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Reliability assessments are pivotal in evaluating system quality and have found extensive application in manufacturing. This research delves into a system comprising five components, one of which is a bridge network. The components are presumed to follow a Bilal lifetime distribution with a failure rate that changes over time. Four distinct methods are employed to enhance the components within the system. This study involves the computation of  δ -fractiles and reliability equivalence factors (REFs). Additionally, a numerical case study is provided to elucidate the theoretical findings. [ABSTRACT FROM AUTHOR]

The significance of applications in count data modeling to real-life scenarios has been emphasized in numerous studies. The present paper focuses on a discrete version of the Bilal distribution, the new discrete Bilal distribution and its structural properties. The proposed distribution's mathematical and statistical characteristics are thoroughly investigated. The model parameter estimation is carried out by maximum likelihood estimation, and a simulation study is performed. The performance of the proposed model is evaluated by real data analysis. The significance of the proposed distribution in time series is demonstrated with the associated INAR(1) process having innovations as the proposed model. The parameter estimation of the INAR(1) process is carried out using conditional maximum likelihood, conditional least squares and Yule–Walker estimation procedures, and real data is used to determine the empirical importance of the proposed process. [ABSTRACT FROM AUTHOR]

Asymmetric distributions, as opposed to symmetric distributions, may be more resilient to extreme values or outliers. Furthermore, when data show substantial skewness, asymmetric distributions can shed light on the underlying processes or phenomena being investigated. In this direction, the size-biased Bilal distribution (SBBD) is suggested in this study as a generalization to the Bilal distribution. The length-biased and area-biased Bilal distributions are discussed in detail as two special cases. The main statistical properties of the distribution including the rth moment, coefficients of variation, skewness, kurtosis, moment generating function, incomplete moments, moments of residual life, harmonic mean, Fisher's information, and the Rényi entropy as a measure of uncertainty are presented. Graphical representations of the cumulative distribution, probability density, odds, survival, hazard, reversed hazard rate, and cumulative hazard functions are presented for further explanation of the distribution behavior. In addition, the methods of moments and maximum likelihood estimates are taken into account for estimating the model parameters. A simulation study is carried out to see the efficiency of the maximum likelihood in terms of standard errors and bias. Real data sets of precipitation and myeloid leukemia patients are considered to show the practical significance of the suggested distributions as an alternative to some well-known distributions such as the Rama, Rani, Bilal, and exponential distributions. It is found that the size-biased Bilal distribution is right-skewed and has a superior fitting performance compared to the other distributions in this study. [ABSTRACT FROM AUTHOR]

In this research work, a new three-parameter lifetime distribution is introduced and studied. It is called the Harris extended Bilal distribution due to its construction from a mixture of the famous Bilal and Harris distributions, resulting from a branching process. The basic properties, such as the moment generating function, moments, quantile function, and Rényi entropy, are discussed. We show that the hazard rate function has ideal features for modeling increasing, upside-down bathtub, and roller-coaster data sets. In a second part, the Harris extended Bilal model is investigated from a statistical viewpoint. The maximum likelihood estimation is used to estimate the parameters, and a simulation study is carried out. The flexibility of the proposed model in a hydrological data analysis scenario is demonstrated using two practical data sets and compared with important competing models. After that, we establish an acceptance sampling plan that takes advantage of all of the features of the Harris extended Bilal model. The operating characteristic values, the minimum sample size that corresponds to the maximum possible defects, and the minimum ratios of lifetime associated with the producer’s risk are discussed.

The aim of this work is to develop a new outstanding lifetime distribution, dubbed the power Bilal (PB) distribution. Both of the probability distribution function (pdf) and the cumlative distribution function (cdf) of the PB distribution have a simple forms. The suggested distribution’s moments, incomplete moments as well as the quantile function are deduced and acquired in explicit forms as a result of its simple forms. Seven estimation methods for estimating the PB distribution parameters are mentioned, and numerical simulations are used to compare the proposed approaches using partial and overall ranks. According to the simulation results of this work, the maximum likelihood estimators are advised to be the best estimation method for estimating the parameters of the PB distribution, when the data genrated from the PB distribution. We shows the importance and flexibility of the PB distribution by comparing it to other existing competing distributions using two different real data sets from COVID-19 mortality rates of two countries.

In this research work, a new three-parameter lifetime distribution is introduced and studied. It is called the Harris extended Bilal distribution due to its construction from a mixture of the famous Bilal and Harris distributions, resulting from a branching process. The basic properties, such as the moment generating function, moments, quantile function, and Rényi entropy, are discussed. We show that the hazard rate function has ideal features for modeling increasing, upside-down bathtub, and roller-coaster data sets. In a second part, the Harris extended Bilal model is investigated from a statistical viewpoint. The maximum likelihood estimation is used to estimate the parameters, and a simulation study is carried out. The flexibility of the proposed model in a hydrological data analysis scenario is demonstrated using two practical data sets and compared with important competing models. After that, we establish an acceptance sampling plan that takes advantage of all of the features of the Harris extended Bilal model. The operating characteristic values, the minimum sample size that corresponds to the maximum possible defects, and the minimum ratios of lifetime associated with the producer's risk are discussed. [ABSTRACT FROM AUTHOR]

Weighted distributions play an important role when observations from a sample are recorded with unequal probabilities. They are useful for the efficient modeling of statistical data when the original distributions are not appropriate. In this paper, a new weighted distribution is proposed. Various statistical properties of the proposed distribution such as survival function, hazard rate function, mean residual life function, moments, moment generating function, Bonferroni curve, Lorenz curve, and order statistic are presented. The Bayesian estimator of the distribution parameter is derived. The behavior of the Bayesian estimator is assessed by a simulation study. Furthermore, a regression model is developed based on the proposed distribution. Some real data applications are analyzed to show the potentiality of the proposed models. [ABSTRACT FROM AUTHOR]

The aim of this work is to develop a new outstanding lifetime distribution, dubbed the power Bilal (PB) distribution. Both of the probability distribution function (pdf) and the cumlative distribution function (cdf) of the PB distribution have a simple forms. The suggested distribution's moments, incomplete moments as well as the quantile function are deduced and acquired in explicit forms as a result of its simple forms. Seven estimation methods for estimating the PB distribution parameters are mentioned, and numerical simulations are used to compare the proposed approaches using partial and overall ranks. According to the simulation results of this work, the maximum likelihood estimators are advised to be the best estimation method for estimating the parameters of the PB distribution, when the data genrated from the PB distribution. We shows the importance and flexibility of the PB distribution by comparing it to other existing competing distributions using two different real data sets from COVID-19 mortality rates of two countries. [ABSTRACT FROM AUTHOR]

This study proposes a new one-parameter discrete distribution, called a discrete Bilal distribution. The structural properties of the proposed distributions, including the shape of the probability mass function, mode, moments, skewness, kurtosis, index of dispersion, mean deviation, stress-strength reliability, and order statistics, are derived. These properties are expressed in closed-forms. The maximum likelihood and method of moments estimation methods are considered to estimate the unknown model parameter. An extensive simulation study is carried out to examine the finite sample performance of estimation methods. The usefulness of the proposed model is illustrated in the first-order integer-valued autoregressive process. The empirical importance of the proposed models is proved through three real data applications.

This study introduces the Poisson-Bilal distribution and its associated two models for modeling the over-dispersed count data sets. The Poisson-Bilal distribution has tractable properties and explicit forms for its statistical properties. A new over-dispersed count regression model and integer-valued autoregressive process with flexible innovation distribution are defined and studied comprehensively. Two real data sets are analyzed to prove empirically the importance of proposed models. Empirical findings show that the Poisson-Bilal distribution has important application fields in time series and regression modeling. [ABSTRACT FROM AUTHOR]