Chernoff's distribution

In probability theory, Chernoff's distribution, named after Herman Chernoff, is the probability distribution of the random variable

${\displaystyle Z={\underset {s\in \mathbf {R} }{\operatorname {argmax} }}\ (W(s)-s^{2}),}$

where W is a "two-sided" Wiener process (or two-sided "Brownian motion") satisfying W(0) = 0. If

${\displaystyle V(a,c)={\underset {s\in \mathbf {R} }{\operatorname {argmax} }}\ (W(s)-c(s-a)^{2}),}$

then V(0, c) has density

${\displaystyle f_{c}(t)={\frac {1}{2}}g_{c}(t)g_{c}(-t)}$

where g_c has Fourier transform given by

${\displaystyle {\hat {g}}_{c}(s)={\frac {(2/c)^{1/3}}{\operatorname {Ai} (i(2c^{2})^{-1/3}s)}},\ \ \ s\in \mathbf {R} }$

and where Ai is the Airy function. Thus f_c is symmetric about 0 and the density ƒ_Z = ƒ₁. Groeneboom (1989) shows that

${\displaystyle f_{Z}(z)\sim {\frac {1}{2}}{\frac {4^{4/3}|z|}{\operatorname {Ai} '({\tilde {a}}_{1})}}\exp \left(-{\frac {2}{3}}|z|^{3}+2^{1/3}{\tilde {a}}_{1}|z|\right){\text{ as }}z\rightarrow \infty }$

where ${\displaystyle {\tilde {a}}_{1}\approx -2.3381}$ is the largest zero of the Airy function Ai and where ${\displaystyle \operatorname {Ai} '({\tilde {a}}_{1})\approx 0.7022}$. In the same paper, Groeneboom also gives an analysis of the process ${\displaystyle \{V(a,1):a\in \mathbf {R} \}}$. The connection with the statistical problem of estimating a monotone density is discussed in Groeneboom (1985). Chernoff's distribution is now known to appear in a wide range of monotone problems including isotonic regression.

The Chernoff distribution should not be confused with the Chernoff geometric distribution (called the Chernoff point in information geometry) induced by the Chernoff information.