  In this paper, we study an asymptotic approximation of the Fisher information
for the estimation of a scalar parameter using quantized measurements. We show
that, as the number of quantization intervals tends to infinity, the loss of
Fisher information induced by quantization decreases exponentially as a
function of the number of quantization bits. A characterization of the optimal
quantizer through its interval density and an analytical expression for the
Fisher information are obtained. A comparison between optimal uniform and
non-uniform quantization for the location and scale estimation problems shows
that non-uniform quantization is only slightly better. As the optimal
quantization intervals are shown to depend on the unknown parameters, by
applying adaptive algorithms that jointly estimate the parameter and set the
thresholds in the location and scale estimation problems, we show that the
asymptotic results can be approximately obtained in practice using only 4 or 5
quantization bits.
