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In this article, we demonstrate how to enhance the Weibull-Weibull (WW) distribution introduced in the earlier literature into a better form for fitting monotone and non-monotone failure rate data, especially the bathtub-shaped failure rate data with or without a flat region. The model is referred to as an improved WW distribution. The model's flexibility enables it to describe lifetime data with various failure rate functions, including increasing, decreasing, U or V-shaped, and bathtub-shaped with a comparatively low and long-flat segment. We provide a thorough Bayesian analysis of the modified model for complete and right-censored data. Additionally, we developed maximum likelihood estimators for the model's parameters for both complete and right-censored data. The Bayesian credible and asymptotic confidence intervals of the estimators are defined, and simulation results validate the estimators' consistency. To illustrate the applications of the improved distribution with the WW and other generalized distributions, we apply one censored and one uncensored failure times data, each with bathtub-shaped failure rates. The numerical results demonstrate that the improved WW model outperforms the WW distribution and other existing models, as indicated by goodness-of-fit statistics and supported by the fitted models' survival and failure rate curves and P-P plots.

In this article, we proposed a new extension of the Topp–Leone family of distributions. Some important properties of the model are developed, such as quantile function, stochastic ordering, model series representation, moments, stress–strength reliability parameter, Renyi entropy, order statistics, and moment of residual life. A particular member called new extended Topp–Leone exponential (NETLE) is discussed. Maximum likelihood estimation (MLE), least-square estimation (LSE), and percentile estimation (PE) are used for the model parameter estimation. Simulation studies were conducted using NETLE to assess the MLE, LSE, and PE performance by examining their bias and mean square error (MSE), and the result was satisfactory. Finally, the applications of the NETLE to two real data sets are provided to illustrate the importance of the NETLG families in practice; the data sets consist of daily new deaths due to COVID-19 in California and New Jersey, USA. The new model outperformed many other existing Topp–Leone’s and exponential related distributions based on the real data illustrations.