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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.

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.

In this paper, we construct a new Lagrangian discrete distribution, named the Lagrangian zero truncated Poisson distribution (LZTPD). It can be presented as a generalization of the zero truncated Poissson distribution (ZTPD) and an alternative to the intervened Poisson distribution (IPD), which was elaborated for modelling both over-dispersed and under-dispersed count datasets. The mathematical aspects of the LZTPD are thoroughly investigated, and its connection to other discrete distributions is crucially observed. Further, we define a finite mixture of LZTPDs and establish its identifiability condition along with some distributional aspects. Statistical work is then performed. The maximum likelihood and method of moment approaches are used to estimate the unknown parameters of the LZTPD. Simulation studies are also undertaken as an assessment of the long-term performance of the estimates. The significance of one additional parameter in the LZTPD is tested using a generalized likelihood ratio test. Moreover, we propose a new count regression model named the Lagrangian zero truncated Poisson regression model (LZTPRM) and its parameters are estimated by the maximum likelihood estimation method. Two real-world datasets are considered to demonstrate the LZTPD’s real-world applicability, and healthcare data are analyzed to demonstrate the LZTPRM’s superiority.

In applied statistical research, a common type of dataset used is count data. However, there are cases where zero events are not observed in the dataset. Consequently, the Poisson distribution, a basic discrete probability model, is inappropriate in such situations. Instead, we need to consider the so-called Zero-Truncated Poisson distribution. Unfortunately, deriving the simplest Method of Moments estimators for the parameter and mean of this distribution in closed form is not feasible. Therefore, estimating the Zero-Truncated Poisson parameter and mean becomes a challenging problem. In this article, the authors used the classical delta method to apply an estimation procedure for the zero-truncated Poisson parameter and the mean and investigated their asymptotic normality. Furthermore, we demonstrated the practicality of our approach through an application to a real-life dataset on unrest events in the southern border area of Thailand.

The problem of estimating the ratio of the means of a two-component Poisson mixture model is considered, when each component is subject to zero-inflation, i.e., excess zero counts. The resulting zero-inflated Poisson mixture (ZIPM) model can be viewed as a three-component Poisson mixture model with one degenerate component. The EM algorithm is applied to obtain frequentist estimators and their standard errors, the latter determined via an explicit expression for the observed information matrix. As an intermediate step, we derive an explicit expression for standard errors in the two-component Poisson mixture model (without zero-inflation), a new result. The ZIPM model is applied to simulated data and real ecological count data of frigatebirds on the Coral Sea Islands off the coast of Northeast Australia.

A new useful version of the Fréchet model is introduced and studied. Some of its properties are derived. The method of maximum likelihood is used for estimating the unknown parameter via two real data applications. The new version is much better than other important competitive Fréchet models in modeling two real data sets.

Several situations interact with count data without zero values, such as the number of deaths associated with road traffic accidents and other factors and the number of European red mites on apple leaves. Recently, the zero-truncated Poisson-Ishita distribution (ZTPID) has been proposed for such data, but its statistical inference, especially confidence interval estimation for the difference between two means, has not been proposed. In this paper, the percentile, simple, and biased-corrected and accelerated bootstrap confidence intervals are proposed and compared the performance in terms of coverage probability and average length, which are estimated from the Monte Carlo simulation method. The parameter values and two population means of ZTPID are varied, resulting in populations with mean differences ranging from small to large values. The simulation results show that small and medium sample sizes are inadequate to attain the nominal level of confidence for all settings and bootstrap methods. When the sample size is large enough, all bootstrap confidence intervals do not substantially differ. Overall, it is observed that the percentile bootstrap confidence interval outperforms the other bootstrap confidence intervals, even with small sample sizes. Lastly, each of the bootstrap confidence intervals is applied to the number of unrest events that occurred in the southern border area of Thailand.

A new version of the Fréchet model is investigated and studied. Some of its are mathematically derived. The well-known method of maximum likelihood is used in estimating the unknown parameters. The new version is much better than other important competitive Fréchet versions in modeling reliability real data.

The analysis and modeling of zero truncated count data is of primary interest in many elds such as engineering, public health, sociology, psychology, epidemiology. Therefore, in this article we have proposed a new and simple structure model, named a zero truncated discrete Lindley distribution. Thedistribution contains some submodels and represents a two-component mixture of a zero truncated geometric distribution and a zero truncated negative binomial distribution with certain parameters. Several properties of the distribution are obtained such as mean residual life function, probability generating function, factorial moments, negative moments, moments of residual life function, Bonferroni and Lorenz curves, estimation of parameters, Shannon and Renyi entropies, order statistics with the asymptotic distribution of their extremes and range, a characterization, stochastic ordering and stress-strength parameter. Moreover, the collective risk model is discussed by considering theproposed distribution as primary distribution and exponential and Erlang distributions as secondary ones. Test and evaluation statistics as well as three real data applications are considered to assess the peformance of the distribution among the most frequently zero truncated discrete probability models.

In this paper, we introduce a new stationary integer-valued autoregressive process of the first order with zero truncated Poisson marginal distribution. We consider some properties of this process, such as autocorrelations, spectral density and multi-step ahead conditional expectation, variance and probability generating function. Stationary solution and its uniqueness are obtained with a discussion to strict stationarity and ergodicity of such process. We estimate the unknown parameters by using conditional least squares estimation, nonparametric estimation and maximum likelihood estimation. The asymptotic properties and asymptotic distributions of the conditional least squares estimators have been investigated. Some numerical results of the estimators are presented and some sample paths of the process are illustrated. Some possible applications of the introduced model are discussed.