The Random Effect Transformation for Three Regularity Classes

We continue the analysis of the influence of the random effect transformation on the regularity of distribution functions. The paper considers three regularity classes: heavy-tailed distributions, distributions with consistently varying tails, and exponential-like-tailed distributions. We apply the random effect transformation to the primary distribution functions from these classes and investigate whether the resulting distribution function remains in the same class. We find that the random effect transformation has the greatest impact on exponential-like-tailed distributions. We establish that any heavy-tailed distribution subjected to a random effect transformation remains heavy-tailed, and any distribution with a consistently varying tail remains with a consistently varying tail after the random effect transformation. Meanwhile, different cases are possible when an exponential-like-tailed class of distributions is subjected to a random effect transformation. Sometimes, depending on the structure of a random effect, the resulting distribution remains exponential-like-tailed, and sometimes that distribution regularly varies. All of the derived theoretical results are illustrated with several examples.