On computing maximum likelihood estimates for the negative binomial distribution

2019 ◽  
Vol 148 ◽  
pp. 54-58 ◽  
Author(s):  
Udika Bandara ◽  
Ryan Gill ◽  
Riten Mitra
Keyword(s):  
Maximum Likelihood ◽  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Negative Binomial ◽  
Maximum Likelihood Estimates

scholarly journals A Bivariate Model based on Compound Negative Binomial Distribution

2018 ◽  
Vol 41 (1) ◽  
pp. 87-108 ◽  
Author(s):  
Maha Ahmad Omair ◽  
Fatimah E AlMuhayfith ◽  
Abdulhamid A Alzaid
Keyword(s):  
Maximum Likelihood ◽  
Method Of Moments ◽  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Traffic Accidents ◽  
Negative Binomial ◽  
Conditional Distributions ◽  
Bivariate Model ◽  
Distributional Properties ◽  
Parameter Estimators

A new bivariate model is introduced by compounding negative binomial and geometric distributions. Distributional properties, including joint, marginal and conditional distributions are discussed. Expressions for the product moments, covariance and correlation coefficient are obtained. Some properties such as ordering, unimodality, monotonicity and self-decomposability are studied. Parameter estimators using the method of moments and maximum likelihood are derived. Applications to traffic accidents data are illustrated.

Modeling earthquake numbers by Negative Binomial Hidden Markov models

2020 ◽  
Author(s):  
Katerina Orfanogiannaki ◽  
Dimitris Karlis
Keyword(s):  
Poisson Distribution ◽  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Negative Binomial ◽  
Time Window ◽  
Hidden Markov ◽  
Maximum Likelihood Estimates ◽  
Ongoing Research ◽  
Real Earthquake ◽  
Earthquake Data

<p>Modeling seismicity data is challenging and it remains a subject of ongoing research. Assumptions about the distribution of earthquake numbers play an important role in seismic hazard and risk analysis. The most common distribution that has been widely used in modeling earthquake numbers is the Poisson distribution because of its simplicity and easy to use. However, the heterogeneity in earthquake data and temporal dependencies that are often present in many real earthquake sequences make the use of the Poisson distribution inadequate. So, we propose the use of a Hidden Markov model (HMM) with state-specific Negative Binomial distributions in which some states are allowed to approach the Poisson distribution. A HMM is a generalization of a mixture model where the different unobservable (hidden) states are related through a Markov process rather than being independent of each other. We parameterize the Negative Binomial distribution in terms of the mean and dispersion (clustering) parameter. Maximum likelihood estimates of the models’ parameters are obtained through an Expectation-Maximization algorithm (EM-algorithm).</p><p>We apply the model to real earthquake data. We have selected the area of Killini Western Greece to test the proposed hypothesis. The area of Killini has been selected based on the fact that in a time window of 17 years three clusters of seismicity associated with strong mainshocks are included in the catalog. Application of the model to the data resulted in three states, representing different levels of seismicity (low, medium, high). The state that corresponds to the low seismicity level approaches the Poisson distribution while the other two states (medium and high) are following the Negative Binomial distribution. This result complies with the nature of the data. The variation within each state that is introduced to the model by the Negative Binomial distribution is greater in the states of medium and high seismicity. </p>

scholarly journals The Influence of the franchise on the number of claims in Motor Insurance

1972 ◽  
Vol 6 (3) ◽  
pp. 191-194 ◽  
Author(s):  
Mogens Muff
Keyword(s):  
Mathematical Model ◽  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Negative Binomial ◽  
Maximum Likelihood Estimates ◽  
Small Claims ◽  
Group A ◽  
One Year ◽  
Group B ◽  
Image Position

1. If you introduce a franchise or raise its amount, the number of claims will decrease. One reason why you will operate a franchise in car damage insurance is to avoid the troublesome and expensive handling of the large number of very small claims.Therefore, the management could use a mathematical model describing the number of claims as related to the size of the franchise. By varying this figure one may find the franchise that provides for the optimal business conditions.2. The mathematical model to be described in this paper was found by studying the distribution of the claims in two portfolios. The general conditions of the two groups were as equal as possible, there was no-claim bonus in neither group, but group B had a franchise of 250 DKr. (about US $ 35) while group A covered the smaller claims also. Each policy had been unaltered for at least one year. The number of claims during 12 successive months was registered as follows:The distribution could be described by a negative binomial distributionwhere the parameters α and γ were estimated byThese estimates are not maximum likelihood estimates but it was not regarded necessary to improve the estimation as the χ2 values of the incomplete χ2-test were found asThe parameters were estimated as follows:3. Special interest is attached to the variations of the γ-parameter. It is well known, that if γ → ∞ and α → ∞ under the condition then the negative binomial distribution will converge towards a Poisson distribution with the parameter λ.

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