A Normal Weighted Inverse Gaussian Distribution for Skewed and Heavy-Tailed Data

High frequency financial data is characterized by non-normality: asymmetric, leptokurtic and fat-tailed behaviour. The normal distribution is therefore inadequate in capturing these characteristics. To this end, various flexible distributions have been proposed. It is well known that mixture distributions produce flexible models with good statistical and probabilistic properties. In this work, a finite mixture of two special cases of Generalized Inverse Gaussian distribution has been constructed. Using this finite mixture as a mixing distribution to the Normal Variance Mean Mixture we get a Normal Weighted Inverse Gaussian (NWIG) distribution. The second objective, therefore, is to construct and obtain properties of the NWIG distribution. The maximum likelihood parameter estimates of the proposed model are estimated via EM algorithm and three data sets are used for application. The result shows that the proposed model is flexible and fits the data well.

Stability of weighted spectral distribution in a pseudo tree-like network model

The comparison of networks with different orders strongly depends on the stability analysis of graph features in evolving systems. In this paper, we rigorously investigate the stability of the weighted spectral distribution（i.e., a spectral graph feature） as the network order increases. First, we use deterministic scale-free networks generated by a pseudo treelike model to derive the precise formula of the spectral feature, and then analyze the stability of the spectral feature based on the precise formula. Except for the scale-free feature, the pseudo tree-like model exhibits the hierarchical and small-world structures of complex networks. The stability analysis is useful for the classification of networks with different orders and the similarity analysis of networks that may belong to the same evolving system.

Acceptance Sampling Plans with Truncated Life Tests for the Length-Biased Weighted Lomax Distribution

In this paper,we considered the Length-biased weighted Lomax distribution and constructed new acceptance sampling plans(ASPs)where the life test is assumed to be truncated at a pre-assigned time.For the new suggested ASPs,the tables of the minimum samples sizes needed to assert a specific mean life of the test units are obtained.In addition,the values of the corresponding operating characteristic function and the associated producer’s risks are calculated.Analyses of two real data sets are presented to investigate the applicability of the proposed acceptance sampling plans;one data set contains the first failure of 20 small electric carts,and the other data set contains the failure times of the air conditioning system of an airplane.Comparisons are made between the proposed acceptance sampling plans and some existing acceptance sampling plans considered in this study based on the minimum sample sizes.It is observed that the samples sizes based on the proposed acceptance sampling plans are less than their competitors considered in this study.The suggested acceptance sampling plans are recommended for practitioners in the field.

Properties, Inference and Applications of Inverse Power Two-Parameter Weighted Lindley Distribution

We proposed “a new extension of three-parametric distribution” called the inverse power two-parameter weighted Lindley (IPWL) distribution capable of modeling a upside-down bathtub hazard rate function. This distribution is studied to get basic structural properties such as reliability measures, moments, inverse moments and its related measures. Simulation studies are done to present the performance and behavior of maximum likelihood estimates of the IPWL distribution parameters. Finally, we perform goodness of fit measures and test statistics using a real data set to show the performance of the new distribution.

Scaling of weighted spectral distribution in weighted small-world networks

Many real-world systems can be modeled by weighted small-world networks with high clustering coefficients. Recent studies for rigorously analyzing the weighted spectral distribution（W SD） have focused on unweighted networks with low clustering coefficients. In this paper, we rigorously analyze the W SD in a deterministic weighted scale-free small-world network model and find that the W SD grows sublinearly with increasing network order（i.e., the number of nodes） and provides a sensitive discrimination for each input of this model. This study demonstrates that the scaling feature of the W SD exists in the weighted network model which has high and order-independent clustering coefficients and reasonable power-law exponents.

On the complete weight distributions of quantum error-correcting codes

In a recent paper, Hu et al. defined the complete weight distributions of quantum codes and proved the Mac Williams identities, and as applications they showed how such weight distributions may be used to obtain the singleton-type and hamming-type bounds for asymmetric quantum codes. In this paper we extend their study much further and obtain several new results concerning the complete weight distributions of quantum codes and applications. In particular, we provide a new proof of the Mac Williams identities of the complete weight distributions of quantum codes. We obtain new information about the weight distributions of quantum MDS codes and the double weight distribution of asymmetric quantum MDS codes. We get new identities involving the complete weight distributions of two different quantum codes. We estimate the complete weight distributions of quantum codes under special conditions and show that quantum BCH codes by the Hermitian construction from primitive, narrow-sense BCH codes satisfy these conditions and hence these estimate applies.

Construction and Weight Distributions of Binary Linear Codes Based on Deep Holes

Deep holes are very important in the decoding of generalized RS codes, and deep holes of RS codes have been widely studied, but there are few works on constructing general linear codes based on deep holes. Therefore, we consider constructing binary linear codes by combining deep holes with binary BCH codes. In this article, we consider the 2-error-correcting binary primitive BCH codes and the extended codes to construct new binary linear codes by combining them with deep holes, respectively. Furthermore, three classes of binary linear codes are constructed, and then we determine the parameters and the weight distributions of these new binary linear codes.

A Finite Mixture of Generalised Inverse Gaussian with Indexes -1/2 and -3/2 as Mixing Distribution for Normal Variance Mean Mixture with Application

Mixture models have become more popular in modelling compared to standard distributions. The mixing distributions play a role in capturing the variability of the random variable in the conditional distribution. Studies have lately focused on finite mixture models as mixing distributions in the mixing mechanism. In the present work, we consider a Normal Variance Mean mixture model. The mixing distribution is a finite mixture of two special cases of Generalised Inverse Gaussian distribution with indexes -1/2 and -3/2. The parameters of the mixed model are obtained via the Expectation-Maximization (EM) algorithm. The iterative scheme is based on a presentation of the normal equations. An application to some financial data has been done.

A Special Weight for Inverse Gaussian Mixing Distribution in Normal Variance Mean Mixture with Application

Normal Variance-Mean Mixture (NVMM) provides a general framework for deriving models with desirable properties for modelling financial market variables such as exchange rates, equity prices, and interest rates measured over short time intervals, i.e. daily or weekly. Such data sets are characterized by non-normality and are usually skewed, fat-tailed and exhibit excess kurtosis. The Generalised Hyperbolic distribution (GHD) introduced by Barndorff-Nielsen (1977) which act as Normal variance-mean mixtures with Generalised Inverse Gaussian (GIG) mixing distribution nest a number of special and limiting case distributions. The Normal Inverse Gaussian (NIG) distribution is obtained when the Inverse Gaussian is the mixing distribution, i.e., the index parameter of the GIG is . The NIG is very popular because of its analytical tractability. In the mixing mechanism, the mixing distribution characterizes the prior information of the random variable of the conditional distribution. Therefore, considering finite mixture models is one way of extending the work. The GIG is a three parameter distribution denoted by and nest several special and limiting cases. When , we have which is called an Inverse Gaussian (IG) distribution. When , , , we have , and distributions respectively. These distributions are related to and are called weighted inverse Gaussian distributions. In this work, we consider a finite mixture of and and show that the mixture is also a weighted Inverse Gaussian distribution and use it to construct a NVMM. Due to the complexity of the likelihood, direct maximization is difficult. An EM type algorithm is provided for the Maximum Likelihood estimation of the parameters of the proposed model. We adopt an iterative scheme which is not based on explicit solution to the normal equations. This subtle approach reduces the computational difficulty of solving the complicated quantities involved directly to designing an iterative scheme based on a representation of the normal equation. The algorithm is easily programmable and we obtained a monotonic convergence for the data sets used.

Study on Enzymatic Hydrolysis of Gadus morrhua Skin Collagen and Molecular Weight Distribution of Hydrolysates

Process parameters on enzymatic hydrolysis and molecular weight （MW） distribution of collagen hydrolysates from Gadus morrhua skin were investigated. The optimal process parameters were obtained by the single-factor and orthogonal experiments. The molecular weight distribution of hydrolysates was determined using both Sephadex G25 partition and high speed liquid chromatography electricity spray mass spectrum （HPLC-ESI-MS）. Collagen hydrolysates were first gained by an alkaline protease ＂alcalase＂ for 3 h at temperature （50~C）, pH （10.0）, substrate concentration （75 g L-~）, and E/S （3%）. The molecular weight distribution of collagen hydrolysates ranged from 300 to 1 500 Da, and most of peptides were under 1 200 Da. Sephadex G25 partition and HPLC-ESI-MS should be successfully employed to determine the molecular weight distribution of collagen hydrolysates.

A HIGHLY ACTIVE NEODYMIUM CHLORIDE ISOPROPANOL COMPLEX/MODIFIED METHYLALUMINOXANE CATALYST FOR PREPARING POLYISOPRENE WITH HIGH CIS-1,4 STEREOSPECIFICITY AND NARROW MOLECULAR WEIGHT DISTRIBUTION

Neodymium chloride isopropanol complex （NdCl3.3＇prOH） activated by modified methylaluminoxane （MMAO） was examined in isoprene polymerization in hexane, with regards to Nd compounds, aluminum （Al） compounds, [Al]/[Nd] ratio, polymerization temperature and time. NdCl3＇3iprOH exhibited high activity producing polymers feasting high cis-l,4 stereospecificity （〉 96%）, very high molecular weight （Mn 〉 1.0 × 10^6） and fairly narrow molecular weight distribution （MWD, Mw/Mn 〈 2.0） simultaneously. In comparison, neodymium isopropoxide also showed high activity providing polymers with narrow MWD （Mw/Mn = 2.07）, but somewhat low cis-1,4 content （ca. 92%）, while neodymium chloride had no activity under present polymerization conditions. The Al compounds affected the polymer yield in the order of Al（i-Bu）3 〉 MMAO 〉 Al（i-Bu）2H. MMAO as cocatalyst afforded polyisoprene with high Mn over 1.0 × 10^6, whereas as stronger chain transfer agent than MMAO, AI（i-Bu）3 and AI（i-Bu）EH yielded polymers with low Mn （1.0 × 10^5-8.0 × 10^5）. NdCl3·3＇PrOH/MMAO catalyst showed a fairly good catalytic activity even at relatively low [Al]/[Nd] ratio of 30, and the produced polymer remained high cis-1,4 content of 95.8% along with high Mn over 1.0× 10^6 even at elevated temperatures up to 70℃. The polymerization rate is of the first order with respect to the concentration of isoprene. The mechanism of active species formation was discussed preliminarily.

SUPPORTED ZIEGLER-NATTA CATALYSTS FOR ETHYLENE SLURRY POLYMERIZATION AND CONTROL OF MOLECULAR WEIGHT DISTRIBUTION OF POLYETHYLENE

The effect of chemical composition of highly active supported Ziegler-Natta catalysts with controlled morphology on the MWD of PE has been studied.It was shown the variation of transition metal compound in the MgCl_2-supported catalyst affect of MWD of PE produced in broad range:Vanadium-magnesium catalyst(VMC)produce PE with broad and bimodal MWD(M_w/M_n=14-21).MWD of PE,produced over titanium-magnesium catalyst(TMC)is narrow or medium depending on Ti content in the catalyst(M_w/M_n=3.1-4.8).The oxidation ...

Molecular Weight and Distribution of Cellulose and Cellulose Microspheres

[ Objective ] This study aimed to provide basic data for studying the relationship between structure and property of cellulose microspheres by measuring molecular weight of cellulose and cellulose microspheres with viscosity method and gel permeation chromatography （GPC） method. [ Method] In viscosity method, cadmium ethylenediamine was used as the solvent, intrinsic viscosity η of the solution was determined at 25 ℃ by using a Ubbelohde viscometer, to calculate the molecular weight of cellulose; in GPC method, 8% LiC1 / N, N-dimethylacetamide （LiC1/DMAc） was used as the solvent and 0.5% LiC1/DMAc was used as the mobile phase to determine the relative molecular weight and distribution of cellulose and cellulose microspheres. In addition, the determination results were analyzed to compare these two methods. [ Result ] Viscosity-average molecular weight Mr/ of cellulose and cellulose microspheres determined with viscosity method were 224,532 and 16,686, respectively; weight-average molecular weight Mw of cellulose and cellulose microspheres determined with GPC method were 284,196 and 22,345, respectively. [ Conclusion] The determination results of （;PC method are relatively close to the actual value and could truly reflect the characteristics of molecular weialat distribution of eellulose and cellulose mierosr, heres.

Weighted average consensus problem in networks of agents with diverse time-delays

This paper studies the weighted average consensus problem for networks of agents with fixed directed asymmetric unbalance information exchange topology. We suppose that the classical distributed consensus protocol is destroyed by diverse time-delays which include communication time-delay and self time-delay. Based on the generalized Nyquist stability criterion and the Gerschgorin disk theorem, some sufficient conditions for the consensus of multi-agent systems are obtained. And we give the expression of the weighted average consensus value for our consensus protocol. Finally, numerical examples are presented to illustrate the theoretical results.

Exact Statistical Distribution of the Body Mass Index (BMI): Analysis and Experimental Confirmation

Body Mass Index (BMI), defined as the ratio of individual mass (in kilograms) to the square of the associated height (in meters), is one of the most widely discussed and utilized risk factors in medicine and public health, given the increasing obesity worldwide and its relation to metabolic disease. Statistically, BMI is a composite random variable, since human weight (converted to mass) and height are themselves random variables. Much effort over the years has gone into attempts to model or approximate the BMI distribution function. This paper derives the mathematically exact BMI probability density function (PDF), as well as the exact bivariate PDF for human weight and height. Taken together, weight and height are shown to be correlated bivariate lognormal variables whose marginal distributions are each lognormal in form. The mean and variance of each marginal distribution, together with the linear correlation coefficient of the two distributions, provide 5 nonadjustable parameters for a given population that uniquely determine the corresponding BMI distribution, which is also shown to be lognormal in form. The theoretical analysis is tested experimentally by gender against a large anthropometric data base, and found to predict with near perfection the profile of the empirical BMI distribution and, to great accuracy, individual statistics including mean, variance, skewness, kurtosis, and correlation. Beyond solving a longstanding statistical problem, the significance of these findings is that, with knowledge of the exact BMI distribution functions for diverse populations, medical and public health professionals can then make better informed statistical inferences regarding BMI and public health policies to reduce obesity.

THE RELATION OF SEQUENCE DISTRIBUTIONS OF S-SBR TO ITS MOLECULAR WEIGHT DISTRIBUTIONS

The relationship between sequence distributions and molecular weight distributions of S-SBR, obtained from styrene and butadiene anionic copolymerization at various conversions with THF/Li^+ as an initiator has been studied by ^(13)C-NMR,GPC. The results showed that the molecular weight distributions of the copolymer couldbe correlated sophisticatedly to the binary sequcne distributions or the monomer unit distributions of the copolymer in a corrected Poisson's distribution from.

Tide forecasting method based on dynamic weight distribution for operational evaluation

Through analysis of operational evaluation factors for tide forecasting, the relationship between the evaluation factors and the weights of forecasters was examined. A tide forecasting method based on dynamic weight distribution for operational evaluation was developed, and multiple-forecaster synchronous forecasting was realized while avoiding the instability cased by only one forecaster. Weights were distributed to the forecasters according to each one＇s forecast precision. An evaluation criterion for the professional level of the forecasters was also built. The eligibility rates of forecast results demonstrate the skill of the forecasters and the stability of their forecasts. With the developed tide forecasting method, the precision and reasonableness of tide forecasting are improved. The application of the present method to tide forecasting at the Huangpu Park tidal station demonstrates the validity of the method.

Effect of Molecular Weight and Molecular Distribution on Skin Structure and Shear Strength Distribution near the Surface of Thin-Wall Injection Molded Polypropylene

In this study, the relationship between skin structure and shear strength distribution of thin-wall injection molded polypropylene (PP) molded at different molecular weight and molecular distribution was investigated. Skin-core structure, cross-sectional morphology, crystallinity, crystal orientation, crystal morphology and molecular orientation were evaluated by using polarized optical microscope, differential scanning calorimeter, X-ray spectroscopic analyzer and laser Raman spectroscopy, respectively, while the shear strength distribution was investigated using a micro cutting method called SAICAS (Surface And Interfacial Cutting Analysis System). The results indicated that the difference of molecular weight and molecular weight distribution showed own skin layer thickness. Especially, high molecular weight sample showed thicker layer of the lamellar orientation and molecular orientation than low molecular weight sample. In addition, wide molecular distribution sample showed large crystal orientation layer.

The Relationship between Bulk Property and Property Distribution in Thin-Wall Injection Molded PP at Different Molecular Weight and Molecular Weight Distribution

Thin-wall injection molded parts have been paid much attention to the lightweight saving from viewpoints of natural resources saving. In the injection molding, skin-core structure can be found in the parts. This skin-core structure affects the property of completed injection molding parts (bulk property) even if in thin-wall injection molding. However, there is a few research about the relationship between bulk property and internal property distribution in the injection molding specimen. In this study, thin-wall injection molded parts of polypropylene (PP) were prepared by 4 different molecular weight and molecular weight distribution to reveal the relationship between bulk property and property distribution. These characteristics were investigated by using tensile test, fracture toughness characterized by Essential Work of Fracture (EWF) method for bulk property and film tensile test by sliced sample for tensile property distribution. The property distribution test results revealed that the highly bulk property sample had thicker highly mechanical property layer on its surface.

Exact Statistical Distribution and Correlation of Human Height and Weight: Analysis and Experimental Confirmation

The statistical relationship between human height and weight is of especial importance to clinical medicine, epidemiology, and the biology of human development. Yet, after more than a century of anthropometric measurements and analyses, there has been no consensus on this relationship. The purpose of this article is to provide a definitive statistical distribution function from which all desired statistics (probabilities, moments, and correlation functions) can be determined. The statistical analysis reported in this article provides strong evidence that height and weight in a diverse population of healthy adults constitute correlated bivariate lognormal random variables. This conclusion is supported by a battery of independent tests comparing empirical values of 1) probability density patterns, 2) linear and higher order correlation coefficients, 3) statistical and hyperstatistics moments up to 6th order, and 4) distance correlation (dCor) values to corresponding theoretical quantities: 1) predicted by the lognormal distribution and 2) simulated by use of appropriate random number generators. Furthermore, calculation of the conditional expectation of weight, given height, yields a theoretical power law that specifies conditions under which body mass index (BMI) can be a valid proxy of obesity. The consistency of the empirical data from a large, diverse anthropometric survey partitioned by gender with the predictions of a correlated bivariate lognormal distribution was found to be so extensive and close as to suggest that this outcome is not coincidental or approximate, but may be a consequence of some underlying biophysical mechanism.