On the empty cells of Poisson histograms

1993 ◽  
Vol 30 (3) ◽  
pp. 561-574
Author(s):  
Wilfrid S. Kendall
Keyword(s):  
Cell Size ◽  
Poisson Distribution ◽  
Finite Groups ◽  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Empirical Bayes ◽  
Negative Binomial ◽  
Almost Sure Convergence ◽  
Unit Cell Size ◽  
Independent Observations

This paper considers the histogram of unit cell size built up from m independent observations on a Poisson (μ) distribution. The following question is addressed: what is the limiting probability of the event that there are no unoccupied cells lying to the left of occupied cells of the histogram? It is shown that the probability of there being no such isolated empty cells (or isolated finite groups of empty cells) tends to unity as the number m of observations tends to infinity, but that the corresponding almost sure convergence fails. Moreover this probability does not tend to unity when the Poisson distribution is replaced by the negative binomial distribution arising when μ is randomized by a gamma distribution. The relevance to empirical Bayes statistical methods is discussed.

On the empty cells of Poisson histograms

1993 ◽  
Vol 30 (03) ◽  
pp. 561-574
Author(s):  
Wilfrid S. Kendall
Keyword(s):  
Cell Size ◽  
Poisson Distribution ◽  
Finite Groups ◽  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Empirical Bayes ◽  
Negative Binomial ◽  
Almost Sure Convergence ◽  
Unit Cell Size ◽  
Independent Observations

This paper considers the histogram of unit cell size built up from m independent observations on a Poisson (μ) distribution. The following question is addressed: what is the limiting probability of the event that there are no unoccupied cells lying to the left of occupied cells of the histogram? It is shown that the probability of there being no such isolated empty cells (or isolated finite groups of empty cells) tends to unity as the number m of observations tends to infinity, but that the corresponding almost sure convergence fails. Moreover this probability does not tend to unity when the Poisson distribution is replaced by the negative binomial distribution arising when μ is randomized by a gamma distribution. The relevance to empirical Bayes statistical methods is discussed.

Patterns of macroparasite aggregation in wildlife host populations

Parasitology ◽  
1998 ◽  
Vol 117 (6) ◽  
pp. 597-610 ◽  
Author(s):  
D. J. SHAW ◽  
B. T. GRENFELL ◽  
A. P. DOBSON
Keyword(s):  
Poisson Distribution ◽  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Goodness Of Fit ◽  
Negative Binomial ◽  
Data Sets ◽  
Frequency Distributions ◽  
System A ◽  
Host Sex ◽  
Host Parasite

Frequency distributions from 49 published wildlife host–macroparasite systems were analysed by maximum likelihood for goodness of fit to the negative binomial distribution. In 45 of the 49 (90%) data-sets, the negative binomial distribution provided a statistically satisfactory fit. In the other 4 data-sets the negative binomial distribution still provided a better fit than the Poisson distribution, and only 1 of the data-sets fitted the Poisson distribution. The degree of aggregation was large, with 43 of the 49 data-sets having an estimated k of less than 1. From these 49 data-sets, 22 subsets of host data were available (i.e. host data could be divided by either host sex, age, where or when hosts were sampled). In 11 of these 22 subsets there was significant variation in the degree of aggregation between host subsets of the same host–parasite system. A common k estimate was always larger than that obtained with all the host data considered together. These results indicate that lumping host data can hide important variations in aggregation between hosts and can exaggerate the true degree of aggregation. Wherever possible common k estimates should be used to estimate the degree of aggregation. In addition, significant differences in the degree of aggregation between subgroups of host data, were generally associated with significant differences in both mean parasite burdens and the prevalence of infection.

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>

The distribution and climbing behaviour of Rhipicephalus appendiculatus Neumann on grass stems

1985 ◽  
Vol 6 (2) ◽  
pp. 213-215 ◽  
Author(s):  
J. W. Chiera
Keyword(s):  
Poisson Distribution ◽  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Negative Binomial ◽  
Adult Tick ◽  
Rhipicephalus Appendiculatus ◽  
Setaria Sphacelata ◽  
Climbing Behaviour ◽  
Marked Preference ◽  
Tick Counts

AbstractThe distribution of Rhipicephalus appendiculatus on Setaria sphacelata flowerheads was investigated. Adult tick counts in a paddock, heavily infested with ticks for experimental purposes, revealed an aggregated distribution by deviating significantly from the expected Poisson distribution, but producing a good fit to the expected negative binomial distribution. A higher proportion of the ticks were on old dry flowerheads than on young green ones. Tests carried out in the laboratory confirmed the marked preference for old flowerheads and the aggregated pattern of distribution. The tick counts in the field revealed no sexual aggregation, though the total number of females collected was significantly higher than that of the males.

scholarly journals KAJIAN SIMULASI OVERDISPERSI PADA REGRESI POISSON DAN BINOMIAL NEGATIF TERBOBOTI GEOGRAFIS UNTUK DATA BALITA GIZI BURUK

2020 ◽  
Vol 4 (3) ◽  
pp. 484-497
Author(s):  
Puput Cahya Ambarwati ◽  
Indahwati Indahwati ◽  
Muhammad Nur Aidi
Keyword(s):  
Poisson Distribution ◽  
Spatial Data ◽  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Negative Binomial ◽  
Negative Binomial Regression ◽  
Real Data ◽  
P Value ◽  
Simulation Data ◽  
Response Variable

Geographic weighted regression (GWR) is one of the regression methods for spatial data. GWR with the response variable following the poisson distribution can use the geographic weighted poisson regression (GWPR). GWPR often does not complete the assumption of dispersion. The classic approach commonly used to overcome overdispersion is related to poisson distribution, which is the approach obtained from poisson and gamma distribution which is similar to negative binomial distribution function. GWR for the response variable following the negative binomial distribution can use the geographical weighted negative binomial regression (GWNBR). The data used in this study are simulation data and real data. The results of the simulation data are the tolerance limits that are still precisely modeled with GWPR are overdispersion approaching 1 based on significant amount and average p-value.. The results of research from real data, the GWNBR is the best model for overdispersion cases in malnourished children in East Java Province in 2017 compared to the GWPR based on comparison of the values ​​of AIC. 

Probabilistic models to analyze psychiatric contacts

1999 ◽  
Vol 8 (1) ◽  
pp. 47-55 ◽  
Author(s):  
Luisa Canal ◽  
Rocco Micciolo
Keyword(s):  
Poisson Distribution ◽  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Degrees Of Freedom ◽  
Probabilistic Models ◽  
Negative Binomial ◽  
Case Register ◽  
Psychiatric Case Register ◽  
Psychiatric Case ◽  
Waring Distribution

SummaryObjective – To present three probabilistic models (Poisson, negative binomial, Waring) to analyze the distribution of the number of contacts of patients followed using a psychiatric case register. Design – Longitudinal to obtain the distribution of the number of contacts (during 91 days following that of the first contact) observed on patients followed using the South-Verona Psychiatric Case Register during the period 1/1/79-31/12/91. Results – There were a total number of 6913 contacts on 3454 subjects. The chi-square test for the goodness of fit yields a significant result both for the Poisson distribution (3580 with 6 degrees of freedom, p < 0.001) and for the negative binomial distribution (65.47 with 18 degrees of freedom, p < 0.001); on the other hand a non significant result was obtained for the Waring distribution (25.31 with 19 degrees of freedom, p = 0.151). Conclusions – The Poisson distribution gave a very poor fit for the distribution of contacts. The negative binomial distribution could be employed to analyze the pattern of contacts when the right tail of the distribution is not important. The Waring distribution is the best of the three presented. Moreover, the variance of the Waring distribution can be decomposed in three components: a random component, a component which accounts for endogenous factors and another component which accounts for esogenous factors. Therefore the Waring distribution is useful when one wants to make comparisons between psychiatric case registers of the same country or of different countries.

Niche Sharing in Intertidal Mollusks and Decapods in Rocky Shore of Easter Island

2019 ◽  
Vol 53 (5) ◽  
pp. 417-422
Author(s):  
P. De los Ríos ◽  
E. Ibáñez Arancibia
Keyword(s):  
Spatial Distribution ◽  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Rocky Shore ◽  
Negative Binomial ◽  
Null Model ◽  
Easter Island ◽  
Published Data ◽  
Distribution Analysis ◽  
Field Works

Abstract The coastal marine ecosystems in Easter Island have been poorly studied, and the main studies were isolated species records based on scientific expeditions. The aim of the present study is to apply a spatial distribution analysis and niche sharing null model in published data on intertidal marine gastropods and decapods in rocky shore in Easter Island based in field works in 2010, and published information from CIMAR cruiser in 2004. The field data revealed the presence of decapods Planes minutus (Linnaeus, 1758) and Leptograpsus variegatus (Fabricius, 1793), whereas it was observed the gastropods Nodilittorina pyramidalis pascua Rosewater, 1970 and Nerita morio (G. B. Sowerby I., 1833). The available information revealed the presence of more species in data collected in 2004 in comparison to data collected in 2010, with one species markedly dominant in comparison to the other species. The spatial distribution of species reported in field works revealed that P. minutus and N. morio have aggregated pattern and negative binomial distribution, L. variegatus had uniform pattern with binomial distribution, and finally N. pyramidalis pascua, in spite of aggregated distribution pattern, had not negative binomial distribution. Finally, the results of null model revealed that the species reported did not share ecological niche due to competition absence. The results would agree with other similar information about littoral and sub-littoral fauna for Easter Island.

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