scholarly journals On the Analysis of the Truncated Generalized Poisson Distribution Using a Bayesian Method

1998 ◽  
Vol 28 (1) ◽  
pp. 135-152 ◽  
Author(s):  
David P.M. Scollnik
Keyword(s):  
Poisson Distribution ◽  
Bayesian Method ◽  
Mass Function ◽  
Probability Mass Function ◽  
Generalized Poisson Distribution ◽  
Generalized Poisson ◽  
Mean And Variance ◽  
Probability Mass ◽  
The Mean ◽  
Definition Of

AbstractThe generalized Poisson distribution with parameters θ and λ was introduced by Consul and Jain (1973) and has recently found several instances of application in the actuarial literature. The most frequently used version of the distribution assumes that θ > 0 and 0 ≤ λ < 1, in which case the mean and variance are θ/(1 − λ) and θ/(1 − λ)3, respectively. These simple moment expressions, along with nearly all of the other theoretical results available for this distribution, fail when λ < 0 or λ > 1 (e.g., Johnson, Kotz, and Kemp, 1992, page 397). In these cases, even the definition of the probability mass function usually given in the literature is not properly normalized so that its values do not sum to unity. For this reason, it is common to truncate the support of the distribution and explicitly normalize the probability mass function. In this paper we discuss the estimation of the parameters of this truncated generalized Poisson distribution using a Bayesian method.

Analysis of generalized Poisson distributions associated with higher Landau levels

2015 ◽  
Vol 18 (04) ◽  
pp. 1550028 ◽  
Author(s):  
Nizar Demni ◽  
Zouhair Mouayn
Keyword(s):  
Probability Distribution ◽  
Characteristic Function ◽  
Poisson Distribution ◽  
Landau Level ◽  
Mass Function ◽  
Probability Mass Function ◽  
Lévy Measure ◽  
Infinitely Divisible ◽  
Generalized Poisson ◽  
Probability Mass

To a higher Landau level corresponds a generalized Poisson distribution arising from generalized coherent states. In this paper, we write down the atomic decomposition of this probability distribution and express its probability mass function as a [Formula: see text]-hypergeometric polynomial. Then, we prove that it is not infinitely divisible in contrast with the Poisson distribution corresponding to the lowest Landau level. We also derive a Lévy–Khintchine-type representation of its characteristic function when the latter does not vanish and deduce that the representative measure is a quasi-Lévy measure. By considering the total variation of this last measure, we obtain the characteristic function of a new infinitely divisible discrete probability distribution for which we also compute the probability mass function.

scholarly journals A finite form for the wrapped Poisson distribution

1992 ◽  
Vol 24 (01) ◽  
pp. 221-222 ◽  
Author(s):  
Frank Ball ◽  
Paul Blackwell
Keyword(s):  
Poisson Distribution ◽  
Mass Function ◽  
Probability Mass Function ◽  
Finite Form ◽  
Probabilistic Proof ◽  
Probability Mass

We give a finite form for the probability mass function of the wrapped Poisson distribution, together with a probabilistic proof. We also describe briefly its connection with existing results.

scholarly journals A finite form for the wrapped Poisson distribution

1992 ◽  
Vol 24 (1) ◽  
pp. 221-222 ◽  
Author(s):  
Frank Ball ◽  
Paul Blackwell
Keyword(s):  
Poisson Distribution ◽  
Mass Function ◽  
Probability Mass Function ◽  
Finite Form ◽  
Probabilistic Proof ◽  
Probability Mass

We give a finite form for the probability mass function of the wrapped Poisson distribution, together with a probabilistic proof. We also describe briefly its connection with existing results.

A study on the injective coloring parameters of certain graphs

2016 ◽  
Vol 08 (03) ◽  
pp. 1650052 ◽  
Author(s):  
N. K. Sudev ◽  
K. P. Chithra ◽  
S. Satheesh ◽  
Johan Kok
Keyword(s):  
Graph Coloring ◽  
Random Variable ◽  
Mass Function ◽  
Probability Mass Function ◽  
Proper Coloring ◽  
Discrete Random Variable ◽  
Injective Coloring ◽  
Random Experiment ◽  
Mean And Variance ◽  
Probability Mass

Coloring the vertices of a graph [Formula: see text] according to certain conditions can be considered as a random experiment and a discrete random variable (r.v.) [Formula: see text] can be defined as the number of vertices having a particular color in the proper coloring of [Formula: see text] and a probability mass function for this random variable can be defined accordingly. In this paper, we extend the concepts of mean and variance to a modified injective graph coloring and determine the values of these parameters for a number of standard graphs.

On certain parameters of equitable coloring of graphs

2017 ◽  
Vol 09 (04) ◽  
pp. 1750054 ◽  
Author(s):  
N. K. Sudev ◽  
K. P. Chithra ◽  
S. Satheesh ◽  
Johan Kok
Keyword(s):  
Random Variable ◽  
Mass Function ◽  
Probability Mass Function ◽  
Proper Coloring ◽  
Discrete Random Variable ◽  
Statistical Parameters ◽  
Equitable Coloring ◽  
Mean And Variance ◽  
Probability Mass ◽  
The Given

Coloring the vertices of a graph [Formula: see text] according to certain conditions is a random experiment and a discrete random variable [Formula: see text] is defined as the number of vertices having a particular color in the given type of coloring of [Formula: see text] and a probability mass function for this random variable can be defined accordingly. An equitable coloring of a graph [Formula: see text] is a proper coloring [Formula: see text] of [Formula: see text] which an assignment of colors to the vertices of [Formula: see text] such that the numbers of vertices in any two color classes differ by at most one. In this paper, we extend the concepts of arithmetic mean and variance, the two major statistical parameters, to the theory of equitable graph coloring and hence determine the values of these parameters for a number of standard graphs.

scholarly journals Modified Recursions for a Class of Compound Distributions

1996 ◽  
Vol 26 (2) ◽  
pp. 213-224 ◽  
Author(s):  
Karl-Heinz Waldmann
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Mass Function ◽  
Cumulative Distribution ◽  
Probability Mass Function ◽  
Initial Value ◽  
Compound Distributions ◽  
Claim Frequency ◽  
Probability Mass ◽  
Stop Loss

AbstractRecursions are derived for a class of compound distributions having a claim frequency distribution of the well known (a,b)-type. The probability mass function on which the recursions are usually based is replaced by the distribution function in order to obtain increasing iterates. A monotone transformation is suggested to avoid an underflow in the initial stages of the iteration. The faster increase of the transformed iterates is diminished by use of a scaling function. Further, an adaptive weighting depending on the initial value and the increase of the iterates is derived. It enables us to manage an arbitrary large portfolio. Some numerical results are displayed demonstrating the efficiency of the different methods. The computation of the stop-loss premiums using these methods are indicated. Finally, related iteration schemes based on the cumulative distribution function are outlined.

Bayesian Estimation of Stimulus Responses in Poisson Spike Trains

2004 ◽  
Vol 16 (7) ◽  
pp. 1325-1343 ◽  
Author(s):  
Sidney R. Lehky
Keyword(s):  
Bayesian Method ◽  
Significant Degree ◽  
Spike Trains ◽  
Neural Responses ◽  
Interval Estimates ◽  
Poisson Statistics ◽  
Likelihood Functions ◽  
Bayesian Estimates ◽  
Mean And Variance ◽  
The Mean

A Bayesian method is developed for estimating neural responses to stimuli, using likelihood functions incorporating the assumption that spike trains follow either pure Poisson statistics or Poisson statistics with a refractory period. The Bayesian and standard estimates of the mean and variance of responses are similar and asymptotically converge as the size of the data sample increases. However, the Bayesian estimate of the variance of the variance is much lower. This allows the Bayesian method to provide more precise interval estimates of responses. Sensitivity of the Bayesian method to the Poisson assumption was tested by conducting simulations perturbing the Poisson spike trains with noise. This did not affect Bayesian estimates of mean and variance to a significant degree, indicating that the Bayesian method is robust. The Bayesian estimates were less affected by the presence of noise than estimates provided by the standard method.

Sign in / Sign up

Export Citation Format

Share Document