Estimation of a discrete monotone distribution

Area of Science:

• Statistics
• Probability Theory
• Nonparametric Statistics

Background:

• Discrete monotone distributions are fundamental in statistical modeling.
• Comparing the performance of different estimators is crucial for robust statistical inference.
• Existing estimators like empirical and rearrangement methods have limitations.

Purpose of the Study:

• To compare three estimators for discrete monotone distributions: empirical, rearrangement, and maximum likelihood.
• To identify conditions under which one estimator strictly dominates others.
• To quantify the performance differences using asymptotic risk.

Main Methods:

• Theoretical analysis of estimator properties.
• Asymptotic risk calculation in the squared ℓ(2) norm.
• Comparative study under specific distribution types (uniform, strictly decreasing).

Main Results:

• The maximum likelihood estimator (MLE) strictly dominates rearrangement and empirical estimators when the distribution has intervals of constancy.
• For a uniform distribution on {0, ..., y}, the MLE's asymptotic risk is of order (log y)/(y + 1), compared to y/(y + 1) for the rearrangement estimator.
• For strictly decreasing distributions, all three estimators are asymptotically equivalent.

Conclusions:

• The MLE is a superior choice for discrete monotone distributions with constant intervals.
• The choice of estimator can significantly impact statistical efficiency.
• Further research can explore other distribution classes and risk metrics.