A characterization of the gamma distribution by the negative binomial distribution

1980 ◽  
Vol 17 (4) ◽  
pp. 1138-1144 ◽  
Author(s):  
Jan Engel ◽  
Mynt Zijlstra
Keyword(s):  
Poisson Process ◽  
Exponential Distribution ◽  
Gamma Distribution ◽  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Negative Binomial ◽  
Random Variable ◽  
One To One

It is proved that for a Poisson process there exists a one-to-one relation between the distribution of the random variable N(Y) and the distribution of the non-negative random variable Y. This relation is used to characterize the gamma distribution by the negative binomial distribution. Furthermore it is applied to obtain some characterizations of the exponential distribution.

A characterization of the gamma distribution by the negative binomial distribution

1980 ◽  
Vol 17 (04) ◽  
pp. 1138-1144 ◽  
Author(s):  
Jan Engel ◽  
Mynt Zijlstra
Keyword(s):  
Poisson Process ◽  
Exponential Distribution ◽  
Gamma Distribution ◽  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Negative Binomial ◽  
Random Variable ◽  
One To One

It is proved that for a Poisson process there exists a one-to-one relation between the distribution of the random variable N(Y) and the distribution of the non-negative random variable Y. This relation is used to characterize the gamma distribution by the negative binomial distribution. Furthermore it is applied to obtain some characterizations of the exponential distribution.

scholarly journals KAJIAN VARIANCE MEAN RATIO PADA SIMULASI SEBARAN DATA BINOMIAL NEGATIF

2020 ◽  
Vol 4 (4) ◽  
pp. 615-626
Author(s):  
Choirun Nisa ◽  
Muhammad Nur Aidi ◽  
I Made Sumertajaya
Keyword(s):  
Data Collection ◽  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Negative Binomial ◽  
Data Distribution ◽  
P Value ◽  
Binomial Data ◽  
Simulation Data ◽  
Count Distribution

The negative binomial distribution is one of the data collection counts that focuses on success and failure events. This study conducted a study of the distribution of negative binomial data to determine the characterization of the distribution based on the value of Variance Mean Ratio (VMR). Simulation data are generated based on negative binomial distribution with a combination of p and n parameters. The results of the VMR study on negative binomial distribution simulation data show that the VMR value will be smaller when the p-value is large and the VMR value is more stable as the sample size increases. Simulation data of negative binomial distribution when p≥0.5 begins to change data distribution to the distribution of Poisson and binomial. The calculation VMR value can be used as a reference for detecting patterns of data count distribution.

scholarly journals On A Model for the Claim Number Process

1988 ◽  
Vol 18 (1) ◽  
pp. 57-68 ◽  
Author(s):  
Matti Ruohonen
Keyword(s):  
Structure Function ◽  
Poisson Process ◽  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Negative Binomial ◽  
Credibility Theory ◽  
Gamma Model ◽  
Function Fitting ◽  
Claim Number ◽  
Two Parameter

AbstractA model for the claim number process is considered. The claim number process is assumed to be a weighted Poisson process with a three-parameter gamma distribution as the structure function. Fitting of this model to several data encountered in the literature is considered, and the model is compared with the two-parameter gamma model giving the negative binomial distribution. Some credibility theory formulae are also presented.

scholarly journals On a Characterization of Zero-Inflated Negative Binomial Distribution

2015 ◽  
Vol 05 (06) ◽  
pp. 511-513
Author(s):  
R. Suresh ◽  
G. Nanjundan ◽  
S. Nagesh ◽  
Sadiq Pasha
Keyword(s):  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Negative Binomial

scholarly journals Testing the odds of inherent vs. observed overdispersion in neural spike counts

2016 ◽  
Vol 115 (1) ◽  
pp. 434-444 ◽  
Author(s):  
Wahiba Taouali ◽  
Giacomo Benvenuti ◽  
Pascal Wallisch ◽  
Frédéric Chavane ◽  
Laurent U. Perrinet
Keyword(s):  
Poisson Process ◽  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Negative Binomial ◽  
Poisson Model ◽  
Probabilistic Methods ◽  
Distribution Model ◽  
Data Sets ◽  
Electrophysiological Recordings ◽  
Neurophysiological Data

The repeated presentation of an identical visual stimulus in the receptive field of a neuron may evoke different spiking patterns at each trial. Probabilistic methods are essential to understand the functional role of this variance within the neural activity. In that case, a Poisson process is the most common model of trial-to-trial variability. For a Poisson process, the variance of the spike count is constrained to be equal to the mean, irrespective of the duration of measurements. Numerous studies have shown that this relationship does not generally hold. Specifically, a majority of electrophysiological recordings show an “overdispersion” effect: responses that exhibit more intertrial variability than expected from a Poisson process alone. A model that is particularly well suited to quantify overdispersion is the Negative-Binomial distribution model. This model is well-studied and widely used but has only recently been applied to neuroscience. In this article, we address three main issues. First, we describe how the Negative-Binomial distribution provides a model apt to account for overdispersed spike counts. Second, we quantify the significance of this model for any neurophysiological data by proposing a statistical test, which quantifies the odds that overdispersion could be due to the limited number of repetitions (trials). We apply this test to three neurophysiological data sets along the visual pathway. Finally, we compare the performance of this model to the Poisson model on a population decoding task. We show that the decoding accuracy is improved when accounting for overdispersion, especially under the hypothesis of tuned overdispersion.

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