Concept of Sub-Independence
8th World Congress in Probability and Statistics
Limit theorems as well as other well-known results in probability and statistics are often based on the distributions of the sums of independent (and often indentically distributed) random variables rather than the joint distribution of the summands. Therefore, the full force of independence of the summands will not be required. In other words, it is the convolution of the marginal distributions which is needed, rather than the joint distribution of the summands which, in the case of independence, is the product of the marginal distributions. This is precisely the reason for the statement:”why assume independence when you can get by with sub-independence”. The concept of sub-independence can help to provide solution for modeling problems where the variable of interest is the sum of a few components. Examples include the household income, the total profit of major firms in an industry, and a regression model Y=g(X)+e where g(X) and e are uncorrelated, however, they may not be independent. For example, in Bazargan et al. (2007), the return value of significant wave height (Y) is modeled by the sum of a cyclic function of random delay, g(X), and a residual term e. They found that the two components are at least uncorrelated but not independent and used sub-independence to compute the distribution of the return value. for the detailed application of the concept of sub-independence in this direction we refer the reader to Bazargan et al. (2007).