Exponentiated power generalized Weibull power series family of distributions: Properties, estimation and applications

Modeling of lifetime scenarios with non-monotonic failure rates

The Weibull distribution is an important continuous distribution that is cardinal in reliability analysis and lifetime modeling. On the other hand, it has several limitations for practical applications, such as modeling lifetime scenarios with non-monotonic failure rates. However, accurate modeling of non-monotonic failure rates is essential for achieving more accurate predictions, better risk management, and informed decision-making in various domains where reliability and longevity are critical factors. For this reason, we introduce a new three parameter lifetime distribution-the Modified Kies Weibull distribution (MKWD)-that is able to model lifetime scenarios with non-monotonic failure rates. We analyze the statistical features of the MKWD, such as the quantile function, median, moments, mean, variance, skewness, kurtosis, coefficient of variation, index of dispersion, moment generating function, incomplete moments, conditional moments, Bonferroni, Lorenz, and Zenga curves, and order statistics. Various measures of uncertainty for the MKWD such as Rényi entropy, exponential entropy, Havrda and Charvat entropy, Arimoto entropy, Tsallis entropy, extropy, weighted extropy and residual extropy are computed. We discuss eight different parameter estimation methods and conduct a Monte Carlo simulation study to evaluate the performance of these different estimators. The simulation results show that the maximum likelihood method leads to the best results. The effectiveness of the newly suggested model is demonstrated through the examination of two different sets of real data. Regression analysis utilizing survival times data demonstrates that the MKWD model offers a superior match compared to other current distributions and regression models.

Strategies of Modelling Incident Outcomes Using Cox Regression to Estimate the Population Attributable Risk

When the Cox model is applied, some recommendations about the choice of the time metric and the model's structure are often disregarded along with the proportionality of risk assumption. Moreover, most of the published studies fail to frame the real impact of a risk factor in the target population. Our aim was to show how modelling strategies affected Cox model assumptions. Furthermore, we showed how the Cox modelling strategies affected the population attributable risk (PAR). Our work is based on data collected in the North-West Province, one of the two PURE study centres in South Africa. The Cox model was used to estimate the hazard ratio (HR) of mortality for all causes in relation to smoking, alcohol use, physical inactivity, and hypertension. Firstly, we used a Cox model with time to event as the underlying time variable. Secondly, we used a Cox model with age to event as the underlying time variable. Finally, the second model was implemented with age classes and sex as strata variables. Mutually adjusted models were also investigated. A statistical test to the multiplicative interaction term the exposures and the log transformed time to event metric was used to assess the proportionality of risk assumption. The model's fitting was investigated by means of the Akaike Information Criteria (AIC). Models with age as the underlying time variable with age and sex as strata variables had enhanced validity of the risk proportionality assumption and better fitting. The PAR for a specific modifiable risk factor can be defined more accurately in mutually adjusted models allowing better public health decisions. This is not necessarily true when correlated modifiable risk factors are 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.

Bivariate Step-Stress Accelerated Life Tests for the Kavya–Manoharan Exponentiated Weibull Model under Progressive Censoring with Applications

A new three-parameter survival model is proposed using the Kavya–Manoharan (KM) transformation family and the exponentiated Weibull (EW) distribution. The shapes of the pdf for the new model can be asymmetric and symmetric shapes, such as unimodal, decreasing, right-skewed and symmetric. In addition, the shapes of the hrf for the suggested model can be increasing, decreasing, constant and J-shaped. Statistical properties are obtained: quantile function, mode, moments, incomplete moments, residual life time, reversed residual life time, probability weighted moments, order statistics and entropy. We discuss the maximum likelihood estimation for the model. The relevance and flexibility of the model are demonstrated using two real datasets. The distribution is very flexible, and it outperforms many known distributions, such as the three-parameter exponentiated Weibull, the modified Weibull model, the Kavya–Manoharan Weibull, the extended Weibull, the odd Weibull inverse Topp–Leone and the extended odd Weibull inverse Nadarajah–Haghigh model. A bivariate step-stress accelerated life test based on progressive type-I censoring (PTIC) using the model is presented. This pattern is noticed when a particular number of lifetime test units are routinely eliminated from the test at the conclusion of each post-test period of time. Minimizing the asymptotic variance of the MLE of the log of the scale parameter at design stress under PTIC yields an expression for the ideal test plan under PTIC.

A New Family of Lifetime Models: Theoretical Developments with Applications in Biomedical and Environmental Data

With the aim of identifying a probability model that not only correctly describes the stochastic behavior of extreme environmental factors such as excess rain, acid rain pH level, and concentrations of ozone, but also measures concentrations of NO2 and leads deliberations, etc., for a specific site or multiple site forms as well as for life testing experiments, we introduced a novel class of distributions known as the Sine Burr X−G family. Some exceptional prototypes of this class are proposed. Statistical assets of the presented class, such as density function, complete and incomplete moments, average deviation, and Lorenz and Bonferroni graphs, are proposed. Parameter estimation is made via the likelihood method. Moreover, the application is explained by using four real data sets. We have also illustrated the significance and elasticity of the proposed class in the above-mentioned stochastic phenomenon.

A new lifetime family of distributions: Theoretical developments and analysis of COVID 19 data

In parametric statistical modeling and inference, it is critical to develop generalizations of existing statistical distributions to make them more flexible in modeling real data sets. Thus , this paper contributes to the subject by investigating a new flexible and versatile generalized family of distributions defined from the alliance of the families known as beta-G and Topp-Leone generated (TL-G), inspiring the name of BTL-G family. The characteristics of this new family are studied through analytical, graphical and numerical approaches. Statistical features of the family such as expansion of density function (pdf), cumulative function (cdf), moments (MOs), incomplete moments (IMOs), mean deviation (MDE), and entropy (ENT) are calculated. The model parameters' maximum likelihood estimates (MaxLEs) and Bayesian estimates (BEs) are provided. Symmetric and Asymmetric Bayesian Loss function have been discussed. A complete simulation study is proposed to illustrate their numerical efficiency. The considered model is also applied to analyze two different kinds of genuine COVID 19 data sets. We show that it outperforms other well-known models defined with the same baseline distribution, proving its high level of adaptability in the context of data analysis.

The Truncated Burr X-G Family of Distributions: Properties and Applications to Actuarial and Financial Data

In this article, the "truncated-composed" scheme was applied to the Burr X distribution to motivate a new family of univariate continuous-type distributions, called the truncated Burr X generated family. It is mathematically simple and provides more modeling freedom for any parental distribution. Additional functionality is conferred on the probability density and hazard rate functions, improving their peak, asymmetry, tail, and flatness levels. These characteristics are represented analytically and graphically with three special distributions of the family derived from the exponential, Rayleigh, and Lindley distributions. Subsequently, we conducted asymptotic, first-order stochastic dominance, series expansion, Tsallis entropy, and moment studies. Useful risk measures were also investigated. The remainder of the study was devoted to the statistical use of the associated models. In particular, we developed an adapted maximum likelihood methodology aiming to efficiently estimate the model parameters. The special distribution extending the exponential distribution was applied as a statistical model to fit two sets of actuarial and financial data. It performed better than a wide variety of selected competing non-nested models. Numerical applications for risk measures are also given.

Box-Cox Gamma-G Family of Distributions: Theory and Applications

This paper is devoted to a new class of distributions called the Box-Cox gamma-G family. It is a natural generalization of the useful Ristić–Balakrishnan-G family of distributions, containing a wide variety of power gamma-G distributions, including the odd gamma-G distributions. The key tool for this generalization is the use of the Box-Cox transformation involving a tuning power parameter. Diverse mathematical properties of interest are derived. Then a specific member with three parameters based on the half-Cauchy distribution is studied and considered as a statistical model. The method of maximum likelihood is used to estimate the related parameters, along with a simulation study illustrating the theoretical convergence of the estimators. Finally, two different real datasets are analyzed to show the fitting power of the new model compared to other appropriate models.

On the Analysis of New COVID-19 Cases in Pakistan Using an Exponentiated Version of the M Family of Distributions

This paper develops the exponentiated Mfamily of continuous distributions, aiming to provide new statistical models for data fitting purposes. It stands out from the other families, as it depends on two baseline distributions, with the use of ratio and power transforms in the definition of the main cumulative distribution function. Thanks to the joint action of the possibly different baseline distributions, flexible statistical models can be created, motivating a complete study in this regard. Thus, we discuss the theoretical properties of the new family, with emphasis on those of potential interest to the overall probability and statistics. Then, a new three-parameter lifetime distribution is derived, with the choices of the inverse exponential and exponential distributions as baselines. After pointing out the great flexibility of the related model, we apply it to analyze an actual dataset of current interest: the daily COVID-19 cases observed in Pakistan from 21 March to 29 May 2020 (inclusive). As notable results, we demonstrate that the proposed model is the best among the 15 top ranked models in the literature, including the inverse exponential and exponential models, several modern extensions of them depending on more parameters, and the “unexponentiated” version of the proposed model as well. As future perspectives, the proposed model can be of interest to analyze data on COVID-19 cases in other countries, for possible comparison studies.

The Exponentiated Truncated Inverse Weibull-Generated Family of Distributions with Applications

In this paper, we propose a generalization of the so-called truncated inverse Weibull-generated family of distributions by the use of the power transform, adding a new shape parameter. We motivate this generalization by presenting theoretical and practical gains, both consequences of new flexible symmetric/asymmetric properties in a wide sense. Our main mathematical results are about stochastic ordering, uni/multimodality analysis, series expansions of crucial probability functions, probability weighted moments, raw and central moments, order statistics, and the maximum likelihood method. The special member of the family defined with the inverse Weibull distribution as baseline is highlighted. It constitutes a new four-parameter lifetime distribution which brightensby the multitude of different shapes of the corresponding probability density and hazard rate functions. Then, we use it for modelling purposes. In particular, a complete numerical study is performed, showing the efficiency of the corresponding maximum likelihood estimates by simulation work, and fitting three practical data sets, with fair comparison to six notable models of the literature.