A note on the median of a distribution

Summary

LetF be a distribution function over the real line. DefineR _p(y)=∫|x−y|^pdF(x) forp≧1. Forp>1 there is a unique minimizer ofR _p(·), sayγ _p. 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.