Logarithmic Series Distribution Research Articles

Extended Sibuya Distribution in Subcritical Markov Branching Processes

The subcritical Markov branching process X(t) starting with one particle as the initial condition has the ultimate extinction probability q = 1. The branching mechanism in consideration is defined by the mixture of logarithmic distributions on the nonnegative integers. The purpose of the present paper is to prove that in this case the random number of particles X(t) alive at time t > 0 follows the shifted extended Sibuya distribution, with parameters depending on the time t > 0. The conditional limit probability is the logarithmic series distribution supported by the positive integers.

On some aspects of a bivariate alternative zero-inflated logarithmic series distribution

In this paper, we discuss some important aspects of the bivariate alternative zero-inflated logarithmic series distribution (BAZILSD) of which the marginals are the alternative zero-inflated logarithmic series distributions of Kumar and Riyaz (2015. An alternative version of zero-inflated logarithmic series distribution and some of its applications. Journal of Statistical Computation and Simulation, 85(6), 1117–1127). We study some important properties of the distribution by deriving expressions for its probability mass function, factorial moments, conditional probability generating functions, and recursion formulae for its probabilities, raw moments and factorial moments. The parameters of the BAZILSD are estimated by the method of maximum likelihood and certain test procedures are also considered. Further certain real-life data applications are cited for illustrating the usefulness of the model. A simulation study is conducted for assessing the performance of the maximum likelihood estimators of the parameters of the BAZILSD.

Flexible multivariate zero to k inflated power series regression model with applications

Inflated distributions are applied in various fields, including insurance, traffic networks and survival analyses. First, they are defined by a baseline discrete distribution, and then, extra masses are added to some points of interest, called inflated points, to achieve more flexible models for data analyses. The baseline distribution is arbitrary and application dependent. Here, the rich family of power series distributions is considered as the baseline, which includes various common discrete distributions such as Poisson, negative binomial, multinomial and logarithmic series distributions. This paper deals with an extension of previous works in two directions. The former is an extension of the univariate inflated distributions to multivariate ones, and the latter is the generalization of the inflated points from the zero single point to the set . Under this setting, various inflated distributions in the literature fall into the proposed family of distributions. These extensions make the proposed model flexible and practically useful in data analyses. To do this, the problem of estimating parameters with various approaches as well as hypotheses testing is studied in detail. Multivariate‐generalized linear models with inflated multivariate discrete responses are also discussed. To assess the performance of the proposed family of inflated distributions, simulation studies are conducted, and a real data set on an Australian health survey study is also analysed.

On bivariate alpha logarithmic series distribution and its properties

Here we propose a bivariate version of the alpha logarithmic series distribution (ALSD) of Kumar and Riyaz (South African Statist. J., 2014) and shown that its marginals are also ALSD. We study some of its important properties by deriving expressions for its probability mass function, factorial moments, conditional probability generating function, covariance etc. The parameters of the model are estimated by the method of maximum likelihood and the distribution has been fitted to certain real life data sets.

The Dirichlet model in marketing

AbstractWe present a review of Gerald Goodhardt's most famous contribution to marketing science—the NBD‐Dirichlet model. This provides a powerful illustration of the complex pathway and useful associated discoveries that over 25 years lead to the specification and application of a key marketing model. We identify the process that started with the negative binomial distribution (NBD) applied to purchase incidence, examined alternatives such as the logarithmic series distribution (LSD) and the beta binomial distribution (BBD), and along the way developed conditional trend analysis (CTA). We then explore the development of brand choice beginning with duplication of purchase as a method for understanding the underlying patterns, before moving into various approaches to modelling choice including a multivariate NBD, and eventually a multivariate BBD—the Dirichlet multinomial distribution (DMD). We discuss key events in modelling consumer behaviour and outline the model's implications for how marketers should think about consumers, or more specifically the Dirichlet consumer. Finally, we provide a survey of the model's applications uncovering a rich recent history and bright future for Goodhardt's legacy.

English

The Generalized Logarithmic Series Distribution (GLSD) adds an extra parameter to the usual logarithmic series distribution and was introduced by Jain and Gupta (1973). This distribution has found applications in various fields. The estimation of parameters of generalized logarithmic series distribution was studied by the methods of maximum likelihood, moments, minimum chi square and weighted discrepancies. The GLSD was fitted to counts of red mites on apple leaves and it was observed that all the estimation techniques perform well in estimating the parameters of generalized logarithmic series distribution but with varying degree of non-significance.

Discrete Distributions in Engineering and the Applied Sciences

This is an introductory book on discrete statistical distributions and its applications. It discusses only those that are widely used in the applications of probability and statistics in everyday life. The purpose is to give a self-contained introduction to classical discrete distributions in statistics. Instead of compiling the important formulas (which are available in many other textbooks), we focus on important applications of each distribution in various applied fields like bioinformatics, genomics, ecology, electronics, epidemiology, management, reliability, etc., making this book an indispensable resource for researchers and practitioners in several scientific fields. Examples are drawn from different fields. An up-to-date reference appears at the end of the book. Chapter 1 introduces the basic concepts on random variables, and gives a simple method to find the mean deviation (MD) of discrete distributions. The Bernoulli and binomial distributions are discussed in detail in Chapter 2. A short chapter on discrete uniform distribution appears next. The next two chapters are on geometric and negative binomial distributions. Chapter 6 discusses the Poisson distribution in-depth, including applications in various fields. Chapter 7 is on hypergeometric distribution. As most textbooks in the market either do not discuss, or contain only brief description of the negative hypergeometric distribution, we have included an entire chapter on it. A short chapter on logarithmic series distribution follows it, in which a theorem to find the kth moment of logarithmic distribution using (k-1)th moment of zero-truncated geometric distribution is presented. The last chapter is on multinomial distribution and its applications. The primary users of this book are professionals and practitioners in various fields of engineering and the applied sciences. It will also be of use to graduate students in statistics, research scholars in science disciplines, and teachers of statistics, biostatistics, biotechnology, education, and psychology.

Bayesian Analysis of Misclassified Generalized Power Series Distributions Under Different Loss Functions

In certain experimental investigations involving discrete distributions external factors may induce measurement error in the form of misclassification. For instance, a situation may arise where certain values are erroneously reported; such a situation termed as modified or misclassified has been studied by many researchers. Cohen (J. Am. Stat. Assoc. 55 (1960), 139–143; Ann. Inst. Stat. Math. 9 (1960), 189–193; Technometrics. 2 (1960), 109–113) studied misclassification in Poisson and the binomial random variables. In this paper, we discuss misclassification in the most general class of discrete distributions, the generalized power series distributions (GPSDs), where some of the observations corresponding to x =c+1;c ≥0 are erroneously observed or at least reported as being x =c with probability α. This class includes among others the binomial, negative binomial, logarithmic series and Poisson distributions. We derive the Bayes estimators of functions of parameters of the misclassified GPSD under different loss functions. The results obtained for misclassified GPSD are then applied to its particular cases like negative binomial, logarithmic series and Poisson distributions. Finally, few numerical examples are provided to illustrate the results.

On A 3-Points Inflated Power Series Distributions Characterizations

Differential equations are used in modelling many disciplines, in engineering, chemistry, physics, biology, economics, and other fields of sciences, hence can be used to understand and to determine the underlying probabilistic behavior of phenomena through their probability distributions. This paper came to use a simple form of differential equations, namely, the linear form, to determine the probabilistic distributions of some of the most important and popular sub class of discrete distributions used in real-life, the Poisson, the binomial, the negative binomial, and the logarithmic series distributions. A class of finite number of inflated points power series distributions, that contains the Poisson, the binomial, the negative binomial, and the logarithmic series distributions as some of its members, was defined and some of its characteristics properties, along with characterization of the 3-points inflated of these four distributions, through a linear differential equation for their probability generating functions were given. Further, some previous known results were shown to be special cases of our results.

Bias-corrected Maximum-Likelihood Estimator for the Parameter of the Logarithmic Series Distribution and its Characterizations

Bias-corrected Maximum-Likelihood Estimator for the Parameter of the Logarithmic Series Distribution and its Characterizations

Logarithmic Lévy process directed by Poisson subordinator

Let $\{L(t),t\geq 0\}$ be a L\'{e}vy process with representative random variable $L(1)$ defined by the infinitely divisible logarithmic series distribution. We study here the transition probability and L\'{e}vy measure of this process. We also define two subordinated processes. The first one, $Y(t)$, is a Negative-Binomial process $X(t)$ directed by Gamma process. The second process, $Z(t)$, is a Logarithmic L\'{e}vy process $L(t)$ directed by Poisson process. For them, we prove that the Bernstein functions of the processes $L(t)$ and $Y(t)$ contain the iterated logarithmic function. In addition, the L\'{e}vy measure of the subordinated process $Z(t)$ is a shifted L\'{e}vy measure of the Negative-Binomial process $X(t)$. We compare the properties of these processes, knowing that the total masses of corresponding L\'{e}vy measures are equal.

Application of finitized power series distributions to accelerated variate generation. Part II: the case of the logarithmic distribution

AbstractThe method of aliasing vastly expands the palette of discrete random variate generation methodologies while providing excellent speed. However, its application is limited to finitely supported distributions. We demonstrate that an application of moment preserving finitization called the Negative Taylor Series Finitization (NTSF) method for the power series family of discrete distributions, when coupled with the method of aliasing, can greatly improve infinitely supported discrete random variate generation speed with certain limitations. We illustrate this with the logarithmic power series distribution, and we compare four published algorithms designed to generate random variates from a logarithmic distribution to the aliasing method of random variate generation from an NTSF version of the same distribution. We compare the accuracy and speed (user‐time) of these various methods for generating variates from a logarithmic distribution.

On generalized logarithmic series distribution and its application to leaf spot grades in mulberry

The generalized logarithmic series distribution (GLSD) adds an extra parameter to the usual logarithmic series distribution and was introduced by Jain and Gupta (1973). This distribution has found applications in various fields. The estimation of parameters of generalized logarithmic series distribution was studied by the methods of maximum likelihood, moments, minimum chi-square and weighted discrepancies. The GLSD was fitted to different leaf spot grades in four varieties of mulberry namely Ichinose, Gosherami, Rokokuvoso and Kokuso-20 and it was observed that all the estimation techniques perform well in estimating the parameters of generalized logarithmic series distribution but with varying degree of non-significance.

Using Cuckoo Algorithm for Estimating Two GLSD Parameters and Comparing it with Other Algorithms

This study introduces and compares different methods for estimating the two parameters of generalized logarithmic series distribution. These methods are the cuckoo search optimization, maximum likelihood estimation, and method of moments algorithms. All the required derivations and basic steps of each algorithm are explained. The applications for these algorithms are implemented through simulations using different sample sizes (n = 15, 25, 50, 100). Results are compared using the statistical measure mean square error.

On some aspects of a generalized alternative zero-inflated logarithmic series distribution

ABSTRACTHere, we consider a generalized form of the alternative zero-inflated logarithmic series distribution of Kumar and Riyaz (J. Statist. Comp. Simul., 2015) and study some of its important aspects. The parameters of the distribution are estimated by the method of maximum likelihood and some test procedures are developed for testing the significance of the additional parameter of the model. All these estimation and testing procedures are illustrated with the help of certain real life datasets. A simulation study is also carried out for assessing the performance of the estimators.

E-Bayesian Estimation of the Parameter of the Logarithmic Series Distribution

E-Bayesian estimation is introduced to estimate the parameter of logarithmic series distribution. In addition, E-Bayesian, Bayesian and maximum likelihood estimation with through applying mean squared error.

An order k version of the alternative zero-inflated logarithmic series distribution and its applications

ABSTRACT Here we introduce an order version of the alternative zero-inflated logarithmic series distribution of Kumar and Riyaz [14] and investigate some of its important properties by deriving an expression for its probability mass function, recurrence relations for its probabilities, raw moments and factorial moments. We discuss the estimation of the parameters of the distribution by the method of maximum likelihood. Certain test procedures are developed for testing the significance of the additional parameters of the model. All these procedures discussed in the paper are illustrated with the help of real life data sets. A simulation study is also considered for assessing the performance of the estimators.

A Modified Version of Logarithmic Series Distribution and Its Applications

In this article, we consider a modified version of logarithmic series distribution and study some of its properties. The maximum likelihood estimation of the parameters of the modified distribution is discussed and the distribution has been fitted to certain real life data sets. Tests are also carried out for justifying the significance of the additional parameter of the modified distribution.

A Discrete Analogue of Reciprocal Coordinate Subtangent and its Role in Characterization Problems

Abtsrcat The concept of reciprocal coordinate subtangent (RCST) has been used in statistical literature as a useful tool to study the monotone behaviours of a continuous density function and for characterizing probability distributions through its functional forms. Motivated by this, in the present paper, we consider a discrete analogue of RCST introduced by Gupta et al. (1997) and study its usefulness in modeling probability mass functions. Characterization results are proved for models viz. geometric, discrete Burr and modified power series. The concept is also extended to the weighted models and proves some useful results as models such as logarithmic series, residual lifetime and partial sum (renewal) distributions. Finally, a new definition for discrete analogue of RCST is introduced and examined its usefulness in modeling proportional hazard models in discrete time.

Type II Family of Bivariate Inflated-Parameter Generalized Power Series Distributions

The family of Inated-parameter Generalized Power Series distributions (IGPSD) was introduced by Minkova in 2002 as a compound Generalized Power Series distributions (GPSD) with geometric compounding distribution. In these notes we introduce a family of compound GPSDs with bivariate geometric compounding distribution. The probability mass function, recursion formulas, conditional distributions and some properties are given. A member of this family is a Type II bivariate Polya-Aeppli distribution, introduced by Minkova and Balakrishanan (2014). In this notes the particular cases of bivariate compound binomial, negative binomial and logarithmic series distributions are analyzed in detail.