Two-sample problem using a random and a biased sample

This study investigates the problem of testing the equality of two population distributions when one sample is obtained through independent random sampling and the other through a design-based biased mechanism. Motivating examples, such as tree diameter data from forest inventory, highlight the practical relevance of this setting. To address the challenge, we develop a graphical diagnostic tool and formal hypothesis tests within the framework of weighted distributions. A Monte Carlo simulation study is used to assess the performance of the proposed methods in terms of size control and power. Finally, the methodology is demonstrated on diameter-at-breast-height data from the Pertouli Forest (Greece), using random and size-biased samples from hybrid fir to validate the method under a known common population, and a further comparison with a random black pine sample to illustrate its behavior when population equivalence is known not to hold.