Problems of Information Transmission, Vol. 39, No. 4, 2003, pp. 352–372. Translated from Problemy Peredachi Informatsii, No. 4, 2003, pp. 41–62.
Original Russian Text Copyright
2003 by Amari, Burnashev.
METHODS OF SIGNAL PROCESSING
On Some Singularities
in Parameter Estimation Problems
and M. V. Burnashev
RIKEN Brain Science Institute, Wako City, Saitama, Japan
Institute for Information Transmission Problems, RAS, Moscow
Received December 17, 2002
Abstract—Assume that there are two identical remote objects and we want to estimate the
distance to the closest of them and the distance between objects. For that purpose, some device
(like a radar) is used, and the observed signal is the sum of signals reﬂected by each object.
Moreover, the observed signal is corrupted by white Gaussian noise. Singularity (i.e., very poor
estimation accuracy) occurs in this problem if objects are very close to each other. Our aim is
to demonstrate this inevitable singularity by the example of the maximum likelihood estimate
and also to show that it takes place for any other estimate.
Assume that, on the time interval [0,T], we observe a signal X(t)oftheform
dX(t)=[f(t− v)+f (t − v − u)] dt + εdW
, 0 ≤ t ≤ T, (1)
is the standard Wiener process. Unknown real-valued parameters v and u are to be
estimated. Assume that v ∈ [0,A]andu ∈ [0, 2B], where A and B are ﬁnite and known. We are
mainly interested in relatively small values of the parameter u. The function f(t) and the value
ε>0 are known (we will consider the “small noise” case, i.e., ε → 0). Moreover, for simplicity,
we assume that the function f(t) has a ﬁnite support such that, for any possible v and u, all nonzero
values of the function f (t − v − u) belong to the observation interval [0,T].
Of course, model (1) is equivalent to the case where the observed signal x(t) is described as
x(t)=f(t− v)+f(t − v − u)+εn(t), 0 ≤ t ≤ T,
where n(t) is a white Gaussian noise of unit intensity.
The origination of the model considered is quite paciﬁc (in a human sense). Imagine that a
frog observes (using its “radar”) two identical ﬂies, which sit on a line R
. The functions f(t− v)
and f (t − v − u) represent the signals reﬂected by the ﬁrst (closest) and second ﬂy respectively.
Parameter v describes the distance to the closest ﬂy, and parameter u describes the distance between
the ﬂies. The frog estimates both parameters (v, u) and, if u is small enough, tries to catch (with
its tongue) both ﬂies simultaneously.
Clearly, the assumption v ∈ [0,A]isequivalenttov ∈ [D, A+D] for any D. We have also assumed
that u ∈ [0, 2D] (in particular, u ≥ 0) in order to resolve possible ambiguity in the problem setting
(note that, for any pair (v,u), the pair (v
= v + u, u
= −u) also satisﬁes model (1).
Supported in part by the INTAS, Grant no. 00-738.
2003 MAIK “Nauka/Interperiodica”