  It is revealed that there is a link between the quantization approach
employed and the dimension of the vector parameter which can be accurately
estimated by a quantized estimation system. A critical quantity called
inestimable dimension for quantized data (IDQD) is introduced, which doesn't
depend on the quantization regions and the statistical models of the
observations but instead depends only on the number of sensors and on the
precision of the vector quantizers employed by the system. It is shown that the
IDQD describes a quantization induced fundamental limitation on the estimation
capabilities of the system. To be specific, if the dimension of the desired
vector parameter is larger than the IDQD of the quantized estimation system,
then the Fisher information matrix for estimating the desired vector parameter
is singular, and moreover, there exist infinitely many nonidentifiable vector
parameter points in the vector parameter space. Furthermore, it is shown that
under some common assumptions on the statistical models of the observations and
the quantization system, a smaller IDQD can be obtained, which can specify an
even more limiting quantization induced fundamental limitation on the
estimation capabilities of the system.
