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    Area of Science:

    • Statistics
    • Information Theory
    • Machine Learning

    Background:

    • Quantization of random variables is crucial in statistical inference.
    • Understanding the properties of quantized data distributions is essential for developing optimal estimation and testing procedures.
    • Deterministic quantizers offer a simplified approach to data processing.

    Purpose of the Study:

    • To establish the optimality of deterministic quantizers based on sufficient statistics for parameter estimation.
    • To demonstrate a connection between parameter estimation and hypothesis testing through quantization.
    • To identify optimal partitioning strategies for sufficient statistics in exponential families.

    Main Methods:

    • Analysis of extreme points of quantized probability distributions.
    • Development of deterministic quantizers utilizing sufficient statistics.
    • Optimization of Fisher information matrix trace for exponential families.
    • Derivation of optimality for likelihood ratio statistics.

    Main Results:

    • Any extreme distribution of a quantized random variable can be achieved by a deterministic quantizer using sufficient statistics.
    • Optimal deterministic quantizers exist for parameter estimation problems.
    • An optimal partitioning of sufficient statistics into K convex polytopes maximizes the Fisher information matrix trace for exponential families.
    • The optimality of the likelihood ratio statistic for simple hypothesis testing is a direct consequence.

    Conclusions:

    • Deterministic quantizers based on sufficient statistics are optimal for parameter estimation.
    • The study establishes a theoretical link between parameter estimation and hypothesis testing via quantization.
    • Optimal quantization strategies involve convex polytope partitions of sufficient statistics, particularly for exponential families.