Power Series Distributions Research Articles

Analysis of joint modeling of longitudinal zero-inflated power series and zero-inflated time to event data

ABSTRACT In longitudinal studies measurements are often collected on different types of responses for each individual. These may contain several longitudinally measured responses (such as the CD4 count) and the time at which an event occurs (e.g., HIV, death, or dropout from the study). These outcomes are often separately analyzed. Compared to separate modeling, joint modeling and simultaneous analysis allows for more coherent, robust analysis and may produce a better insight into the process under study. However, there has always been difficulty to the analyst that finding a proper multi-variable joint distribution for linking responses. In this article, we survey the zero-inflated property for longitudinal count and time to event data. We apply a member of the family of power series distributions (PSDs) and the Cox proportional hazard regression model (Cox PH) with Weibull baseline hazard rate, respectively, for these correlated responses. Also we consider both right and left censoring mechanisms in time to event process. This modeling strategy leads to expand the class of joint models and presents some new joint models which, as far as we know, have not yet been investigated by other researchers. The parameters in the joint model are estimated by using likelihood techniques.

On some properties of the hazard rate function for compound distributions

The hazard rate function plays a crucial role to extract some informations of several random life phenomena. In this paper, we investigate important properties satisfied by the hazard rate function for a general power series class of distributions, as monotonicity properties, sharp bounds and convexity properties. In particular, we highlight how these properties are related to those of the baseline distribution and the corresponding probability-generating function of discrete power series distributions.

Exponentiated quasi power Lindley power series distribution with applications in medical science

Abstract The present paper introduces an advanced five parameter lifetime model which is obtained by compounding exponentiated quasi power Lindley distribution with power series family of distributions. The EQPLPS family of distributions contains several lifetime sub-classes such as quasi power Lindley power series, power Lindley power series, quasi Lindley power series and Lindley power series. The proposed distribution exhibits decreasing, increasing and bathtub shaped hazard rate functions depending on its parameters. It is more flexible as it can generate new lifetime distributions as well as some existing distributions. Various statistical properties including closed form expressions for density function, cumulative function, limiting behaviour, moment generating function and moments of order statistics are brought forefront. The capability of the quantile measures in terms of Lambert W function is also discussed. Ultimately, the potentiality and the flexibility of the new class of distributions has been demonstrated by taking three real life data sets by comparing its sub-models.

Exponentiated power generalized Weibull power series family of distributions: Properties, estimation and applications.

In this paper, we introduce the exponentiated power generalized Weibull power series (EPGWPS) family of distributions, obtained by compounding the exponentiated power generalized Weibull and power series distributions. By construction, the new family contains a myriad of new flexible lifetime distributions having strong physical interpretations (lifetime system, biological studies…). We discuss the characteristics and properties of the EPGWPS family, including its probability density and hazard rate functions, quantiles, moments, incomplete moments, skewness and kurtosis. The main vocation of the EPGWPS family remains to be applied in a statistical setting, and data analysis in particular. In this regard, we explore the estimation of the model parameters by the maximum likelihood method, with accuracy supported by a detailed simulation study. Then, we apply it to two practical data sets, showing the applicability and competitiveness of the EPGWPS models in comparison to some other well-reputed models.

On Generalized Moment Exponential Distribution and Power Series Distribution

In this research paper, a new life time family is introduced. Sadaf [1] proposed a moment exponential power series (MEPS) distribution. Generalized moment exponential power series (GMEPS) distribution is a general form of MEPS distribution. It is characterized by compounding GME distribution and power series (PS) distribution. This new family has some new sub models such as GME geometric distribution, GME Poisson (GMEP) distribution, GME logarithmic (GMEL) distribution and GME binomial (GMEB) distribution. We provide statistical properties of GMEPS family of distributions. We find here expression of quantile function based on Lambert W function, the density function of rth order statistic and moments of GMEPS distribution. Descriptive expressions of Shannon entropy and Rényi entropy of new general model are found. We provide special sub-models of the GMEPS family of distributions. The maximum likelihood (ML) estimation method is used to find estimates of the parameters of GMEPS distribution. Simulation study is carried out to check the convergence of new estimators. We apply GMEPS family of distributions on two sets of real data.

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.

Linear recurrent relations, power series distributions, and generalized allocation scheme

Рассматриваются распределения типа степенного ряда, которые определяются производящими функциями чисел, удовлетворяющих линейным рекуррентным соотношениям с неотрицательными коэффициентами, эти функции разлагаются в степенные ряды в некоторых конечных интервалах с центром в нуле. Доказана интегральная предельная теорема о сходимости таких распределений к экспоненциальному распределению. Рассмотрена порожденная этими линейными соотношениями обобщенная схема размещения, для которой доказана локальная нормальная теорема для общего числа компонент. В качестве следствия из более общих результатов автора сформулирована предельная теорема, указывающая достаточные условия сходимости распределений чисел компонент заданного объема к распределению Пуассона.

Application of zero inflated power series distributions for modelling agricultural data related to insects

Application of zero inflated power series distributions for modelling agricultural data related to insects

On the distribution of multiple power series regularly varying at the boundary point

Abstract Let B(x) be a multiple power series with nonnegative coefficients which is convergent for all x ∈ (0, 1)n and diverges at the point 1 = (1, …, 1). Random vectors (r.v.)ξx such that ξx has distribution of the power series B(x) type is studied. The integral limit theorem for r.v. ξx as x ↑ 1 is proved under the assumption that B(x) is regularly varying at this point. Also local version of this theorem is obtained under the condition that the coefficients of the series B(x) are one-sided weakly oscillating at infinity.

A New Generalized Family of Lifetime Distributions Motivated by Parallel and Series Structures

Any given system can be represented as a parallel arrangement of series structures. Motivated by this fact, a general family of distributions is introduced, by adding two extra parameters to a distribution (called baseline distribution),twice compounding with power series distribution. The new family can allow various hazard rate curves that compete well with other alternatives in fitting real data. We derive formal expressions for its moments, generating function, mean residual lifetime and other reliability functions. Certain characterizations of the new family are presented in terms of the ratio of two truncated moments as well as based on the hazard rate function. The maximum likelihood estimation technique is used to estimate the model parameters and a simulation study is conducted to investigate the performance of the maximum likelihood estimates. Finally, two applications of the model with real data sets are presented to illustrate the usefulness of the proposed distribution.

Exponential intervened Poisson distribution

We introduce a new family of distributions using intervened power series distribution. It is a rich family containing distributions such as Marshall–Olkin extended families of distributions, families of distributions generated through truncated negative binomial, and families of distributions generated through truncated binomial distribution. Also, the proposed distribution is a generalization of the families of distributions generated through zero truncated Poisson distribution. As a particular case, we study exponential intervened Poisson (EIP) distribution. This is a generalization of exponential-Poisson distribution. The shape properties of the pdf and hazard rate are proved. The model identifiability is established. The expression for moments, mean deviation, distribution of order statistics, and residual life function of EIP distribution are derived. The unknown parameters of the distribution are estimated using the method of maximum likelihood. The existence and uniqueness of maximum likelihood estimates (MLEs) are proved. Simulation study is carried out to show the performance of MLEs. EIP distribution is fitted to two real data sets and it is shown that the distribution is more appropriate than other competitive models. The adequacy of the EIP model for the data set is established using parametric bootstrap approach.

On estimating reliability function for the family of power series distribution

Explicit expression of the Stress - Strength reliability function, is derived, when the stress (X) and strength (Y) distributions are different members of the Power series family of distributions. Minimum Variance Unbiased (MVU) estimator of R, variance of the MVU estimator together with its MVU estimator are derived for Power series family of distributions. Maximum Likelihood (ML) estimator of R along with related large sample results are also derived. Analytic expressions of the above mentioned quantities are provided for different members of the Power series family of distributions. Further, an application, in the context of real Soccer games, is enumerated.

Some connections between various classes of analytic functions associated with the power series distribution

The primary motivation of the paper is to investigate the power series distribution (Pascal model) for the analytic function classes TG, TGS and TGC. Furthermore, we give necessary and sufficient conditions for the Pascal distribution series belonging to these classes.

Joint modeling for longitudinal set-inflated continuous and count responses

A joint mixture model for analyzing mixed longitudinal continuous and count data is presented. The continuous response is inflated in a set , and a set-inflated normal (SIN) distribution is used as its distribution. The count response is inflated in a set . includes one or more points of sample space and a set-inflated power series (SIPS) distribution is used as its distribution. A full likelihood-based approach is used to obtain the maximum likelihood estimates of parameters via the EM algorithm. A random effects approach is applied to investigate the correlated longitudinal responses and correlated inflation mechanisms of each subject through time. Also, to consider the correlation between the mixed continuous and count responses of each individual at each time, the correlated random effects are used. In order to assess the performance of the model, some simulation studies are performed. An application of our models is illustrated for joint analysis of (1) number of days in the last month that the individual drank alcohol, and (2) weight of respondent for the first two waves of the American’s Changing Lives survey.

Power series cure rate model for spatially correlated interval-censored data based on generalized extreme value distribution

In this work, we propose a flexible cure rate model to allow for spatial correlations by including spatial frailty in the interval-censored data setting. The proposed model is quite flexible and generalizes the Bernoulli, geometric, Poisson, and logarithmic models. It can be tested for the best fit in a straightforward way. Our approach enables different underlying activation mechanisms that lead to the event of interest, and the number of competing causes that can be responsible for the occurrence of the event of interest follows a flexible exponential discrete power series distribution. MCMC methods are used in Bayesian inference for the proposed models and Bayesian comparison criteria are used for model comparison. Moreover, we conduct influence diagnostics through the diagnostic measures in order to detect possible influential or extreme observations that can cause distortions in the analysis results. Finally, the proposed models are used to analyze a real dataset.

Generalized inverse Lindley power series distributions: modeling and simulation

Generalized inverse Lindley power series distributions: modeling and simulation

The Odd Log-Logistic Power Series Family of Distributions: Properties and Applications

A new family of continuous distributions obtained by compounding the odd log-logistic and power series distributions is introduced. The mathematical properties of the proposed family are discussed. The estimation of the parameters is considered by the maximum likelihood method. In order to assess the finite sample performance of maximum likelihood estimators, simulation studies are performed. Finally, the potentiality of the family is illustrated by means of applications to two real data sets.

Acoustic Emission in a Layer of Geomaterial under Deformation by Shear

The new method is proposed for interpreting data of acoustic emission during initiation and growth of dynamic breakaways. The method is based on the analysis of wave form of the emitted acoustic pulses. Clustering of the pulses by the wave form criterion shows that in the localization zone of strains different-scale processes described with various scaling relations take place. All classes of acoustic pulses obey the power-series amplitude-frequency distribution. The sharp-arrival acoustic pulses posses unaltered scaling relations in the period of nucleation and growth of dynamic breakaways whereas the smooth-arrival pulses demonstrate the nonlinear change in the scaling relations. At the final stage of the dynamic breakaway formation, the proportion and amplitude of acoustic pulses with smooth arrival increase.

A New Family of Lifetime Distributions: Theory, Application and Characterizations

A new class of distributions with increasing, decreasing, bathtub-shaped and unimodal hazard rate forms called generalized quadratic hazard rate-power series distribution is proposed. The new distribution is obtained by compounding the generalized quadratic hazard rate and power series distributions. This class of distributions contains several important distributions appeared in the literature, such as generalized quadratic hazard rate-geometric, -Poisson, -logarithmic, -binomial and -negative binomial distributions as special cases. We provide comprehensive mathematical properties of the new distribution. We obtain closed-form expressions for the density function, cumulative distribution function, survival and hazard rate functions, moments, mean residual life, mean past lifetime, order statistics and moments of order statistics; certain characterizations of the proposed distribution are presented as well. The special distributions are studied in some details. The maximum likelihood method is used to estimate the unknown parameters. We propose to use EM algorithm to compute the maximum likelihood estimators of the unknown parameters. It is observed that the proposed EM algorithm can be implemented very easily in practice. One data set has been analyzed for illustrative purposes. It is observed that the proposed model and the EM algorithm work quite well in practice.

A new long-term survival model with dispersion induced by discrete frailty.

Frailty models are generally used to model heterogeneity between the individuals. The distribution of the frailty variable is often assumed to be continuous. However, there are situations where a discretely-distributed frailty may be appropriate. In this paper, we propose extending the proportional hazards frailty models to allow a discrete distribution for the frailty variable. Having zero frailty can be interpreted as being immune or cured (long-term survivors). Thus, we develop a new survival model induced by discrete frailty with zero-inflated power series distribution, which can account for overdispersion. A numerical study is carried out under the scenario that the baseline distribution follows an exponential distribution, however this assumption can be easily relaxed and some other distributions can be considered. Moreover, this proposal allows for a more realistic description of the non-risk individuals, since individuals cured due to intrinsic factors (immune) are modeled by a deterministic fraction of zero-risk while those cured due to an intervention are modeled by a random fraction. Inference is developed by the maximum likelihood method for the estimation of the model parameters. A simulation study is performed in order to evaluate the performance of the proposed inferential method. Finally, the proposed model is applied to a data set on malignant cutaneous melanoma to illustrate the methodology.