Zero-truncated Poisson Distribution Research Articles

One-Inflated Zero-Truncated Poisson Distribution: Statistical Properties and Real Life Applications

One-Inflated Zero-Truncated Poisson Distribution: Statistical Properties and Real Life Applications

Bayesian Inference for the Gamma Zero-Truncated Poisson Distribution with an Application to Real Data

This article presents Bayesian estimation methods applied to the gamma zero-truncated Poisson (GZTP) and the complementary gamma zero-truncated Poisson (CGZTP) distributions, encompassing both one-parameter and two-parameter models. These distributions are notably flexible and useful for modeling lifetime data. In the one-parameter model case, the Jeffreys prior is mathematically derived. The use of informative and noninformative priors, combined with the random walk Metropolis algorithm within a Bayesian framework, generates samples from the posterior distributions. Bayesian estimators’ effectiveness is examined through extensive simulation studies, in comparison with the maximum likelihood method. Results indicate that Bayesian estimators provide more precise parameter estimates, even with smaller sample sizes. Furthermore, the study and comparison of the coverage probabilities (CPs) and average lengths (ALs) of the credible intervals with those from Wald intervals suggest that Bayesian credible intervals typically yield shorter ALs and higher CPs, thereby demonstrating the effectiveness of Bayesian inference in the context of GZTP and CGZTP distributions. Lastly, Bayesian inference is applied to real data.

WEIBULL-POISSON DISTRIBUTION AND THEIR APPLICATION TO SYSTEMATIC PARALLEL RISK

The Weibull-Poisson distribution represents a continuous distribution type applicable to various forms of hazard, including monotone up, monotone down, and upside-down bathtub shapes that ascend. The distribution characterizes lifetimes and can effectively model failures within a series of systems, which evolves from the Exponential-Poisson distribution. This distribution emerges through the compounding of the Weibull Distribution and Zero Truncated Poisson Distribution. The compounding itself integrates several mathematical properties, such as statistical order and Taylor’s number expansion, to reach its final form. Alongside the formulation of the Weibull-Poisson distribution, this paper includes the probability density function, distribution function, rth moment, rth central moment, mean, and variance. For illustration, the Weibull-Poisson distribution is applied to guinea pig survival data after being infected with Turblece virus Bacilli.

Designing Of Double Sampling Plan Indexed Through Six Sigma Quality Level – 1 Using Truncated Binomial Distribution

Among the various probability distributions utilized to characterize situations where an observational apparatus becomes active only upon the occurrence of at least one event, the Zero Truncated Poisson Distribution (ZTPD) stands out. Shanmugam (1985) demonstrated the applicability of the ZTPD in modeling scenarios such as second quality lots, where there exists the possibility of at least one defective item in the sample. In this paper, a systematic procedure for constructing Double Sampling Plans, indexed through Six Sigma Quality Level – 1 (SSQL - 1), is presented. The methodology employs the Truncated Binomial Distribution (TBD) as the baseline distribution. Furthermore, to facilitate the selection of these plans, tables generated using an Excel packages are provided, ensuring ease of use and accessibility.

On the Normal Approximations to the Method of Moments Point Estimators of the Parameter and Mean of the Zero-Truncated Poisson Distribution

On the Normal Approximations to the Method of Moments Point Estimators of the Parameter and Mean of the Zero-Truncated Poisson Distribution

On Surprise Indices Related to Univariate Discrete and Continuous Distributions: A Survey

The notion that the occurrence of an event is surprising has been discussed in the literature without adequate details. By definition, a surprise index is an index by which how surprising an event is may be determined. Since its inception, this index has been evaluated for univariate discrete probability models, such as the binomial, negative binomial, and Poisson probability distributions. In this article, we derive and discuss using numerical studies, in addition to the above-mentioned probability models, surprise indices for several other univariate discrete probability models, such as the zero-truncated Poisson, geometric, Hermite, and Skellam distributions, by adopting a established strategy and using the Mathematica, version 12 software. In addition, we provide symbolical expressions for the surprise index for several univariate continuous probability models, which has not been previously discussed. For illustrative purposes, we present some possible real-life applications of this index and potential challenges to extending the notion of the surprise index to bivariate and higher dimensions, which might involve ubiquitous normalizing constants.

Complementary Gamma Zero-Truncated Poisson Distribution and Its Application

Numerous lifetime distributions have been developed to assist researchers in various fields. This paper proposes a new continuous three-parameter lifetime distribution called the complementary gamma zero-truncated Poisson distribution (CGZTP), which combines the distribution of the maximum of a series of independently identical gamma-distributed random variables with zero-truncated Poisson random variables. The proposed distribution’s properties, including proofs of the probability density function, cumulative distribution function, survival function, hazard function, and moments, are discussed. The unknown parameters are estimated using the maximum likelihood method, whose asymptotic properties are examined. In addition, Wald confidence intervals are constructed for the CGZTP parameters. Simulation studies are conducted to evaluate the efficacy of parameter estimation, and three real-world data applications demonstrate that CGZTP can be an alternative distribution for fitting data.

Economic Valuation of Lake Tana: A Recreational Use Value Estimation through the Travel Cost Method

Lake Tana is a well-known tourist spot in northern Ethiopia that lures both domestic and foreign tourists. The lake’s value is still underrated, despite the site’s immense potential for recreation and tourism. In this study, the recreational value of Lake Tana is estimated, and the consumer characteristics associated with recreational demand are analyzed. The study employs a Zero-Truncated Poisson (ZTP) regression model for data generated by the Individual Travel Cost Method (ITCM) and draws information from 1094 on-site surveys. According to ZTP regression results, variables that are significantly and positively associated with demand for Lake Tana recreation include the monthly income and age of the visitors and their preference for other recreation destinations such as Tis-Abay and Gondar Fasiledes Royal Castle. However, respondents’ leisure time, overall cost, and residential distance from the recreational site are all negatively associated with the demand for recreation. It is also evident that Lake Tana is endowed with a wealth of attributes, ranging from natural beauty to ancient religious and cultural heritages, making the lake a highly desirable recreation destination. According to the estimation, Lake Tana has a yearly recreational value of around USD 68.5 million. However, it also demonstrates that if lake settings were to improve in quality, the value of Lake Tana would rise considerably, reaching USD 151 million. The value of sentimental attachment implies that determining Lake Tana’s recreational value is a key element in the resource’s sustainable utilization and management. To make greater use of Lake Tana’s recreational amenities, however, uncontrolled waste disposal, favorable recreation facilities, and the spread of water hyacinth should be addressed immediately. Furthermore, it is imperative to maintain the lake’s diverse attributes, as the integration of these attributes is what gives the lake its primary appeal.

Initial Coefficients Estimates and Fekete–Szegö Inequality Problem for a General Subclass of Bi-Univalent Functions Defined by Subordination

In the present work, we aim to introduce and investigate a novel comprehensive subclass of normalized analytic bi-univalent functions involving Gegenbauer polynomials and the zero-truncated Poisson distribution. For functions in the aforementioned class, we find upper estimates of the second and third Taylor–Maclaurin coefficients, and then we solve the Fekete–Szegö functional problem. Moreover, by setting the values of the parameters included in our main results, we obtain several links to some of the earlier known findings.

Zero Truncated Poisson - Pareto Distribution: Application and Estimation Methods

This article introduces and discusses a new three-parameter lifespan distribution called Zero-Truncated Poisson Pareto distribution ZTPP. that is built on compounding Pareto distribution as a continuous distribution and Zero-Truncated Poisson distribution as a discrete distribution. Various statistical properties and reliability characteristics of the proposed distribution have been investigated including explicit expressions for the moments, moment generating function, quantile function, and median. With three parameters, the suggested distribution has an advantage over other distributions in that it makes estimating the model parameters simpler. To estimate the unknown parameters of the ZTPP distribution, the maximum likelihood method, and L. Moments method are employed. Moreover, a real data set is used to evaluate the significance and ensure the applicability of the proposed distribution as compared to other probability distributions. The derived model proved to be the best compared to other fitted models, where the criteria values of (AIC), (CAIC), and (BIC) are minimum values by using the ZTPP distribution. The proposed model is hoped to attract a wider application.

Zero Truncated Poisson - Pareto Distribution: Application and Estimation Methods

This article introduces and discusses a new three-parameter lifespan distribution called Zero-Truncated Poisson Pareto distribution ZTPP. that is built on compounding Pareto distribution as a continuous distribution and Zero-Truncated Poisson distribution as a discrete distribution. Various statistical properties and reliability characteristics of the proposed distribution have been investigated including explicit expressions for the moments, moment generating function, quantile function, and median. With three parameters, the suggested distribution has an advantage over other distributions in that it makes estimating the model parameters simpler. To estimate the unknown parameters of the ZTPP distribution, the maximum likelihood method, and L. Moments method are employed. Moreover, a real data set is used to evaluate the significance and ensure the applicability of the proposed distribution as compared to other probability distributions. The derived model proved to be the best compared to other fitted models, where the criteria values of (AIC), (CAIC), and (BIC) are minimum values by using the ZTPP distribution. The proposed model is hoped to attract a wider application.

A Novel Flexible Class of Intervened Poisson Distribution by Lagrangian Approach

The zero-truncated Poisson distribution (ZTPD) generates a statistical model that could be appropriate when observations begin once at least one event occurs. The intervened Poisson distribution (IPD) is a substitute for the ZTPD, in which some intervention processes may change the mean of the rare events. These two zero-truncated distributions exhibit underdispersion (i.e., their variance is less than their mean). In this research, we offer an alternative solution for dealing with intervention problems by proposing a generalization of the IPD by a Lagrangian approach called the Lagrangian intervened Poisson distribution (LIPD), which in fact generalizes both the ZTPD and the IPD. As a notable feature, it has the ability to analyze both overdispersed and underdispersed datasets. In addition, the LIPD has a closed-form expression of all of its statistical characteristics, as well as an increasing, decreasing, bathtub-shaped, and upside-down bathtub-shaped hazard rate function. A consequent part is devoted to its statistical application. The maximum likelihood estimation method is considered, and the effectiveness of the estimates is demonstrated through a simulated study. To evaluate the significance of the new parameter in the LIPD, a generalized likelihood ratio test is performed. Subsequently, we present a new count regression model that is suitable for both overdispersed and underdispersed datasets using the mean-parametrized form of the LIPD. Additionally, the LIPD’s relevance and application are shown using real-world datasets.

Estimating the recreational use value of Tis-Abay Waterfall in the upstream of the Blue Nile River, North-West Ethiopia

Tis-Abay Waterfall is a famous tourist destination in northern Ethiopia, attracting both domestic and international visitors. Although the site's recreation and tourism potential are enormous, the value of the waterfall remains underestimated; an application of economic valuation methods can provide information to better utilize the resource. This study estimates the recreational value of Tis-Abay Waterfall and analyzes the consumer characteristics associated with recreational demand. Drawing on data from 1044 on-site surveys, the study uses the Individual Travel Cost Method (ITCM) with a Zero-Truncated Poisson (ZTP) regression model. Results of the ZTP regression suggest that visitors' age, monthly income, and interest in alternative recreation sites like Lake Tana and Gondar Fasiledes Royal Castle are variables significantly and positively related with recreational demand for the Tis-Abay Waterfall. However, visitors' recreational demand is negatively associated with respondents' distance from the site, leisure time, and total cost of site access. The appraisal suggests that the Tis-Abay Waterfall has a significant annual recreational value of $9.5 million. But it also shows that the waterfall's value would increase significantly, up to $17.3 million, with hypothetical quality improvements in the waterfall settings. The value attachment suggests that estimating the recreation value for Tis-Abay Waterfall is a central component in the sustainable use and management of the resource. However, the presence of unfavorable trade-offs with the electric power plant and the inadequate infrastructure and services for reaching the site are major concerns that require immediate attention to make better use of Tis-Abay Waterfall's recreational services.

SOME RESULTS ON r-TRUNCATED DEGENERATE POISSON RANDOM VARIABLES

The zero-truncated Poisson distributions are certain discrete probability distributions whose supports are the set of positive integers, which are also known as the conditional Poisson distributions or the positive Poisson distributions. Recently, as a natural extension of those distributions, Kim–Kim studied the zero-truncated degenerate Poisson distributions. In this paper, we introduce the [Formula: see text]-truncated degenerate Poisson random variable with parameter [Formula: see text], whose probability mass function is given by [Formula: see text], and investigate various properties of this random variable.

Mean regression model for the zero-truncated Poisson distribution and its generalization

Zero-truncated count data (e.g., days of staying in hospital; survival weeks of female patients with breast cancer) often arise in various fields such as medical studies. To model such data, the zero-truncated Poisson (ZTP) distribution is commonly utilized to investigate the relationship between the response counts and a set of covariates. For existing ZTP regression models, it is very hard to explain the regression coefficients β or it is quite difficult to perform a constrained optimization to calculate the maximum likelihood estimates (MLEs) of β. This paper aims to introduce a new mean regression model for the ZTP distribution with a clear interpretation about the regression coefficients. Because of a challenge that the original Poisson mean parameter λi cannot be expressed explicitly by the ZTP mean parameter μi, an embedded Newton–Raphson algorithm is developed to calculate the MLEs of regression coefficients. The construction of bootstrap confidence intervals is presented and three hypothesis tests (i.e., the likelihood ratio test, the Wald test and the score test) are considered. Furthermore, the ZTP mean regression model is generalized to the mean regression model for the k-truncated Poisson distribution. Simulation studies are conducted and two real data are analyzed to illustrate the proposed model and methods.

An Avant-Garde Construction for Subclasses of Analytic Bi-Univalent Functions

The zero-truncated Poisson distribution is an important and appropriate model for many real-world applications. Here, we exploit the zero-truncated Poisson distribution probabilities to construct a new subclass of analytic bi-univalent functions involving Gegenbauer polynomials. For functions in the constructed class, we explore estimates of Taylor–Maclaurin coefficients a2 and a3, and next, we solve the Fekete–Szegő functional problem. A number of new interesting results are presented to follow upon specializing the parameters involved in our main results.

Consolidation of a Certain Discrete Probability Distribution with a Subclass of Bi-Univalent Functions Involving Gegenbauer Polynomials

In this work, we introduce and investigate a new subclass of analytic bi-univalent functions based on subordination conditions between the zero-truncated Poisson distribution and Gegenbauer polynomials. More precisely, we will estimate the first two initial Taylor–Maclaurin coefficients and solve the Fekete–Szegö functional problem for functions belonging to this new subclass.

Double Moving Average Control Chart for Zero-Truncated Poisson Distribution

The objective of this paper is to present an explicit formula for Double Moving Average (DMA) control charts when the observation follows a Zero-Truncated Poisson (ZTP) distribution. The popular characteristics of a control chart are Average Run Length (ARL0 ) for in-control process and Average Run Length (ARL1 ) for out-of-control process. Whereas ARL0 is the mean of observations taken before, a system gives an out-of-control signalled. For ARL1 is the expected number of observations taken from an out-of-control process until the control chart is shown the outof-control signal. The ARL results indicated that the DMA control chart for Zero-Truncated Poisson distribution (DMAZTP ) is superior to the MA control chart for Zero-Truncated Poisson distribution (MAZTP ) for detecting all magnitudes of shifts. This explicit formula is accurate, as well as easy to be implemented and used by practitioners

Zero truncated Poisson model: an alternative approach for analyzing count data with excess zeros

The inference procedure under generalized linear model breaks down when there exists too many zeros in count data that the parent distribution cannot accommodate. We argue that the process of separating ‘bad' zeros through so-called zero inflated count model will lead to a less optimal estimators of the regression parameters than that obtained by fitting zero truncated distribution after discarding ‘all’ zeros. In this paper extensive simulation studies have been carried out to compare performance of fitting zero-truncated Poisson (ZTP), Poisson hurdle (PH) and zero-inflated Poisson (ZIP) models when the data generating process is either ZIP or PH. This study reveals the fact that for analyzing Poisson count data subject to excess zeros, instead of fitting a zero-inflated model, the traditional ZTP would be the best choice. As an illustration, results have been compared by analyzing antenatal care seeking behavior data extracted from a demographic health survey of Bangladesh.

“The Road not taken” statistically taken: A probabilistic analysis of Frost’s poem

“The Road Not Taken” is an exemplary poem composed by Robert Frost in 1916. The focal point of the paper is a careful statistical study of this poem intensifying several features such as: Whether the number of words per line follows uniform distribution or not. Studying the frequency distribution of word length and fitting a suitable probability model. Do the number of vowels follow a binomial or a quasi binomial model? Are the number of letters per line uniformly distributed? Computing the empirical probability of each alphabet in the poem. Analysis of rhythm in the poem. Analysis of parts of speech used in the poem. From the statistical analysis it is found that a Zero truncated Poisson distribution is a good fit for the frequency distribution of word length. The number of words per line may be taken as distributed uniformly in the poem which is tested by Chi Square goodness of fit test. Binomial distribution turns out to be a good fit for the number of vowels per stanza in the poem. The number of letters per line are distributed uniformly in the poem which is also tested by Chi Square goodness of fit test. The poem is following the rhythmic scheme abaab cdccd efeef ghggh. The number of words between two successive rhythmic words can be trichohotomized. The proportion of the first 13 letters (A to M) is more and significantly different as compared to that of the last 13 letters (N to Z). Finally verb is the maximum used part of speech in the poem.