No empirical probability measure can converge in the total variation sense for all distributions. (English) Zbl 0707.60026

Devroye, Luc; Györfi, László

Ann. Stat. 18, No. 3, 1496-1499 (1990).

Summary: For any sequence of empirical probability measures \(\{\mu_ n\}\) on the Borel sets of the real line and any \(\delta >0\), there exists a singular continuous probability measure \(\mu\) such that \[ \inf_{n} \sup_{A}| \mu_ n(A)-\mu (A)| \geq -\delta \quad almost\quad surely. \]

MSC:

60E99	Distribution theory
62G05	Nonparametric estimation

Keywords:

empirical probability measures; singular continuous probability measure