P.C. Consul

Bio: P.C. Consul is an academic researcher from University of Calgary. The author has contributed to research in topics: Poisson distribution & Negative binomial distribution. The author has an hindex of 18, co-authored 45 publications receiving 1845 citations.

A Generalization of the Poisson Distribution

Abstract: A new generalization of the Poisson distribution, with two parameters λ1 and λ2, is obtained as a limiting form of the generalized negative binomial distribution. The variance of the distribution is greater than, equal to or smaller than the mean according as λ2 is positive, zero or negative. The distribution gives a very close fit to supposedly binomial, Poisson and negative-binomial data and provides with a model suitable to most unimodel or reverse J-shaped distributions. Diagrams showing the variations in the form of the distribution for different values of λ1 and λ2 are given.

Generalized Poisson regression model

Abstract: The generalized Poisson distribution has been found useful in fitting over-dispersed as well as under-dispersed count data. Since a number of models and methods have been proposed for the regression analysis of count data either with under-dispersion or with over-dispersion, we define and study a generalized Poisson regression (GPR) model which is useful in predicting a response variable affected by one or more covariates. This regression model is suitable for both types of dispersions. The methods of maximum likelihood and moments are given for the estimation of parameters. Approximate tests for the adequacy of the model are considered. Asymptotic tests are given for the significance of regression parameters. The GPR model has been applied to four observed data sets to which other regression models were applied earlier.

A Generalized Negative Binomial Distribution

Abstract: A generalized negative binomial (GNB) distribution with an additional parameter $\beta $ has been obtained by using Lagrange’s expansion. The parameter is such that both mean and variance tend to increase or decrease with an increase or decrease in its value but the variance increases or decreases faster than the mean. For $\beta = \frac{1} {2}$, the mean and variance are approximately equal and so the GNB distribution resembles the Poisson distribution. When $\beta = 0$ or 1, the GNB distribution reduces to the binomial or negative binomial distribution respectively. It has been shown that the generalized negative binomial distribution converges to a Poisson-type distribution in which the variance may be more or less than the mean, depending upon the value of a parameter. Expected frequencies have been calculated for a number of examples to show that the distribution provides a very satisfactory fit in different practical situations. Its convolution property together with other properties are quite inter...

Use of Lagrange Expansion for Generating Discrete Generalized Probability Distributions

Abstract: Considering $g( t )$ and $f( t )$ as two probability generating functions defined on nonnegative integers such that $g( 0 ) e 0$, we use Lagrange’s expansion, together with the transformation $t = u \cdot g( t )$, to define families of discrete generalized probability distributions by the name of Lagrange distributions as \[ \begin{gathered} \Pr \,[ {X = 0} ] = L( {g;f;0} ) = f( 0 ), \hfill \\ \Pr \,[ {X = x} ] = L( {g;f;x} ) = \frac{1}{{x!}}\frac{{d^{x - 1} }}{{dt^{x - 1} }}\{ {( {g( t )} )^x \cdot f'( t )} \}|_{t = 0} \hfill \\ \end{gathered} \]for $x = 1,2,3, \cdots $, where the different families are generated by assigning different values to $g( t )$ and $f( t )$. General formulas for writing down the central moments of Lagrange distributions are obtained and it is shown that they satisfy the convolution property. The double binomial family of Lagrange distributions is studied in greater detail as it gives a large number of discrete distributions, including Borel-Tanner distribution, Haight’s distr...

Traffic and related self-driven many-particle systems

Abstract: Since the subject of traffic dynamics has captured the interest of physicists, many surprising effects have been revealed and explained. Some of the questions now understood are the following: Why are vehicles sometimes stopped by ``phantom traffic jams'' even though drivers all like to drive fast? What are the mechanisms behind stop-and-go traffic? Why are there several different kinds of congestion, and how are they related? Why do most traffic jams occur considerably before the road capacity is reached? Can a temporary reduction in the volume of traffic cause a lasting traffic jam? Under which conditions can speed limits speed up traffic? Why do pedestrians moving in opposite directions normally organize into lanes, while similar systems ``freeze by heating''? All of these questions have been answered by applying and extending methods from statistical physics and nonlinear dynamics to self-driven many-particle systems. This article considers the empirical data and then reviews the main approaches to modeling pedestrian and vehicle traffic. These include microscopic (particle-based), mesoscopic (gas-kinetic), and macroscopic (fluid-dynamic) models. Attention is also paid to the formulation of a micro-macro link, to aspects of universality, and to other unifying concepts, such as a general modeling framework for self-driven many-particle systems, including spin systems. While the primary focus is upon vehicle and pedestrian traffic, applications to biological or socio-economic systems such as bacterial colonies, flocks of birds, panics, and stock market dynamics are touched upon as well.

Negative Binomial Regression

Abstract: Preface 1. Introduction 2. The concept of risk 3. Overview of count response models 4. Methods of estimation and assessment 5. Assessment of count models 6. Poisson regression 7. Overdispersion 8. Negative binomial regression 9. Negative binomial regression: modeling 10. Alternative variance parameterizations 11. Problems with zero counts 12. Censored and truncated count models 13. Handling endogeneity and latent class models 14. Count panel models 15. Bayesian negative binomial models Appendix A. Constructing and interpreting interactions Appendix B. Data sets and Stata files References Index.

Regression analysis of count data

Abstract: Three nonlinear count models, Poisson R.egression (PR), Negative Binomial Regression (NBR), and Generalized Poisson Regression (GPR) are used for assessing the effects of risk factors on agricultural injuries from farm injury data. A sample of 1,322 respondents who participated in the farm safety/injury baseline survey in nine rural counties in Alabama and Mississippi, aged 18 years and older are considered for analysis. The dispersion parameter estimates and their standard errors for GPR models were consistently smaller than that of NBR models. Estimated dispersion parameters in the NBR and GPR models were positive and significantly different from zero. Estimated goodness-of-fit measures showed that GPR models outperformed the NBR and PR models.

Conditions under Which Mean Square Ratios in Repeated Measurements Designs Have Exact F-Distributions

Abstract: Investigation is made of the character of the covariance matrix which will result in exact F-distributions for the treatments and interaction variance ratios in repeated measurements designs. It is shown, assuming multivariate normality, that the matrix may exhibit a more general character than is typically implied to be essential. Equality of variances and equality of covariances, with identical matrices for all levels of a second treatment factor, are sufficient but not necessary conditions. The necessary and sufficient condition is the equality of variances of differences for all pairs of treatment measures assumed to be correlated. An alternative statement is that the Box-Geisser-Greenhouse parameter e = 1.0. A test is described which bears on the tenability of this condition.