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)^[1] 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).^[2] Chernoff's distribution is now known to appear in a wide range of monotone problems including isotonic regression.^[3]

The Chernoff distribution should not be confused with the Chernoff geometric distribution^[4] (called the Chernoff point in information geometry) induced by the Chernoff information.