Abstract

In practical scenarios, units are often selected with probabilities proportional to predetermined weight functions based on unit values. For instance, Cox illustrates this with the example of sampling fibers in textile manufacturing, where the probability of selecting a fiber is proportional to its length. Similarly, in natural animal count surveys, larger individuals or groups tend to be observed more frequently. In customer satisfaction surveys at shopping malls, patrons who spend more time shopping are more likely to be sampled.
This phenomenon, known as length-bias or more broadly as size-bias, is widely observed across various domains. Under the assumption of independent sampling, numerous estimators have been developed and studied within this framework. This study specifically focuses on size-biased sampling from finite populations, investigates the properties of various estimators, and discusses their practical applications.