Optimal Scalar Quantization for Parameter Estimation

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.