Inference under unequal probability sampling with the Bayesian exponentially tilted empirical likelihood.

Area of Science:

• Statistics
• Statistical Inference
• Computational Statistics

Background:

• Bayesian inference with unequal probability sampling demands strong assumptions.
• Frequentist semiparametric methods offer robustness but may lack small-sample advantages.
• Existing Bayesian approaches may require specific priors or variance corrections.

Purpose of the Study:

• To develop a method combining Bayesian inference benefits with frequentist robustness.
• To enhance small-sample inference and evidence synthesis in complex sampling designs.
• To create semiparametric models using moment constraints from unbiased estimating equations.

Main Methods:

• Utilizing Bayesian exponentially tilted empirical likelihood.
• Constructing semiparametric models via moment constraints.
• Proving Bernstein-von Mises theorems for posterior normality.

Main Results:

• The proposed method yields approximately normal posteriors centered at the chosen estimator.
• The posterior exhibits double robustness and matching asymptotic variance.
• Credible sets achieve frequentist coverage approximately equal to their credibility.

Conclusions:

• The Bayesian exponentially tilted empirical likelihood offers a robust approach to statistical inference.
• It provides modified estimators with improved properties, including parameter space guarantees.
• Simulations indicate superior frequentist performance compared to existing Bayesian methods.