On Positive Inflated Geometric Distribution: Properties and Applications

We consider data modelling under one inflation for zero-truncated count data, as they typically arise in capture-recapture modelling. One-inflation in zero-truncated count data has recently found considerable attention. In this regard, zero-truncated New Discrete distribution and a distribution to a point mass at one are used to create a one-inflated model namely one-inflated zero-truncated New Discrete distribution. Its reliability characteristics, generating functions, and distributional properties are investigated in some detail. which includes survival function, hazard rate function, probability generating function, characteristic function, variance, skewness, and kurtosis. Monte Carlo Simulation have been undertaken to evaluate the effectiveness of the maximum likelihood estimators. To test the compatibility of our proposed model, the baseline model and the proposed model are distinguished by using the two different test procedures. The adaptability of the suggested model is demonstrated using two real-life datasets from separate domains by taking various performance measures into consideration.

A weighted version of Lindley Distribution has been presented in this paper, offering a new single-parameter lifetime distribution. The expressions for the moment generating function (mgf), cumulant generating function (cgf), characteristic function, moments, reliability and hazard rate functions of the new weighted Lindley distribution have also been obtained. Tierney and Kadane approximation method has been used to derive Bayes estimators for its parameter (θ), reliability function R(t), and hazard rate function h(t) under three priors namely uniform, exponential and gamma. The findings have been illustrated using various randomly produced data sets from the proposed model using simulation technique, with each sample replicated 10,000 times. The Bayes Risks have been estimated under Squared Error Loss Function (SELF). Two real life data sets have further been used to illustrate its utility. It is finally concluded that gamma prior outperforms uniform and exponential priors for computing the Bayes estimates of the parameter θ, reliability function R(t) and hazard rate function h(t) of the proposed weighted Lindley distribution.

As an extension of the standard Weibull distribution, a new crucial distribution termed alpha power Weibull distribution has been presented. It can model decreasing, increasing, bathtub, and upside-down bathtub failure rates. This research investigates the estimation of model parameters and some of its reliability characteristics using progressively Type-II censored data. To get estimates of unknown parameters, reliability, and hazard rate functions, the maximum likelihood, and Bayesian estimation approaches are studied. To acquire estimated confidence intervals for unknown parameters and reliability characteristics, the maximum likelihood asymptotic properties are used. The Markov chain Monte Carlo approach is used in Bayesian estimation to provide Bayesian estimates under squared error and LINEX loss functions. Furthermore, the highest posterior density credible intervals of the parameters and reliability characteristics are determined. A Monte Carlo simulation study is used to investigate the accuracy of various point and interval estimators. In addition, various optimality criteria are used to choose the best progressive censoring schemes. Two real data from the engineering field are analyzed to demonstrate the applicability and significance of the proposed approaches. Based on numerical results, the Bayesian procedure for estimating the parameters and reliability characteristics of alpha power Weibull distribution is recommended. The analysis of two real data sets showed that the alpha power Weibull distribution is a good model to investigate engineering data in the presence of progressive Type-II censoring.

In this research, the statistical inference of unknown lifetime parameters is proposed in the presence of independent competing risks using a progressive Type-II censored dataset. The lifetime distribution associated with a failure mode is assumed to follow the new Pareto distribution, with consideration given to two distinct competing failure reasons. Maximum likelihood estimators (MLEs) for the unknown model parameters, as well as reliability and hazard functions, are derived, noting that they are not expressible in closed form. The Newton–Raphson, expectation maximization (EM), and stochastic expectation maximization (SEM) methods are employed to generate maximum likelihood (ML) estimations. Approximate confidence intervals for the unknown parameters, reliability, and hazard rate functions are constructed using the normal approximation of the MLEs and the normal approximation of the log-transformed MLEs. Additionally, the missing information principle is utilized to derive the closed form of the Fisher information matrix, which, in turn, is used with the delta approach to calculate confidence intervals for reliability and hazards. Bayes estimators are derived under both symmetric and asymmetric loss functions, with informative and non-informative priors considered, including independent gamma distributions for informative priors. The Monte Carlo Markov Chain sampling approach is employed to obtain the highest posterior density credible intervals and Bayesian point estimates for unknown parameters and reliability characteristics. A Monte Carlo simulation is conducted to assess the effectiveness of the proposed techniques, with the performances of the Bayes and maximum likelihood estimations examined using average values and mean squared errors as benchmarks. Interval estimations are compared in terms of average lengths and coverage probabilities. Real datasets are considered and examined for each topic to provide illustrative examples.

In this paper, we adopt the Ishita lifetime distribution to analyze biomedical science and engineering lifetime data under an accelerated life test (ALT) model. This data is exposed concerning the mechanism of a type-I generalized hybrid censoring scheme under a partially step-stress ALT model. The model parameters and the parameters of life (survival and hazard rate function) are estimated using maximum likelihood and Bayesian estimation. Also, the interval estimators are formulated with respect to the normal distribution of the maximum likelihood estimate, two parametric bootstrap confidence techniques, and Bayesian credible intervals. Two real data sets are analyzed to illustrate the proposed methods. Monte Carlo simulation is used to compare various methods.

In this paper, a new two-parameter discrete distribution is introduced. It belongs to the family of the weighted geometric distribution (GD), with the feature of using a particular trigonometric weight. This configuration adds an oscillating property to the former GD which can be helpful in analyzing the data with over-dispersion, as developed in this study. First, we present the basic statistical properties of the new distribution, including the cumulative distribution function, hazard rate function and moment generating function. Estimation of the related model parameters is investigated using the maximum likelihood method. A simulation study is performed to illustrate the convergence of the estimators. Applications to two practical datasets are given to show that the new model performs at least as well as some competitors.

In this paper, we discuss the problem of constant-stress accelerated life test when the failure data are progressive type-I interval censored. Both classical and Bayesian inferential approaches of the distribution parameters and reliability characteristics are discussed. In the classical scenario, the maximum likelihood estimates are approximated using the EM algorithm and the mid-point approximation method. Furthermore, the model's parameters are estimated by method of moments. Next in the Bayesian framework, the point estimates of unknown parameters are obtained using Tierney-Kadane's technique and Markov Chain Monte Carlo (MCMC) method. In addition, both approximate and credible confidence intervals (CIs) of the estimators are constructed. For illustration purpose, a Monte Carlo simulation is conducted to investigate the performance of the proposed estimators and a real data set is analysed.

We study some mathematical properties of a new generator of continuous distributions with two extra parameters called the exponentiated half-logistic family. We present some special models. We investigate the shapes of the density and hazard rate function. We derive explicit expressions for the ordinary and incomplete moments, quantile and generating functions, probability weighted moments, Bonferroni and Lorenz curves, Shannon and Rényi entropies, and order statistics, which hold for any baseline model. We introduce two bivariate extensions of this family. We discuss the estimation of the model parameters by maximum likelihood and demonstrate the potentiality of the new family by means of two real data sets.

In this article, we introduce a discrete analogue of odd Lomax generator of distributions. The new discrete class can be utilized as a probabilistic tool to generalize any discrete baseline model. After proposing the new class, two special discrete models are investigated and discussed in detail. Some mathematical and statistical properties including, probability mass function, hazard rate function, quantile, crude moments, index of dispersion, entropies, order statistics, and L-moment statistics, are derived. It is found that the presented discrete class can be used to model symmetric and asymmetric data under different types of kurtosis shapes. It can be utilized to explain and analyze overdispersion data with extreme, zero-inflated or outliers? observations. Furthermore, it can be applied to discuss various shapes of hazard rates including monotone increasing, monotone decreasing, unimodal, bathtub, unimodal-bathtub, among others. We discuss the estimation of the class parameters by the maximum likelihood approach. The performance of the estimation method is tested via Markov chain Monte Carlo (MCMC) simulation technique. Finally, to demonstrate the proposed methodology in a real-life scenario, three real data sets are considered to show the applicability of the proposed generator.

This article deals with the estimation of parameters and reliability characteristics of Lindley distribution underrandom censoring. Expected time on test based on randomly censored data is obtained. The maximum likelihood estimators of the unknown parameters and reliability characteristics are derived. The asymptotic, bootstrap p and bootstrap t confidence intervals of the parameters are constructed. The Bayes estimators of the parameters and reliability characteristics under squared error loss function using non-informative and gamma informative priors are obtained. For computing of Bayes estimates, Lindley approximation and MCMC methods are considered. Highest posterior density (HPD) credible intervals of the parameters are obtained using MCMC method. Various estimation procedures are compared using a Monte Carlo simulation study. Finally, a real data set is analyzed for illustration purposes.

The continuous development of flexible probability distributions has become essential for accurately describing complex stochastic phenomena. In this study, a new theoretical based on the New Sine Type II Generalized Topp-Leone distribution is investigated. The model is constructed by integrating the sine trigonometric transformation with the Type II Topp-Leone generator applied to an arbitrary baseline distribution. This combination introduces additional shape parameters that enhance the adaptability of the resulting family of distributions. Several mathematical properties of the proposed model are derived. These include explicit expressions for the cumulative distribution function (cdf), probability density function (pdf), survival function, and hazard rate function. In addition, analytical forms of the quantile function, moments, moment generating function, and order statistics are established. The author’s adopted two estimation techniques in this work: the maximum likelihood estimation (MLE) and the maximum product of spacing (MPLS) estimation methods. The structural characteristics of the distribution reveal that it is capable of producing a wide range of density and hazard rate shapes depending on the selected baseline model and parameter values. The theoretical results presented in this work provide a general framework for constructing new members of the family and offer a foundation for further statistical developments involving estimation techniques and practical implementations. Plots of the pdf, cdf as well as the validity check on the pdf of the new model indicate that the model is adequate capable of handling all the intricate associated with real-life datasets.

Despite the existence of various competing risk (CR) models with varied formulations, their practical implementations do not align with some present CR problems. This is because certain CR data necessitates constructing CR models that are based on hazard rate (HR) patterns associated with cause‐specific failure times. To circumvent this obstacle, this study proposes an easy‐to‐handle methodology that involves constructing a minimum variable following Weibull or Dhillon failures. The model is called the Weibull–Dhillon competing risk (WDCR) model. Compared to many existing literature, the proposed WDCR model can analyze failure times that exhibit increasing and bathtub HRs and effectively characterize failure times linked to inverted bathtub and modified bathtub HRs. We examined several reliability characteristics and demonstrated the relationship between the model's mean residual life and HR function. Due to its complicated joint posterior distribution, the Hamiltonian Monte Carlo (HMC) procedure is employed for Bayesian inference of WDCR model parameters and reliability features. We provide two illustrations to illustrate the proposed model's practical value. The outcomes were better than those of the other contenders, indicating a valuable approach to modeling different CR data, particularly in the context of reliability analyses or other quantitative studies.

This paper presents a new generalized bounded distribution called the Kumaraswamy unit-Gompertz (KUG) distribution. Some of the Mathematical properties which include; the density function, cumulative distribution function, survival and hazard rate functions, quantile, mode, median, moment, moment generating function, Renyi entropy and distribution of order statistics are derived. We employ the maximum likelihood estimation method to estimate the unknown parameters of the proposed KUG distribution. A Monte Carlo simulation study is carried out to investigate the performance of the maximum likelihood estimates of the unknown parameters of the proposed distribution. Two real datasets are used to illustrate the applicability of the proposed KUG distribution in lifetime data analysis.

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.

This article introduces a two-parameter exponentiated Teissier distribution. It is the main advantage of the distribution to have increasing, decreasing and bathtub shapes for its hazard rate function. The expressions of the ordinary moments, identifiability, quantiles, moments of order statistics, mean residual life function and entropy measure are derived. The skewness and kurtosis of the distribution are explored using the quantiles. In order to study two independent random variables, stress–strength reliability and stochastic orderings are discussed. Estimators based on likelihood, least squares, weighted least squares and product spacings are constructed for estimating the unknown parameters of the distribution. An algorithm is presented for random sample generation from the distribution. Simulation experiments are conducted to compare the performances of the considered estimators of the parameters and percentiles. Three sets of real data are fitted by using the proposed distribution over the competing distributions.