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Cressie Read Power Divergence for Moment-Based Estimation: Hyperparameter and Finite Sample Behavior

Jieun Lee, Anil K. Bera

Abstract

We study Cressie Read power divergence (CRPD) estimation for moment based models, focusing on finite sample behavior. While generalized empirical likelihood estimators, dual to CRPD, are known to outperform generalized method of moments estimators in small to moderate samples, the power parameter is typically chosen arbitrarily by the researcher, serving mainly as an index. We interpret it as a hyperparameter that determines the loss function and governs the learning procedure, shaping the curvature of the objective and influencing finite sample performance. Using second order asymptotics, we show that it affects both the structural estimator and the associated Lagrange multipliers, governing robustness, bias, and sensitivity to sampling variation. Monte Carlo simulations illustrate how estimator performance varies with the choice of the power parameter and underlying distributional features, with implications for second order bias and coverage distortion. An empirical illustration based on Owen (2001)s classical example highlights the practical relevance of tuning the power parameter.

Cressie Read Power Divergence for Moment-Based Estimation: Hyperparameter and Finite Sample Behavior

Abstract

We study Cressie Read power divergence (CRPD) estimation for moment based models, focusing on finite sample behavior. While generalized empirical likelihood estimators, dual to CRPD, are known to outperform generalized method of moments estimators in small to moderate samples, the power parameter is typically chosen arbitrarily by the researcher, serving mainly as an index. We interpret it as a hyperparameter that determines the loss function and governs the learning procedure, shaping the curvature of the objective and influencing finite sample performance. Using second order asymptotics, we show that it affects both the structural estimator and the associated Lagrange multipliers, governing robustness, bias, and sensitivity to sampling variation. Monte Carlo simulations illustrate how estimator performance varies with the choice of the power parameter and underlying distributional features, with implications for second order bias and coverage distortion. An empirical illustration based on Owen (2001)s classical example highlights the practical relevance of tuning the power parameter.
Paper Structure (26 sections, 11 theorems, 78 equations, 2 figures, 6 tables, 1 algorithm)

This paper contains 26 sections, 11 theorems, 78 equations, 2 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The power parameter and the loss geometry
  3. Moment-based CRPD estimation
  4. Moment-based CRPD estimator
  5. Asymptotic analysis and finite sample properties
  6. Monte Carlo simulations
  7. Discussion
  8. Empirical Application
  9. Empirical specification.
  10. Data-driven selection of $\gamma$.
  11. Estimation results.
  12. Conclusion
  13. Proof for Lemma \ref{['lem:population_solution']}
  14. Proof for Theorem \ref{['thm:uniform_conv_profiled']}
  15. Proof for Lemma \ref{['lem:exist_unique']}
  16. ...and 11 more sections

Key Result

Lemma 3.1

(Population solution under correct specification) Under correct specification, the solution to the CRPD problem lies in a neighborhood of

Figures (2)

  • Figure 1: Second-order bias and coverage distortion ($n = 50$)
  • Figure 2: Estimated observation weights $\hat{\boldsymbol{\pi}}$ for alternative choices of $\gamma$

Theorems & Definitions (11)

  • Lemma 3.1
  • Theorem 3.1
  • Lemma 3.2: Local existence and uniqueness of multipliers
  • Theorem 3.2: Consistency of $\hat{\theta}$
  • Corollary 3.1: Consistency of multipliers
  • Theorem 3.3: First-order expansion for $\hat{\lambda}(\hat{\theta})$ and rate for $\hat{\delta}(\hat{\theta})$
  • Theorem 3.4: Nondegenerate limit for the probability multiplier
  • Theorem 3.5: Asymptotic normality of $\hat{\theta}$
  • Lemma 3.3: Second-order expansion of the moment equation
  • Theorem 3.6: Second-order expansion for $\hat{\lambda}$
  • ...and 1 more