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.

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.

Spatial distribution of snow depth based on geographically weighted regression kriging in the Bayanbulak Basin of the Tianshan Mountains, China

Snow depth is a general input variable in many models of agriculture,hydrology,climate and ecology.This study makes use of observational data of snow depth and explanatory variables to compare the accuracy and effect of geographically weighted regression kriging(GWRK)and regression kriging(RK)in a spatial interpolation of regional snow depth.The auxiliary variables are analyzed using correlation coefficients and the variance inflation factor(VIF).Three variables,Height,topographic ruggedness index(TRI),and land surface temperature(LST),are used as explanatory variables to establish a regression model for snow depth.The estimated spatial distribution of snow depth in the Bayanbulak Basin of the Tianshan Mountains in China with a spatial resolution of 1 km is obtained.The results indicate that 1)the result of GWRK's accuracy is slightly higher than that of RK(R^2=0.55 vs.R^2=0.50,RMSE(root mean square error)=0.102 m vs.RMSE=0.077 m);2)for the subareas,GWRK and RK exhibit similar estimation results of snow depth.Areas in the Bayanbulak Basin with a snow depth greater than 0.15m are mainly distributed in an elevation range of 2632.00–3269.00 m and the snow in this area comprises 45.00–46.00% of the total amount of snow in this basin.However,the GWRK resulted in more detailed information on snow depth distribution than the RK.The final conclusion is that GWRK is better suited for estimating regional snow depth distribution.

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.

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.

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.

Analysis of Molecular Weight Distribution and Antioxidant Activities of Cirrhinus molitorella Skin Collagen Hydrolysates

[Objective] The aim of this work was to identify molecular weight （MW） distribution and antioxidant activity of fish skin col agen hydrolysates. [Method] The MW distribution of hydrolysates was determined using both size exclusion chromatography and matrix-assisted laser desorption ionization time-of-flight mass spec-trometry （MALDI-TOF-MS）. Fish skin were treated by the alkaline protease 2709. [Result] The optional conditions for hyerolysis were time 3 h, temperature 55 ℃, pH 10.0, substrate concentration 80 g/L and E/S 4%. The results of both methods indi-cated that the molecular weight of col agen hydrolysates was from 400 to 1 800 Da, and the peptides’ molecular weight was less than 1 400 Da mostly. The reducing power and antioxidant/radical scavenging activity [1, 1-diphenyl-2-picryl-hydrazyl （DPPH） free radical scavenging activity] were determined. [Conclusion] The results reveal that the fishskin hydrolysate is a potential source of antioxidants.

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.

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.

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 ...

Study on the distribution of active centers in novel low Ti-loading MgCl2-supported Ziegler-Natta catalyst

Novel MgCl2-supported Ziegler-Natta (Z-N) catalysts prepared using a new one-pot ball milling method can effectively control the amounts of Ti-loading in the catalysts. Complex GPC data on polypropylene synthesized by these novel catalysts were analyzed using the method of fitting the molecular weight distribution (MWD) curves with a multiple Flory-Schulz function. It was found that multiple active centers exist in these novel catalysts. Detailed study of the effects of the Ti-loadings in the catalysts on the distribution of the active centers showed that the Ti-loadings in the novel MgCl2-supported Z-N catalysts might affect the proportion of each type of active centers; and might be the main factor responsible for the effect of the Ti-loadings on the microstructure, the molecular weight and molecular weight distribution width of the resultant polymer, the catalytic activity and polymerization kinetics.

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.

Development of a mass model in estimating weight-wise particle size distribution using digital image processing

Particle size distribution of coarse aggregates through mechanical sieving gives results in terms of cumu- lative mass percent. But digital image processing generated size distribution of particles, while being fast and accurate, is often expressed in terms of area function or number of particles. In this paper, a mass model is developed which converts the image obtained size distribution to mass-wise distribution, mak- ing it readily comparable to mechanical sieving data. The concept of weight/particle ratio is introduced for mass reconstruction from 2D images of particle aggregates. Using this mass model, the effects of several particle shape parameters （such as major axis, minor axis, and equivalent diameter） on sieve-size of the particles is studied. It is shown that the sieve-size of a particle strongly depend upon the shape param- eters, 91% of its variation being explained by major axis, minor axis, bounding box length and equivalent diameter. Furthermore, minor axis gives an overall accurate estimate of particle sieve-size, error in mean size （D-50） being just 0.4%. However, sieve-size of smaller particles （〈20 ram） strongly depends upon the length of the smaller arm of the bounding box enclosing them and sieve-sizes of larger particles （〉20 mm） are highly correlated to their equivalent diameters. Multiple linear regression analysis has been used to generate overall mass-wise particle size distribution, considering the influences of all these shape parameters on particle sieve-size. Multiple linear regression generated overall mass-wise particle size distribution shows a strong correlation with sieve generated data. The adjusted R-square value of the regression analysis is found to be 99 percent （w.r,t cumulative frequency）. The method proposed in this paper provides a time-efficient way of producing accurate （up to 99%） mass-wise PSD using digital image processing and it can be used effectively to renlace the mechanical sieving.

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.

Molecular Weight Distribution of Polystyrene Produced in a Starved Feed Reactor

A starved feed reactor (SFR) is a semi-batch polymerization reactor where initiator and monomer are fed slowly into a fixed amount of solvent. The polymerization is carried out isothermally at elevated temperatures. The added initiator decomposes instantaneously and the added monomer polymerizes immediately. The molecular weight (MW) and molecular weight distribution (MWD) of the product polymer can be effectively controlled by the feed ratio of monomer to initiator. This paper presents a study on the MWD of styrene polymerization in a SFR. The MWD model parameters are regressed with experimental data. Although the solids fraction in the SFR is high (higher than 50%), viscosity is not too high and the 'gel effect' is weak due to the low molecular weight of the products. It is found that the termination rate constant is a power function of molecular weight, radicals terminate via 100% combination, the thermal initiation can be neglected even at high reaction temperature studied. And calculated results indicate that in the SFR, the validity of the long chain assumption becomes doubted. It appears that other alterative assumption should be found for an improved model.