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To assess radiation damage in steel for reactor pressure vessels in the nuclear industry, specimens are subjected to the Charpy test, which measures how much energy a specimen can absorb at a given test temperature before cracking. The resulting Charpy impact energy data are well represented by a three-parameter Burr curve as a function of test temperature, in which the parameters of the Burr curve are themselves dependent on irradiation dose. The resulting non-linear model function, combined with heteroscedastic random errors, gives rise to complicated likelihood surfaces that make conventional statistical techniques difficult to implement. To compute estimates of parameters of practical interest, Markov chain Monte Carlo sampling-based techniques are implemented. The approach is applied to 40 data sets from specimens subjected to no irradiation or one or two doses of irradiation. The influence of irradiation dose on the amount of energy absorbed is investigated.