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Hybrid combinations of parametric and empirical likelihoods

Nils Lid Hjort, Ian W. McKeague, Ingrid Van Keilegom

TL;DR

A hybrid likelihood (HL) method based on a compromise between parametric and nonparametric likelihoods is developed and asymptotic normality of the corresponding HL estimator and a version of the Wilks theorem are established.

Abstract

This paper develops a hybrid likelihood (HL) method based on a compromise between parametric and nonparametric likelihoods. Consider the setting of a parametric model for the distribution of an observation $Y$ with parameter $θ$. Suppose there is also an estimating function $m(\cdot,μ)$ identifying another parameter $μ$ via $E\,m(Y,μ)=0$, at the outset defined independently of the parametric model. To borrow strength from the parametric model while obtaining a degree of robustness from the empirical likelihood method, we formulate inference about $θ$ in terms of the hybrid likelihood function $H_n(θ)=L_n(θ)^{1-a}R_n(μ(θ))^a$. Here $a\in[0,1)$ represents the extent of the compromise, $L_n$ is the ordinary parametric likelihood for $θ$, $R_n$ is the empirical likelihood function, and $μ$ is considered through the lens of the parametric model. We establish asymptotic normality of the corresponding HL estimator and a version of the Wilks theorem. We also examine extensions of these results under misspecification of the parametric model, and propose methods for selecting the balance parameter $a$.

Hybrid combinations of parametric and empirical likelihoods

TL;DR

A hybrid likelihood (HL) method based on a compromise between parametric and nonparametric likelihoods is developed and asymptotic normality of the corresponding HL estimator and a version of the Wilks theorem are established.

Abstract

This paper develops a hybrid likelihood (HL) method based on a compromise between parametric and nonparametric likelihoods. Consider the setting of a parametric model for the distribution of an observation with parameter . Suppose there is also an estimating function identifying another parameter via , at the outset defined independently of the parametric model. To borrow strength from the parametric model while obtaining a degree of robustness from the empirical likelihood method, we formulate inference about in terms of the hybrid likelihood function . Here represents the extent of the compromise, is the ordinary parametric likelihood for , is the empirical likelihood function, and is considered through the lens of the parametric model. We establish asymptotic normality of the corresponding HL estimator and a version of the Wilks theorem. We also examine extensions of these results under misspecification of the parametric model, and propose methods for selecting the balance parameter .
Paper Structure (14 sections, 5 theorems, 45 equations, 3 figures)

This paper contains 14 sections, 5 theorems, 45 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Behaviour of HL under the parametric model
  3. Hybrid likelihood outside model conditions
  4. Fine-tuning the balance parameter
  5. An illustration: Roman era Egyptian life-lengths
  6. Further developments and the Supplementary Material
  7. Proof of Lemma \ref{['lemma:lemma1']}
  8. Proof of Theorem \ref{['theorem:theorem1']}
  9. Proof of Corollary \ref{['cor:cor1']}
  10. Proof of Theorem \ref{['theorem:theorem2']}
  11. The HL for regression models
  12. Confidence curve for a focus parameter
  13. An implied goodness-of-fit test for the parametric model
  14. A related hybrid estimation method

Key Result

Lemma 1

For a compact $S\subset \mathbb{R}^p$, suppose that (i) $\sup_{s \in S} \|V_n(s)\| = O_{\rm pr}(1)$; (ii) $\sup_{s \in S}\|W_n(s)-W\| \rightarrow_{\rm pr} 0$, where $W = {\rm Var}\,m(Y,\mu(\theta_0))$ is of full rank; (iii) $n^{-1/2} \sup_{s \in S} \max_{i\le n}\|m_{i,n}(s)\|\rightarrow_{\rm pr}0$.

Figures (3)

  • Figure 1: (a) Peter at the occasion of his honorary doctorate at the Institute of Statistics in Louvain-la-Neuve in 1997; (b) Peter and Ingrid Van Keilegom in Tidbinbilla Nature Reserve near Canberra in 2002 (picture taking by Jeannie Hall).
  • Figure 2: (a) The dotted horizontal line indicates the root-mse for the ML estimator, and the full curve the root-mse for the HL estimator, as a function of the balance parameter $a$ in the HL construction. (b) The root-${\rm fic}(a)$, as a function of the balance parameter $a$, constructed on the basis of $n=100$ simulated observations, from a case where $\gamma=1+\delta/\sqrt{n}$, with $\delta$ described in the text.
  • Figure 3: (a) The q-q plot shows the ordered life-lengths $y_{(i)}$ plotted against the ML-estimated gamma quantile function $F^{-1}(i/(n+1),\widehat{b},\widehat{c})$. (b) The curve $\widehat{p}_a$, with the probability $p=P\{Y\in[9.5,20.5]\}$ estimated via the HL estimator, is displayed, as a function of the balance parameter $a$. At balance position $a=0.61$, the efficiency loss is 10% compared to the ML precision under ideal gamma model conditions.

Theorems & Definitions (9)

  • Example 1
  • Example 2
  • Example 3
  • Lemma 1
  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Theorem 3
  • Remark 1