Extending Statistics of Extremes to Distributions Varying in Position and Scale and the Implications for Race Models

Race models are characterized by the largest or smallest of samples from n distributions. The asymptotic theory of extremes has demonstrated that for identically distributed, independent, and lower-bounded random variables, whose left tail approximates a power function, the distribution of the minimum tends toward a Weibull distribution as n increases. In this article, we remove the restriction of identically distributed random variables by letting the lower bound or the scale of the random variables be random variables themselves. We prove that the Weibull distribution is still the asymptotic distribution of the minimum and relate its parameters to the parameters of the input distributions. We discuss the potential use of such findings in models of psychological processes.