A note on the median of a distribution

Abstract

LetF be a distribution function over the real line. DefineRp(y)=∫|x−y|pdF(x) forp≧1. Forp>1 there is a unique minimizer ofRp(·), sayγp. Under mild conditions onF it is shown that\(\mathop {\lim }\limits_{p \to 1} \gamma _p \) exists and that the limit is a median. Thus, one could use this limit as a definition of a unique median. Also it is shown that\(\mathop {\lim }\limits_{p \to 1} \gamma _p = {{\left( {R + L} \right)} \mathord{\left/ {\vphantom {{\left( {R + L} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}\) whereR is the right extremity ofF andL is the left extremity ofF provided that −∞<L≦R<∞. A similar result is available ifL=−∞,R=∞, yetF has symmetric tails.