Power Series Distributions Research Articles

Maximum Likelihood Estimation for Generalized Inflated Power Series Distributions

Maximum Likelihood Estimation for Generalized Inflated Power Series Distributions

A Modified Cure Rate Model Based on a Piecewise Distribution with Application to Lobular Carcinoma Data

A novel cure rate model is introduced by considering, for the number of concurrent causes, the modified power series distribution and, for the time to event, the recently proposed power piecewise exponential distribution. This model includes a wide variety of cure rate models, such as binomial, Poisson, negative binomial, Haight, Borel, logarithmic, and restricted generalized Poisson. Some characteristics of the model are examined, and the estimation of parameters is performed using the Expectation–Maximization algorithm. A simulation study is presented to evaluate the performance of the estimators in finite samples. Finally, an application in a real medical dataset from a population-based study of incident cases of lobular carcinoma diagnosed in the state of São Paulo, Brazil, illustrates the advantages of the proposed model compared to other common cure rate models in the literature, particularly regarding the underestimation of the cure rate in other proposals and the improved precision in estimating the cure rate of our proposal.

The Type II Exponentiated Half Logistic-Gompertz-G Power Series Class of Distributions: Properties and Applications

We propose and study a new generalized class of distributions called the Type II Exponentiated Half Logistic-Gompertz-G Power Series (TIIEHL-Gom-GPS) distribution. Some structural properties including expansion of density,ordinary and conditional moments, generating function, order statistics and entropy are derived. We present some specialcases of the proposed distribution. The maximum likelihood method is used for estimating the model parameters. Theimportance of the new class of distributions are illustrated by means of two applications to real data sets.

Statistical Inference for the Unit Gompertz Power Series Distribution Using Ranked Set Sampling with Applications

Statistical Inference for the Unit Gompertz Power Series Distribution Using Ranked Set Sampling with Applications

The type II exponentiated half logistic-odd Burr X-G power series class of distributions with applications

A new distribution, the type II exponentiated half logistic-odd Burr X-G power series (TII-EHL-OBX-GPS), is introduced in this study. This distribution combines the type II exponentiated half logistic-odd Burr X-G family of distributions with power series distributions. We discuss its mathematical characteristics, maximum likelihood estimates and simulation experiments, along with practical applications in the type II exponentiated half logistic-odd Burr X-log-logistic Poisson distribution.

On the Distribution of a Random Power Series on the Dyadic Half-Line

On the Distribution of a Random Power Series on the Dyadic Half-Line

Certain Class of Close-to-Convex Univalent Functions

The purpose of this paper was to define a new class of close-to-convex function, denoted by CV(δ,α), which is a subclass of all functions that are univalent in D and have positive coefficients normalized by the conditions f(0)=0, f′(0)=1, if it satisfies such a condition that is dependent on positive real part. Furthermore, we proved how the power series distribution is essential for determining the sufficient and necessary condition on any function f in class CV(δ,α).

Kn-out-of-n: F System with a Random Number of Components

k-out-of-n systems are among the most important models in practical engineering, they have been extensively studied in the past decades. The system fails if and only if there exist at least k failed components. Generally these systems are considered with a fixed number of components. However, many systems have a variable number of components which can be considered a random variable. In the present paper, we studied the [Formula: see text]-out-of-[Formula: see text]: F system, we assume that the system has an unknown number of components, and we search for this number, assuming it is a random variable, by optimizing the system’s cost. This variable is supposed following a power series distribution such as geometric, Poisson and logarithmic distributions. This work discusses the case where [Formula: see text] for optimization problems. Optimal replacement policies, the optimal number of components and the optimal replacement time are determined to make maintenance and avoid failure, this will minimize the mean cost rates. We started with the case of constant number of components, where we give real and approximate values of the MTTF, optimal number of components and optimal replacement time, then, we considered variable number of components, where the results are compared to the constant case. Numerical examples are also given to illustrate our results.

A class of power inverted Topp-Leone distribution: Properties, different estimation methods & applications

The power inverted Topp Leone power series (PITLPS) class is a novel three-parameter version of the power inverted Topp Leone (PITL) distribution. The PITLPS class is created by combining a power series distribution with the PITL distribution. This compounding process allows for the creation of flexible classes of distributions with strong physical interpretations in fields such as biology and engineering. The PITLPS class has a number of different hazard rate shapes. This class includes some novel lifetime distributions as well as a new compound class. The formula for the density function, limiting behaviour, moments and incomplete moments, and the Rényi entropy measure are all examined as important distributional features. Furthermore, for one elective model, five distinct estimating approaches are considered. To compare the performance of various estimates, a simulation analysis of the PITL Poisson distribution as a one sub-model is provided. In-depth simulation results have been arranged using partial and overall ranks to find the best estimation technique. Finally, the PITL Poisson model is applied to three real data sets, and the fit is compared to that of some other well-known distributions for explanatory purposes. The proposed model is seen to have certain advantages in interpreting lifetime data when compared to other common models in that it has more shapes and flexibility than many already existing distributions.

Cure rate models for heterogeneous competing causes.

Cure rate models have been widely studied to analyze time-to-event data with a cured fraction of patients. In this type of model, the number of concurrent causes is assumed to be a random variable. However, in practice, it is natural to admit that the distribution of the number of competing causes is different from individual to individual. Our proposal is to assume that the number of competing causes belongs to a class of a finite mixture of competing causes distributions. We assume the number of malignant cells follow a mixture of two power series distributions and suppose that the time to the event of interest follows a Weibull distribution. We consider the proportion of the cured number of competing causes depending on covariates, allowing direct modeling of the cure rate. The proposed model includes several well-known models as special cases and defines many new special models. An expectation-maximization algorithm is proposed for parameter estimation, where the expectation step involves the computation of the expected number of concurrent causes for each individual. A Monte Carlo simulation is performed to assess the behavior of the estimation method. In order to show the potential for the practice of our model, we apply it to the real medical data set from a population-based study of incident cases of cutaneous melanoma diagnosed in the state of São Paulo, Brazil, illustrating that the model proposed can outperform traditional models in terms of model fitting.

A Compound Class of Unit Burr XII Model: Theory, Estimation, Fuzzy, and Application

The current research offers an enhanced three-parameter lifetime model that combines the unit Burr XII distribution with a power series distribution. The novel class of distribution is named the unit Burr XII power series (UBXIIPS). This compounding technique allows for the production of flexible distributions with strong physical meanings in domains such as biology and engineering. The UBXIIPS class includes the unit Burr XII Poisson (UBXIIP) distribution, the unit Burr XII binomial distribution, the unit Burr XII geometric distribution, and the unit Burr XII negative binomial distribution. The statistical properties of the class include formulas for the density and cumulative distribution functions, and limiting behaviour, moments and incomplete moments, entropy measures, and quantile function are provided. For estimating population parameters and fuzzy reliability for the UBXIIP model, maximum likelihood and Bayesian approaches are studied by the Metropolis–Hastings algorithm. For maximum likelihood estimators, the length of asymptotic confidence intervals is specified, whereas, for Bayesian estimators, the length of credible confidence intervals is assigned. A simulation investigation of the UBXIIP model was established to evaluate the performance of suggested estimates. In addition, the UBXIIP distribution is explored using real-world data. The UBXIIP distribution appears to offer some benefits in understanding lifetime data when compared to unit Weibull, beta, Kumaraswamy, Kumaraswamy Kumaraswamy, Marshall-Olkin Kumaraswamy, and Topp–Leone Weibull Lomax distributions.

A Compound Class of Inverse-Power Muth and Power Series Distributions

This paper introduces the inverse-power Muth power series model, which is a composition of the inverse-power Muth and the class of power series distributions. The use of the Bell distribution in this context is emphasized for the first time in the literature. Probability density, survival and hazard functions are studied, as well as their moments. Using the stochastic representation of the model, the maximum-likelihood estimators are implemented by the use of the expectation-maximization algorithm, while standard errors are calculated using Oakes’ method. Monte Carlo simulation studies are conducted to show the performance of the maximum-likelihood estimators in finite samples. Two applications to real datasets are shown, where our proposal is compared with some models based on power series compositions.

Zero-and-One Integer-Valued AR(1) Time Series with Power Series Innovations and Probability Generating Function Estimation Approach

Zero-and-one inflated count time series have only recently become the subject of more extensive interest and research. One of the possible approaches is represented by first-order, non-negative, integer-valued autoregressive processes with zero-and-one inflated innovations, abbr. ZOINAR(1) processes, introduced recently, around the year 2020 to the present. This manuscript presents a generalization of ZOINAR processes, given by introducing the zero-and-one inflated power series (ZOIPS) distributions. Thus, the obtained process, named the ZOIPS-INAR(1) process, has been investigated in terms of its basic stochastic properties (e.g., moments, correlation structure and distributional properties). To estimate the parameters of the ZOIPS-INAR(1) model, in addition to the conditional least-squares (CLS) method, a recent estimation technique based on probability-generating functions (PGFs) is discussed. The asymptotic properties of the obtained estimators are also examined, as well as their Monte Carlo simulation study. Finally, as an application of the ZOIPS-INAR(1) model, a dynamic analysis of the number of deaths from the disease COVID-19 in Serbia is considered.

PRANAV-POWER SERIES DISTRIBUTION: PROPERTIES AND APPLICATIONS TO SURVIVAL AND WAITING TIME DATA

This research article presents a new Pranav power series class of distributions which is obtained by compounding one parameter Pranav and power series distributions. Various special cases of the new model have been unfolded. Numerous statistical properties of the proposed model are investigated including closed-form expressions for the density function, cumulative distribution function, survival function, hazard rate function, the moments of order statistics and MLEs. Finally, the flexibility and potentiality of the PPS distribution has been demonstrated by means of two real life data sets.

On the non-existence of a discrete power series distribution with a constant coefficient of variation

Non-existence of distributions with constant coefficient of variation(CV) is investigated within the discrete Power Series and Modified Power Series families of distributions. The development is used to revisit and comment on the problem of existence of a better but biased estimator.

Analysis of Milk Production and Failure Data: Using Unit Exponentiated Half Logistic Power Series Class of Distributions

The unit exponentiated half logistic power series (UEHLPS), a family of compound distributions with bounded support, is introduced in this study. This family is produced by compounding the unit exponentiated half logistic and power series distributions. In the UEHLPS class, some interesting compound distributions can be found. We find formulas for the moments, density and distribution functions, limiting behavior, and other UEHLPS properties. Five well-known estimating approaches are used to estimate the parameters of one sub-model, and a simulation study is created. The simulated results show that the maximum product of spacing estimates had lower accuracy measure values than the other estimates. Ultimately, three real data sets from various scientific areas are used to analyze the performance of the new class.

The Slash Half-Normal Distribution Applied to a Cure Rate Model with Application to Bone Marrow Transplantation

This paper proposes, for the first time, the use of an asymmetric positive and heavy-tailed distribution in a cure rate model context. In particular, it introduces a cure-rate survival model by assuming that the time-to-event of interest follows a slash half-normal distribution and that the number of competing causes of the event of interest follows a power series distribution, which defines six new cure rate models. Several properties of the model are derived and an alternative expression for the cumulative distribution function of the model is presented, which is very useful for the computational implementation of the model. A procedure based on the expectation–maximization algorithm is proposed for the parameter estimation. Two simulation studies are performed to assess some properties of the estimators, showing the good performance of the proposed estimators in finite samples. Finally, an application to a bone marrow transplant data set is presented.

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.

Exponentiated Half Logistic-Generalized G Power Series Class of Distributions: Properties and Applications

A new generalized class of distributions called the Exponentiated Half Logistic-Generalized G Power Series (EHL-GGPS) distribution is proposed. We present some special cases in the proposed distribution. Several mathematical properties of the EHL-GGPS distribution were also derived including order statistics, moments and maximum likelihood estimates. A simulation study for selected parameter values is presented to examine the consistency of the maximum likelihood estimates. Finally, some real data applications of the EHL-GGPS distribution are presented to illustrate the usefulness of the proposed class of distributions.

On a generalization of class of negative binomial distributions

Abstract A class of one-dimensional discrete power series distributions containing negative binomial distributions is considered. Properties of distributions of this class are investigated. Limit theorems generalizing similar theorems for the negative binomial distributions are proved. The proofs are based both on elementary asymptotic methods and on a modification of saddle-point method.