Detecting changes in the rate Poisson process

CUSUM test and Lorden’s criterion
Discrete time, i.i.d. observations.
Pdf before and after the change: f ∞ (ξ n ), f 0 (ξ n )
Since change time τ is unknown
sup 06τ 6t
n=τ+1
t
Σn=1
t
f 0 (ξ n )
Σlog( )
f ∞ (ξ n )
-inf 06τ6t Σn=1
> ν
f 0 (ξ n )
log( )
f 0 (ξ n )
f ∞ (ξ n )
f ∞ (ξ n )
τ
log( )
> ν
u t
– m t
> ν
G.V. Moustakides: Detecting changes in the rate of a Poisson process 8