Two Types of Size-Biased Samples When Modeling Extreme Phenomena

The present research deals with two possible sources of bias that arise naturally from the selection procedure when modeling extreme phenomena. More specifically, the first type of bias arises when an r-size-biased sample from a set of maximum values is selected, while the second one occurs when a random sample of maxima is observed where each observation is obtained by a series of r-size-biased samples. The concept of weighted distributions is used, not only to describe both cases but also as an adjustment methodology. The differences between the two types of bias are discussed, while the impact of ignoring the bias on the estimation of the unknown parameters is revealed both theoretically and with the use of a simulation study, under the assumption that the parent distribution belongs to the Fréchet maximum domain of attraction. Finally, numerical results indicate that ignorance of the bias or misspecification of r results in inconsistent estimators.