoidal dissipation ratio in the near-wall region, which represents a major deviation from high-Mach-number perfect gas boundary layers. Other significant deviations are represented by the more symmetric probability distributions of fluctuating quantities such as the density and velocity divergence, due to the more balanced occurrence of strong expansion and compression events.

Asymmetric generalizations of symmetric univariate probability distributions obtained through quantile splicing

Balakrishnan et al. proposed a two-piece skew logistic distribution by making use of the cumulative distribution function (CDF) of half distributions as the building block, to give rise to an asymmetric family of two-piece distributions, through the inclusion of a single shape parameter. This paper proposes the construction of asymmetric families of two-piece distributions by making use of quantile functions of symmetric distributions as building blocks. This proposition will enable the derivation of a general formula for the L-moments of two-piece distributions. Examples will be presented, where the logistic, normal, Student’s t(2) and hyperbolic secant distributions are considered.