The Need for Teaching Weighted Distribution Theory: Illustrated with Applications in Environmental Statistics

I review weighted distribution theory and applications in observational studies where biased data arise. Recorded observations will be biased and not have the original distribution unless every observation is given an equal chance of being recorded. G. P. Patil and C. R. Rao in 1977 wrote “Although the situations that involve weighted distributions seem to occur frequently in various fields, the underlying concept of weighted distributions as a major stochastic concept does not seem to have been widely recognized.” The same quote is applicable today. Our profession is missing an opportunity to provide structure and understanding to a collection of isolated statistical methods for analysis of data from observational studies. Specifically, I present applications in study of wildlife and fish populations that should motivate undergraduate and graduate students of statistics who have an interest in environmental issues.