The Riemannian Structure of the Three-Parameter Gamma Distribution

In this paper, we will utilize the results already known in differential geometry and provide an intuitive understanding of the Gamma Distribution. This approach leads to the definition of new concepts to provide new results of statistical importance. These new results could explain Chen [1-3] experienced difficulty when he attempts to simulate the sampling distribution and power function of Cox’s [4,5] test statistics of separate families of hypotheses. It may also help simplify and clarify some known statistical proofs or results. These results may be of particular interest to mathematical physicists. In general, it has been shown that the parameter space is not of constant curvature. In addition, we calculated some invariant quantities, such as Sectional curvature, Ricci curvature, mean curvature and scalar curvature.

Tuning mechanical behaviors of highly entangled hydrogels with the random distribution of mobile entanglements

Highly entangled hydrogels exhibit excellent mechanical properties,including high toughness,high stretchability,and low hysteresis.By considering the evolution of randomly distributed entanglements within the polymer network upon mechanical stretches,we develop a constitutive theory to describe the large stretch behaviors of these hydrogels.In the theory,we utilize a representative volume element(RVE)in the shape of a cube,within which there exists an averaged chain segment along each edge and a mobile entanglement at each corner.By employing an explicit method,we decouple the elasticity of the hydrogels from the sliding motion of their entanglements,and derive the stress-stretch relations for these hydrogels.The present theoretical analysis is in agreement with experiment,and highlights the significant influence of the entanglement distribution within the hydrogels on their elasticity.We also implement the present developed constitutive theory into a commercial finite element software,and the subsequent simulations demonstrate that the exact distribution of entanglements strongly affects the mechanical behaviors of the structures of these hydrogels.Overall,the present theory provides valuable insights into the deformation mechanism of highly entangled hydrogels,and can aid in the design of these hydrogels with enhanced performance.

Magnetized magma intrusions being sources of a weak and a strong lunar magnetic anomaly revealed by 3D distribution of magnetization

In this work,we aim to investigate the origin of the magnetic carriers in the lunar crust and the intensity of the ancient dynamo field.The magnetization and depth range of magnetic carriers are studied under a weak and a strong magnetic anomaly in Mare Tranquillitatis and in Oceanus Procellarum,respectively,where the surface ages are 3.6 and 3.3 billion years.A sophisticated three-dimensional amplitude inversion software program from a geophysical survey is used to reconstruct the distributions of magnetization in the lunar crust.Because no globally measured surface magnetic field exists for the Moon,a crustal magnetic anomaly model with a grid resolution of 0.2°is used.The depth range of the magnetic source is fixed by the boundary identified by a relative criterion,which is 20%of the recovered maximum magnetization.The central burial depths of the magnetic carriers are approximately 15 km and 25 km under Reiner Gamma and Mare Tranquillitatis,respectively.The volumes of the two magnetic sources are at scales of 104 and 105 km3,respectively.The aforementioned differences may imply a hotter crust under Reiner Gamma than Mare Tranquillitatis by 3.3 billion years.The results support the view that the magma intrusions magnetized by an ancient magnetic field could be the origin of magnetic anomalies under Reiner Gamma and Mare Tranquillitatis.Compared with previous works,the maximum magnetization of 3 A/m under Reiner Gamma supports the intensity of the field being several microteslas.

Designing Bayesian Two-Sided Group Chain Sampling Plan for Gamma Prior Distribution

Acceptance sampling is used to decide either the whole lot will be accepted or rejected,based on inspection of randomly sampled items from the same lot.As an alternative to traditional sampling plans,it is possible to use Baye-sian approaches using previous knowledge on process variation.This study pre-sents a Bayesian two-sided group chain sampling plan(BTSGChSP)by using various combinations of design parameters.In BTSGChSP,inspection is based on preceding as well as succeeding lots.Poisson function is used to derive the probability of lot acceptance based on defective and non-defective products.Gamma distribution is considered as a suitable prior for Poisson distribution.Four quality regions are found,namely:(i)quality decision region(QDR),(ii)probabil-istic quality region(PQR),(iii)limiting quality region(LQR)and(iv)indifference quality region(IQR).Producer’s risk and consumer’s risk are considered to esti-mate the quality regions,where acceptable quality level(AQL)is associated with producer’s risk and limiting quality level(LQL)is associated with consumer’s risk.Moreover,AQL and LQL are used in the selection of design parameters for BTSGChSP.The values based on all possible combinations of design parameters for BTSGChSP are presented and inflection points’values are found.Thefinding exposes that BTSGChSP is a better substitute for the existing plan for industrial practitioners.

Does the Redshift Distribution of Swift Long GRBs Trace the Star-Formation Rate?

Gamma-ray bursts (GRBs) are extremely powerful explosions that have been traditionally classified into two categories: long bursts (LGRBs) with an observed duration T90 > 2 s, and short bursts (SGRBs) with an observed duration T90 T90 is the time interval during which 90% of the fluence is detected. LGRBs are believed to emanate from the core-collapse of massive stars, while SGRBs are believed to result from the merging of two compact objects, like two neutron stars. Because LGRBs are produced by the violent death of massive stars, we expect that their redshift distribution should trace the star-formation rate (SFR). The purpose of our study is to investigate the extent to which the redshift distribution of LGRBs follows and reflects the SFR. We use a sample of 370 LGRBs taken from the Swift catalog, and we investigate different models for the LGRB redshift distribution. We also carry out Monte Carlo simulations to check the consistency of our results. Our results indicate that the SFR can describe the LGRB redshift distribution well for high redshift bursts, but it needs an evolution term to fit the distribution well at low redshift.

Earthquake Magnitude Probability and Gamma Distribution

The earthquake magnitude probability distribution is one of the underlying input data for certain earthquake analyses, such as probabilistic seismic hazard analysis. Nowadays, the method proposed by McGuire and Arabasz (1990) is commonly used for obtaining the (simulated) earthquake magnitude probability distributions. However, based on the observed earthquake data in 5 regions (Taiwan, Japan, California, Turkey, and Greece), the model did not fit the observation well. Instead, all of the case studies show that using the newly proposed gamma distribution can improve the simulation significantly compared to the conventional method.

Two Probability Plots of the Three-Parameter Lognormal Distribution

The two-parameter lognormal distribution is a variant of the normal distribution and the three-parameter lognormal distribution is an extension of the two-parameter lognormal distribution by introducing a location parameter. The Q-Q plot of the three-parameter lognormal distribution is widely used. To obtain the Q-Q plot one needs to iteratively try different values of the shape parameter and subjectively judge the linearity of the Q-Q plot. In this paper,a mathematical method was proposed to determine the value of the shape parameter so as to simplify the generation of the Q-Q plot. Then a new probability plot was proposed,which was more easily obtained and provided more accurate parameter estimates than the Q-Q plot. These are illustrated by three realworld examples.

广义Gamma分布弱湍流环境含指向误差串行中继水下无线光通信系统误码性能分析

鉴于目前普遍选用的水下弱湍流模型无法准确拟合实测数据的缺陷,选取经实验测试验证的广义Gamma分布来表征海洋弱湍流,以期合理评估受湍流等因素影响的水下无线光通信系统的误码性能。提出一种包含广义Gamma分布弱湍流、零/非零视轴指向误差以及无衰落信道冲激响应隐路径损耗和多径效应的混合衰落信道模型,据此推导混合衰落串行中继无线光通信系统平均误码率的数学闭型表达;通过数值仿真验证所推导的误码率理论表达式的准确性,并考察不同系统核心参数对误码率的影响。研究结果表明:串行中继节点的引入可有效提高系统的误码性能,以零视轴指向误差、最终传输距离达45 m、误码率需求10-3为例,引入1至3个串行中继节点后,系统从一开始的无法工作,变为所需的发射功率分别降低至24 dBm、12 dBm和6 dBm;而光源初始发散角增大导致的无衰落信道冲激响应时延扩展,即符号间干扰,则会严重降低此性能改善。

One-sided empirical Bayes test for location parameter in Gamma distribution

In this paper, we devote to constructing the one-sided empirical Bayes(EB) test for the location parameter in the Gamma distribution by nonparametric method. Under some mild conditions, we prove that the EB test is asymptotically optimal with the rate of the order O(n^(-δs/(2s+1))), where 1/2 ≤ δ < 1 and s > 1 is a given natural number. An example is also given to illustrate that the conditions of the main theorems are easily satisfied.

一类折叠Gamma分布的尖峰厚尾性质及应用

组合分布思想的出现及扩展为研究现实数据分布提供了优良方案。鉴于许多数据的尖峰后尾特征,文章从Gamma分布出发,构造了具有尖峰厚尾性质的折叠Gamma分布,进而讨论了折叠Gamma分布的性质,并在Newton-Raphson算法下给出了分布参数的矩估计和极大似然估计,同时以数值模拟证明了估计的可行性,最后以实际数据对比研究折叠Gamma分布和正态分布的拟合效果。结果表明:折叠Gamma分布能更好地拟合尖峰厚尾数据,且其拟合效果随着数据样本量的增大而变好。

Empirical Bayesian Approach to Testing Homogeneity of Several Means of Inflated Poisson Distributions (IPD)

Objectives: We introduce a special form of the Generalized Poisson Distribution. The distribution has one parameter, yet it has a variance that is larger than the mean a phenomenon known as “over dispersion”. We discuss potential applications of the distribution as a model of counts, and under the assumption of independence we will perform statistical inference on the ratio of two means, with generalization to testing the homogeneity of several means. Methods: Bayesian methods depend on the choice of the prior distributions of the population parameters. In this paper, we describe a Bayesian approach for estimation and inference on the parameters of several independent Inflated Poisson (IPD) distributions with two possible priors, the first is the reciprocal of the square root of the Poisson parameter and the other is a conjugate Gamma prior. The parameters of Gamma distribution are estimated in the empirical Bayesian framework using the maximum likelihood (ML) solution using nonlinear mixed model (NLMIXED) in SAS. With these priors we construct the highest posterior confidence intervals on the ratio of two IPD parameters and test the homogeneity of several populations. Results: We encountered convergence problem in estimating the hyperparameters of the posterior distribution using the NLMIXED. However, direct maximization of the predictive density produced solutions to the maximum likelihood equations. We apply the methodologies to RNA-SEQ read count data of gene expression values.

Thorium distribution on the Moon:new insights from Chang’E-2 gamma-ray spectrometer

We present the thorium distribution on the lunar surface derived from observations by the Chang’E-2 gamma-ray spectrometer(CE-2 GRS). This new map shows a similar thorium distribution to previous observations. In combination with this new thorium map and impact cratering model, we investigate the origination of thorium on the Moon’s highlands, which was previously thought to be contributed from Imbrium ejecta. We found that the Imbrium ejecta has a small contribution(~20%–30%) to the thorium on the lunar highlands but most thorium is likely to be indigenous before the deposition of the Imbrium ejecta. This new thorium map also confirms that the eastern highlands have a relatively higher thorium concentration than the western highlands. We propose that the thin crust and large basins on the eastern highlands are responsible for this difference in thorium.

What Is the Difference between Gamma and Gaussian Distributions?

An inequality describing the difference between Gamma and Gaussian distributions is derived. The asymptotic bound is much better than by existing uniform bound from Berry-Esseen inequality.

A Note on Finding Geodesic Equation of Two Parameters Gamma Distribution

Engineers commonly use the gamma distribution to describe the life span or metal fatigue of a manufactured item. In this paper, we focus on finding a geodesic equation of the two parameters gamma distribution. To find this equation, we applied both the well-known Darboux Theorem and a pair of differential equations taken from Struik [1]. The solution proposed in this note could be used as a general solution of the geodesic equation of gamma distribution. It would be interesting if we compare our results with Lauritzen’s [2].

On Robustness of a Sequential Test for Scale Parameter of Gamma and Exponential Distributions

The main aim of the present paper is to study the robustness of the developed sequential probability ratio test (SPRT) for testing the hypothesis about scale parameter of gamma distribution with known shape parameter and exponential distribution with location parameter. The robustness of the SPRT for scale parameter of gamma distribution is studied when the shape parameter has undergone a change. The similar study is conducted for the scale parameter of exponential distribution when the location parameter has undergone a change. The expressions for operating characteristic and average sample number functions are derived. It is found in both the cases that the SPRT is robust only when there is a slight variation in the shape and location parameter in the respective distributions.

The Kolmogorov-Smirnov test for three redshift distributions of long gamma-ray bursts in the Swift Era

We investigate redshift distributions of three long burst samples, with the first sample containing 131 long bursts with observed redshifts, the second including 220 long bursts with pseudo-redshifts calculated by the variability-luminosity relation, and the third including 1194 long bursts with pseudo-redshifls calculated by the lag-luminosity relation, respectively. In the redshift range 0-1 the Kolmogorov-Smirnov probability of the observed redshift distribution and that of the variability-luminosity relation is large. In the redshift ranges 1-2, 2-3, 3-6.3 and 0-37, the Kolmogorov-Smirnov probabilities of the redshift distribution from lag-luminosity relation and the observed redshift distribution are also large. For the GRBs, which appear both in the two pseudo-redshift burst samples, the KS probability of the pseudo-redshift distribution from the lag-luminosity relation and the observed reshift distribution is 0.447, which is very large. Based on these results, some conclusions are drawn： i） the V-Liso relation might be more believable than the τ-Liso relation in low redshift ranges and the τ-Liso relation might be more real than the V-Liso relation in high redshift ranges; ii） if we do not consider the redshift ranges, the τ-Liso relation might be more physical and intrinsical than the V-Liso relation.

Investigations into Some Simple Expressions of the Gamma Function in Wind Power Theoretical Estimate by the Weibull Distribution

The Weibull distribution is a probability density function (PDF) which is widely used in the study of meteorological data. The statistical analysis of the wind speed v by using the Weibull distribution leads to the estimate of the mean wind speed , the variance of v around and the mean power density in the wind. The gamma function Γ is involved in those calculations, particularly Γ (1+1/k), Γ (1+2/k) and Γ (1+3/k). The paper reports the use of the Weibull PDF f(v) to estimate the gamma function. The study was performed by looking for the wind speeds related to the maximum values of f(v), v2 f(v) and v3 f(v). As a result, some approximate relationships were obtained for Γ (1+1/k), Γ (1+2/k) and Γ (1+3/k), that use some fitting polynomial functions. Very good agreements were found between the exact and the estimated values of Γ (1+n/k) that can be used for the estimation of the mean wind speed , the variance σ2 of the wind speed v;around the mean speed and the average wind power density.

A Distributional Representation of Gamma Function with Generalized Complex Domian

In this paper, we present a new representation of gamma function as a series of complex delta functions. We establish the convergence of this representation in the sense of distributions. It turns out that the gamma function can be defined over a space of complex test functions of slow growth denoted by Z. Some properties of gamma function are discussed by using the properties of delta function.

Content subscribing mechanism in P2P streaming based on gamma distribution prediction

P2P systems are categorized into tree-based and mesh-based systems according to their topologies. Mesh-based systems are considered more suitable for large-scale lnternet applications, but require optimization on latency issue. This paper proposes a content subscribing mechanism （CSM） to eliminate unnecessary time delays during data relaying. A node can send content data to its neighbors as soon as it receives the data segment. No additional time is taken during the interactive stages prior to data segment transmission of streaming content. CSM consists of three steps. First, every node records its historical segments latency, and adopts gamma distribution, which possesses powerful expression ability, to express latency statistics. Second, a node predicts subscribing success ratio of every neighbor by comparing the gamma distribution parameters of the node and its neighbors before selecting a neighbor node to subscribe a data segment. The above steps would not increase latency as they are executed before the data segments are ready at the neighbor nodes. Finally, the node, which was subscribed to, sends the subscribed data segment to the subscriber immediately when it has the data segment. Experiments show that CSM significantly reduces the content data transmission latency.

Analytical Evaluation of Non-Elementary Integrals Involving Some Exponential, Hyperbolic and Trigonometric Elementary Functions and Derivation of New Probability Measures Generalizing the Gamma-Type and Gaussian-Type Distributions

The non-elementary integrals involving elementary exponential, hyperbolic and trigonometric functions, and where α,η and β are real or complex constants are evaluated in terms of the confluent hypergeometric function 1F1 and the hypergeometric function 1F2. The hyperbolic and Euler identities are used to derive some identities involving exponential, hyperbolic, trigonometric functions and the hypergeometric functions 1F1 and 1F2. Having evaluated, these non-elementary integrals, some new probability measures generalizing the gamma-type and Gaussian distributions are also obtained. The obtained generalized probability distributions may, for example, allow to perform better statistical tests than those already known (e.g. chi-square (x2) statistical tests and other statistical tests constructed based on the central limit theorem (CLT)), while avoiding the use of computational approximations (or methods) which are in general expensive and associated with numerical errors.