Chernoff's distribution - Alchetron, the free social encyclopedia

Z = argmax s ∈ R ( 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 ) = argmax s ∈ R ( W ( s ) − c ( s − a ) 2 ) ,

then V(0, c) has density

f c ( t ) = 1 2 g c ( t ) g c ( − t )

where g_c has Fourier transform given by

g ^ c ( s ) = ( 2 / c ) 1 / 3 Ai ⁡ ( i ( 2 c 2 ) − 1 / 3 s ) , s ∈ 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 ) ∼ 1 2 4 4 / 3 | z | Ai ′ ⁡ ( a ~ 1 ) exp ⁡ ( − 2 3 | z | 3 + 2 1 / 3 a ~ 1 | z | ) as z → ∞

where a ~ 1 ≈ − 2.3381 is the largest zero of the Airy function Ai and where Ai ′ ⁡ ( a ~ 1 ) ≈ 0.7022 .

References

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