Stochastic Orders Comparisons of Negative Binomial Distribution with Negative Binomial—Lindley Distribution

The purpose of this study is to compare a negative binomial distribution with a negative binomial—Lindley by using stochastic orders. We characterize the comparisons in usual stochastic order, likelihood ratio order, convex order, expectation order and uniformly more variable order based on theorem and some numerical example of comparisons between negative binomial random variable and negative binomial—Lindley random variable.

Wigner Function of Density Operator for Negative Binomial Distribution

By using the technique of integration within an ordered product （IWOP） of operator we derive Wigner function of density operator for negative binomial distribution of radiation field in the mixed state case, then we derive the Wigner function of squeezed number state, which yields negative binomial distribution by virtue of the entangled state representation and the entangled Wigner operator.

Negative Binomial Distribution Based Reputation Direct Update for Wireless Sensor Networks

In the reputation modeling of wireless sensor networks（WSNs） many literatures have proposed creative reputation indirect update methods,such as reputation integration,discounting,aging to eliminate,and filtering malicious reputation information. However,few have discussed the reputation direct update. In this paper,based on sound statistical theories,a negative binominal distribution method in the reputation direct update for WSNs is proposed. Results show that the proposed method is more suitable and time-saving for the reputation update of the resource constraint WSNs and can improve the computation power efficiency as well.

Zero-inflated non-central negative binomial distribution

In this article, the zero-inflated non-central negative binomial(ZINNB) distribution is introduced. Some of its basic properties are obtained. In addition, we use the maximum likelihood estimation method to estimate the parameters of the ZINNB distribution, and illustrate its application by fitting the actual data sets.

On a Characterization of Zero-Inflated Negative Binomial Distribution

Zero-inflated negative binomial distribution is characterized in this paper through a linear differential equation satisfied by its probability generating function.

A Characterization of the Members of a Subfamily of Power Series Distributions

This paper discusses a characterization of the members of a subfamily of power series distributions when their probability generating functions satisfy the functional equation where a, b and c are constants and is the derivative of f.

Large Deviation Results for Generalized Compound Negative Binomial Risk Models

In this paper we extend and improve some results of the large deviation for random sums of random variables. Let {Xn;n 〉 1} be a sequence of non-negative, independent and identically distributed random variables with common heavy-tailed distribution function F and finite mean μ ∈R^＋, {N（n）; n ≥0} be a sequence of negative binomial distributed random variables with a parameter p C （0, 1）, n ≥ 0, let {M（n）; n ≥ 0} be a Poisson process with intensity λ 〉 0. Suppose {N（n）; n ≥ 0}, {Xn; n≥1} and {M（n）; n ≥ 0} are mutually independent. Write S（n） =N（n）∑i=1 Xi-cM（n）.Under the assumption F ∈ C, we prove some large deviation results. These results can be applied to certain problems in insurance and finance.

Minimum Density Power Divergence Estimator for Negative Binomial Integer-Valued GARCH Models

In this paper,we study a robust estimation method for the observation-driven integervalued time-series models in which the conditional probability mass of current observations is assumed to follow a negative binomial distribution.Maximum likelihood estimator is highly affected by the outliers.We resort to the minimum density power divergence estimator as a robust estimator and showthat it is strongly consistent and asymptotically normal under some regularity conditions.Simulation results are provided to illustrate the performance of the estimator.An application is performed on data for campylobacteriosis infections.

Ruin Probability and Joint Distributions of Some Actuarial Random Vectors in the Compound Pascal Model

The compound negative binomial model, introduced in this paper, is a discrete time version. We discuss the Markov properties of the surplus process, and study the ruin probability and the joint distributions of actuarial random vectors in this model. By the strong Markov property and the mass function of a defective renewal sequence, we obtain the explicit expressions of the ruin probability, the finite-horizon ruin probability, the joint distributions of T, U（T - 1）, ｜U（T）｜ and inf U（n） （i.e., the time of ruin, the surplus immediately before ruin, the deficit at ruin and maximal deficit from ruin to recovery） and the distributions of some actuarial random vectors.

CONDITIONAL RECURSIVE EQUATIONS ON EXCESS-OF-LOSS REINSURANCE

The rharginal recursive equations on excess-of-loss reinsurance treaty are investignted, under the assumption that the number of claims belongs to the family consisting of Poisson, binomial and negative binomial, and that the severity distribution has bounded continuous density function. On conditional of the numbers of claims associated with the reinsurer and the cedent, some recursive equations are obtained for the marginal distributions of the total payments of the reinsurer and the cedent.

A spatially-explicit count data regression for modeling the density of forest cockchafer(Melolontha hippocastani) larvae in the Hessian Ried(Germany)

Background： In this paper, a regression model for predicting the spatial distribution of forest cockchafer larvae in the Hessian Ried region （Germany） is presented. The forest cockchafer, a native biotic pest, is a major cause of damage in forests in this region particularly during the regeneration phase. The model developed in this study is based on a systematic sample inventory of forest cockchafer larvae by excavation across the Hessian Ried. These forest cockchafer larvae data were characterized by excess zeros and overdispersion. Methods： Using specific generalized additive regression models, different discrete distributions, including the Poisson, negative binomial and zero-inflated Poisson distributions, were compared. The methodology employed allowed the simultaneous estimation of non-linear model effects of causal covariates and, to account for spatial autocorrelation, of a 2-dimensional spatial trend function. In the validation of the models, both the Akaike information criterion （AIC） and more detailed graphical procedures based on randomized quantile residuals were used. Results： The negative binomial distribution was superior to the Poisson and the zero-inflated Poisson distributions, providing a near perfect fit to the data, which was proven in an extensive validation process. The causal predictors found to affect the density of larvae significantly were distance to water table and percentage of pure clay layer in the soil to a depth of I m. Model predictions showed that larva density increased with an increase in distance to the water table up to almost 4 m, after which it remained constant, and with a reduction in the percentage of pure clay layer. However this latter correlation was weak and requires further investigation. The 2-dimensional trend function indicated a strong spatial effect, and thus explained by far the highest proportion of variation in larva density. Conclusions： As such the model can be used to support forest practitioners in their decision making for regeneration and forest protection planning in the Hessian predicting future spatial patterns of the larva density is still comparatively weak. Ried. However, the application of the model for somewhat limited because the causal effects are

Modeling the Browsing Behavior of World Wide Web Users

The World Wide Web is essential to general public nowadays. From a data analysis viewpoint, it provides rich opportunities to gather observational data on a large-scale. This paper focuses on modeling the behavior of visitors to an academic website. Although the conventional probability models, which were used by other literature for fitting in a commercial web site, capture the power law behavior in our data, they fail to capture other important features like the long tail. We propose a new model based on the identities of the users. Qualitative and quantitative tests, which are used for comparing the model fitting to our data, show that the new model outperforms other two conventional probability models.

Some Aspects of Multiparticle Production in Relativistic Nuclear Collisions

An attempt is made to study various types of multiplicity distributions and multiplicity correlations amongst secondary charged particles produced in 4.5 and 14.5 A GeV/c 28Si-nucleus interactions. The results reveal that multiplicity correlations of the type , where i, j = b, g, s and h with i ≠ j, are linear. The observed multiplicity correlations are nicely fitted by the method of least squares fitting of the type, = b + ani. Furthermore, the multiplicity distributions of relativistic charged particles and compound multiplicity at 4.5 A GeV/c are nicely fitted by Poisson distributions with peaks at relatively lower multiplicities. However, in the case of multiplicity distributions of various types of secondary charged particles produced in 14.5 A GeV/c 28Si-nucleus collisions definite trends are not discernible.

Statistical Data Analyses on Aircraft Accidents in Japan: Occurrences, Causes and Countermeasures

We investigate the major characteristics of the occurrences, causes of and counter measures for aircraft accidents in Japan. We apply statistical data analysis and mathematical modeling techniques to determine the relations among economic growth, aviation demand, the frequency of aircraft/helicopter accidents, the major characteristics of the occurrence intervals of accidents, and the number of fatalities due to accidents. The statistical model analysis suggests that the occurrence intervals of accidents and the number of fatalities can be explained by probability distributions such as the exponential distribution and the negative binomial distribution, respectively. We show that countermeasures for preventing accidents have been developed in every aircraft model, and thus they have contributed to a significant decrease in the number of accidents in the last three decades. We find that the major cause of accidents involving large airplanes has been weather, while accidents involving small airplanes and helicopters are mainly due to the pilot error. We also discover that, with respect to accidents mainly due to pilot error, there is a significant decrease in the number of accidents due to the aging of airplanes, whereas the number of accidents due to weather has barely declined. We further determine that accidents involving small and large airplanes mostly occur during takeoff and landing, whereas those involving helicopters are most likely to happen during flight. In order to decrease the number of accidents, i) enhancing safety and security by further developing technologies for aircraft, airports and air control radars, ii) establishing and improving training methods for crew including pilots, mechanics and traffic controllers, iii) tightening public rules, and iv) strengthening efforts made by individual aviation-related companies are absolutely necessary.

Weather Impact on Heat-Related Illness in a Tropical City State, Singapore

In this article we propose a novel hurdle negative binomial (HNB) regression combined with a distributed lag nonlinear model (DLNM) to model weather factors’ impact on heat related illness (HRI) in Singapore. AIC criterion is adopted to help select proper combination of weather variables and check their lagged effect as well as nonlinear effect. The process of model selection and validation is demonstrated. It is observed that the predicted occurrence rate is close to the observed one. The proposed combined model can be used to predict HRI cases for mitigating HRI occurrences and provide inputs for related public health policy considering climate change impact.

Slow Particle Production in Nucleus-Nucleus Collisions at Relativistic Energies

In this paper an effort has been made to study the general characteristics of slow particles produced in the interactions of 32S-Em at 200 AGeV to extract the information about the mechanism of particle production. The results have been compared with the experimental results obtained by other workers. The multiplicity distributions of the slow target associated particles (black, grey and heavy tracks) produced by 32S-beam with different targets have been studied. Also several types of correlations among them have been investigated. The variation of the produced particles with projectile mass number and target size has been studied. Also the multiplicity distributions of slow particles with NBD fits are presented and scaling multiplicity distributions of slow particles produced have been studied in order to check the validity of KNO-scaling.

Energy-Saving Reputation Method for Wireless Sensor Networks

Reputation mechanism has been recently introduced into wireless sensor networks(WSNs) to overcome the deficiencies incurred by cryptography alone.Unlike most current reputation mechanisms that are based on binomial distribution and to some extent,do not give due attention to the aspect of energy constraint of WSNs,this research deduces and proves the feasibility that negative binomial distribution can well be used in the reputation update with distinctive energy-saving features.Comparison tests with respect to the energy consumption in terms of reputation computing frequencies are done between the traditional reputation method and the one in this study.Results show that our method can save more energy for the reputation update and thus can better meet the power constraints of WSNs.

Approximation for counts of head runs

In this paper, we study the count of head runs up to a fixed time in a two-state stationary Markov chain. We prove that in total variance distance, the negative binomial, Poisson and binomial distributions are appropriate approximations according to the relation of the variance and mean of the count, generalizing earlier results in previous literatures. The proof is based on Stein's method and coupling.

The Gamma/Weibull Customer Lifetime Model

This paper proposes a new customer lifetime model:the Gamma/Weibull distribution(G/W).Similar to the Pareto/NBD model,we propose a G/W/NBD model by combining the G/W distribution with a negative binomial distribution(NBD)and study its properties such as(i)the probability that a customer to be alive at a time point;(ii)the expectation and variance of the number of transactions for a customer during a fixed time period;(iii)the conditional expectation and conditional variance of the number of future transactions for a customer during a fixed time period.Several simulation studies are conducted to investigate the forecasting accuracy and flexibility of the proposed model.A CDNOW data set is analyzed by the proposed model.

Approximation for Counts of 2-runs in a Two State Markov Chain

We consider a Markov chain X = {Xi, i = 1, 2,...} with the state space {0, 1}, and define W = ∑1=1^x XiXi＋1, which is the number of 2-runs in X before time n ＋ 1. In this paper, we prove that the negative binomial distribution is an appropriate approximation for LW when VarW is greater than EW. The error estimate obtained herein improves the corresponding result in previous literatures.