Chernoff's distribution

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

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

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

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

then V(0, c) has density

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

where g_c has Fourier transform given by

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

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

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

where ${\tilde {a}}_{1}\approx -2.3381$ is the largest zero of the Airy function Ai and where $\operatorname {Ai} '({\tilde {a}}_{1})\approx 0.7022$.

References

• Groeneboom, Piet (1989). "Brownian motion with a parabolic drift and Airy functions". Probability Theory and Related Fields. 81: 79–109. doi:10.1007/BF00343738. MR 0981568.
• Groeneboom, Piet; Wellner, Jon A. (2001). "Computing Chernoff's Distribution". Journal of Computational and Graphical Statistics. 10: 388–400. doi:10.1198/10618600152627997. MR 1939706.
• Piet Groeneboom (1985). Estimating a monotone density. In: Le Cam, L.E., Olshen, R. A. (eds.), Proceedings of the Berkeley conference in honor of Jerzy Neyman and Jack Kiefer, vol. II, pp. 539–555. Wadsworth.