Introduction
Flexible lifetime distributions play a central role in both the theoretical development and practical application of survival and reliability analysis. Emerging experimental lifetime data in engineering and biological sciences often display skewness, heavy tails, and non-monotonic hazard rates that classical lifetime distributions fail to capture. Motivated by these limitations, this study introduces a new four-parameter lifetime distribution called the Transmuted Exponential–Weibull–Exponential (TE-W-E) distribution that extends the existing Weibull–exponential distribution. The TE-W-E is proposed to enhance the flexibility of the Weibull–exponential distribution for positive data.

Method
The TE-W-E distribution is constructed using the transmuted exponential-G generator to the Weibull–exponential distribution. The flexibility of the Weibull–exponential distribution was increased by inducing scale and transmuted parameters. Comprehensive derivations of its mathematical and statistical properties are presented. Expressions for the probability density, cumulative distribution, ordinary and central moments, moment generating, survival, and hazard functions were analytically derived. Model parameters were estimated using the maximum likelihood method, and the observed Fisher information matrix was derived to establish the asymptotic inference. Monte Carlo simulations were conducted to evaluate estimator performance under different sample sizes in terms of bias, variance, and mean squared error.

Results
The TE-W-E distribution is highly flexible, capable of modelling increasing, decreasing, and bathtub-shaped hazard rate functions. Simulation findings confirmed the consistency and efficiency of the maximum likelihood estimator. Application to two real datasets was demonstrated. The parameter estimates were obtained using the optim function in R software. The proposed TE-W-E model was compared with some well-established competing models: Weibull, Weibull–Exponential, and Weibull–Gamma. It outperforms the models based on likelihood and goodness-of-fit criteria: AIC, BIC, and CAIC. The new parameters contributed to the tail thickness and asymmetry.

Conclusion
The TE-W-E distribution is a robust model for lifetime data. Its flexibility and strong empirical performance make it a valuable contribution to survival and reliability analysis.