Optimal Choice of Observation Window for Poisson Observations

We consider the possibility of optimal choice of observation window in the problem of parameter estimation by the observations of an inhomogeneous Poisson process. A minimax lower bound is proposed for the risk of estimation under an arbitrary choice of observation window. Then the adaptive procedure is proposed which is asymptotically e cient in the sense of this bound. Let X be a separable metric space, B, the -algebra of its Borelian subsets, the set A 2 B and a family of Poisson processes of mean measures #, # 2 ; IR observed n times on the set A. We suppose that the value of the parameter # is unknown to the observer and he have to estimate it by n independent observations of the Poisson process. If the set A is xed then under the regularity conditions the maximum likelihood estimator (MLE) #̂n is asymptotically normal with the limit variance 2 equal to the inverse Fisher information, i.e., p n(#̂n #) =)N (0; ); 2 = I( ) = Z A _ S(#; #; x) #(dx) where dot means the derivation with respect to # and S(#1; #2; x) = #1(dx)= #2(dx); _ S(#; #; x) = @S(y; #; x)=@y y=# (see [5], Theorem 2.4). Let us call the set A an observation window and consider the problem of its optimal choice. We write I(#) = I(#;A) and note that a reasonable solution to this problem is to maximize I(#;A) over some class of sets fAg. For instance, one may consider the class A de ned by Am = fA : A A; (A) mg where is some measure on X (it can be one of the measures f #; # 2 g or, in nite-dimensional case, the Lebesgue measure), A is some (rather large) set from B and m > 0 is a given number. We see that the information matrix I(#;A) depends generally on the unknown parameter # and therefore there is no any universal optimal choice of the observation window A .