We explore the cosmic evolution of the accelerating universe in the framework of dynamical Chern–Simons modified gravity in an interacting scenario by taking the flat homogeneous and isotropic model. For this purpose, we take some parametrizations of the equation of state parameter. This parametrization may be a Taylor series extension in the redshift, a Taylor series extension in the scale factor or any other general parametrization of $\omega$ . We analyze the interaction term which calculates the action of interaction between dark matter and dark energy. We explore various cosmological parameters such as deceleration parameter, squared speed of sound, Om-diagnostic and statefinder via graphical behavior.

The weighted distributions are useful when the sampling is done using an unequal probability of the sampling units. The Weighted Power function distribution (WPFD) has applications in the fields of reliability engineering, management sciences and survival analysis. WPFD is more beneficial in Statistical process control (SPC). SPC is defined as the use of statistical techniques to control a process or production method. SPC tools and procedures can help to monitor process behaviour, discover problems in internal systems, and find solutions for production issues. To identify and remove the variation in different reliability processes and also to monitor the reliability of machines where the number of errors follows WPFD, we develop control charts to keep the process in control. A memory-based control chart like an exponentially weighted moving average (EWMA) control chart and an extended exponentially weighted moving average (EEWMA) control chart are discussed and compared each other. The proposal of these control charts is based on the modified maximum likelihood estimator (MMLE) under the shape parameter of WPFD. We have presented Monte Carlo simulation technique and a real-life application to compare the proposed control charts. This study shows that an EEWMA control chart based on MMLE performs better than EWMA control chart, when the underlying distribution of the errors in process monitoring follows WPFD. These findings can be useful for researchers and practitioners in dealing with production errors and optimizing the output.

A fully parametric family of distributions called the new generalized family (NGF) is proposed. The cumulative distribution function of the proposed NGF is constructed based on the reciprocal of the hazard rate function. Furthermore, a special sub-family called the new generalized Pareto (NGP) family is also addressed. A special sub-model of the NGP class called the new Kumaraswamy Pareto (NKP) distribution is introduced. The NKP model provides flexible shapes for its hazard and density functions. The properties of the NKP distribution are derived. In addition, the inferences under complete and bootstrapped samples of the NKP parameters are addressed. Monte Carlo simulation studies are also performed to explore the behavior of different estimators. Three real-life datasets are fitted using the NKP distribution, showing the superiority fit of the proposed statistical model.

In statistical modeling, lifetime datasets play an important and significant role. Weighted distributions can be useful to attain the essential purposes relating to the developed distributions’ flexibility and to improve further compliances in data forming; they provide the extension of distributions. The current study suggests a new best-fit probability distribution model based on Pareto distribution. The new model is called Survival Weighted Pareto (SWP) distribution model. Furthermore, the statistical expressions for different properties, which include moments, inverse moments, quantile function, and order statistics, are reported. Moreover, to check the performance and the behavior of different estimators using different sample sizes, a simulation study, which is based on SWP distribution, is performed. According to the simulation results, it is established that the proposed model is a most useful fitted probability model. In addition, an application of the SWP model for three datasets is provided to check the goodness of fit measures of the model.