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The paper introduces a new failure time distribution called additive Teissier–Weibull distribution. Among its features, it is capable of modeling increasing and bathtub hazard rates. It is thus useful in modeling heterogeneous populations and can be viewed as the failure time model of the system having two modes of failures that follow the Teissier and Weibull distributions, respectively. Some of its important properties are obtained, such as quantiles, moments and shapes of the probability density and hazard functions. Four different methods of estimation such as the maximum likelihood, least squares, weighted least squares and maximum product spacing are proposed to estimate the unknown parameters. To compare the performance of the estimates, an extensive simulation study is carried out with varying sample sizes. Finally, failure times of primary reactor pumps and power generators are fitted and analyzed to show that the new additive hazard rate model may give a better fit than many other existing models.