Profile likelihood estimation of the vulnerability P(X > v) and the mixing proportion p parameters in the gumbel mixture model

This paper concerns to the problem of making inferences about the vulnerability θ = P(X > v) and the mixing proportion p parameters, when the random variable X is distributed as a mixture of two Gumbel distributions and v is a known fixed value. A profile likelihood approach is proposed for the estimation of these parameters. This approach is a powerful though simple method for separately estimating a parameter of interest in the presence of unknown nuisance parameters. Inferences about θ, p or (θ, p) are given in terms of profile likelihood regions and can be easily obtained on a computer. This methodology is illustrated through a real problem where the main purpose is to model the size of non-metallic inclusions in steel.