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 981568.
• Groeneboom, Piet and 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.