Proposing new families of probability models for data modeling in applied sectors is a prominent research topic. This paper also proposes a new method based on the trigonometric function to derive the updated form of the existing probability models. The proposed family is called the cotangent trigonometric-G family of distributions. Based on the cotangent trigonometric-G method, a new version of the Weibull model, namely, the cotangent trigonometric Weibull distribution, is studied. Certain mathematical properties of the cotangent trigonometric-G family are derived. The estimators of the cotangent trigonometric-G distributions are obtained via the maximum likelihood method. The Monte Carlo simulation study is conducted to assess the performances of the estimators. Finally, two applications from the health sector are considered to illustrate the cotangent trigonometric-G method. Based on seven evaluating criteria, it is observed that the cotangent trigonometric-G significantly improves the fitting power of the existing models.

Probability distributions play a pivotal and significant role in modeling real-life data in every field. For this activity, a series of probability distributions have been introduced and exercised in applied sectors. This paper also contributes a new method for modeling continuous data sets. The proposed family is called the exponent power sine-G family of distributions. Based on the exponent power sine-G method, a new model, namely, the exponent power sine-Weibull model is studied. Several mathematical properties such as quantile function, identifiability property, and $r^{th}$ moment are derived. For the exponent power sine-G method, the maximum likelihood estimators are obtained. Simulation studies are also presented. Finally, the optimality of the exponent power sine-Weibull model is shown by taking two applications from the healthcare sector. Based on seven evaluating criteria, it is demonstrated that the proposed model is the best competing distribution for analyzing healthcare phenomena.

Probability models are frequently used in numerous healthcare, sports, and policy studies. These probability models use datasets to identify patterns, analyze lifetime scenarios, predict outcomes of interest, etc. Therefore, numerous probability models have been studied, introduced, and implemented. In this paper, we also propose a novel probability model for analyzing data in different sectors, particularly in biomedical and sports sciences. The probability model is called a new modified exponential-Weibull distribution. The heavy-tailed characteristics along with some other mathematical properties are derived. Furthermore, the estimators of the new modified exponential-Weibull are derived. A simulation study of the new modified exponential-Weibull model is also provided. To illustrate the new modified exponential-Weibull model, a practical dataset is analyzed. The dataset consists of seventy-eight observations and represents the recovery time after the injuries in different basketball matches.

Statistical modeling is a crucial phase for decision-making and predicting future events. Data arising from engineering-related fields have most often complex structures whose failure rate possesses mixed state behaviors (i.e., non-monotonic shapes). For the data sets whose failure rates are in the mixed state, the utilization of the traditional probability models is not a suitable choice. Therefore, searching for more flexible probability models that are capable of adequately describing the mixed state failure data sets is an interesting research topic for researchers. In this paper, we propose and study a new statistical model to achieve the above goal. The proposed model is called a new beta power very flexible Weibull distribution and is capable of capturing five different patterns of the failure rate such as uni-modal, decreasing-increasing-decreasing, bathtub, decreasing, increasing-decreasing-increasing shapes. The estimators of the new beta power very flexible Weibull distribution are obtained using the maximum likelihood method. The evaluation of the estimators is assessed by conducting a simulation study. Finally, the usefulness and applicability of the new beta power very flexible Weibull distribution are shown by analyzing two engineering data sets. Using four information criteria, it is observed that the new beta power very flexible Weibull distribution is the best-suited model for dealing with failure times data sets.