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Semiparametric inference for inequality measures under nonignorable nonresponse using callback data

Xinyu Wang, Chunlin Wang, Tao Yu, Pengfei Li

TL;DR

The paper tackles nonignorable nonresponse in survey data, which biases inequality measures, by combining callback data with a semiparametric full-likelihood model that leaves the distribution $F(y)$ unspecified. It develops semiparametric estimators for $F$ and for a broad class of inequality measures, including quantiles, the Theil index, and the Gini index, and derives explicit asymptotic variances to support Wald-type inference. An efficient EM algorithm is proposed for practical computation, with theoretical guarantees such as monotonicity of the algorithm. Simulation studies and a Consumer Expenditure Survey application demonstrate bias correction and near-benchmark efficiency, confirming the practical value of incorporating callback information for distributional inequality inference.

Abstract

This paper develops semiparametric methods for estimation and inference of widely used inequality measures when survey data are subject to nonignorable nonresponse, a challenging setting in which response probabilities depend on the unobserved outcomes. Such nonresponse mechanisms are common in household surveys and invalidate standard inference procedures due to selection bias and lack of population representativeness. We address this problem by exploiting callback data from repeated contact attempts and adopting a semiparametric model that leaves the outcome distribution unspecified. We construct semiparametric full-likelihood estimators for the underlying distribution and the associated inequality measures, and establish their large-sample properties for a broad class of functionals, including quantiles, the Theil index, and the Gini index. Explicit asymptotic variance expressions are derived, enabling valid Wald-type inference under nonignorable nonresponse. To facilitate implementation, we propose a stable and computationally convenient expectation-maximization algorithm, whose steps either admit closed-form expressions or reduce to fitting a standard logistic regression model. Simulation studies demonstrate that the proposed procedures effectively correct nonresponse bias and achieve near-benchmark efficiency. An application to Consumer Expenditure Survey data illustrates the practical gains from incorporating callback information when making inference on inequality measures.

Semiparametric inference for inequality measures under nonignorable nonresponse using callback data

TL;DR

The paper tackles nonignorable nonresponse in survey data, which biases inequality measures, by combining callback data with a semiparametric full-likelihood model that leaves the distribution unspecified. It develops semiparametric estimators for and for a broad class of inequality measures, including quantiles, the Theil index, and the Gini index, and derives explicit asymptotic variances to support Wald-type inference. An efficient EM algorithm is proposed for practical computation, with theoretical guarantees such as monotonicity of the algorithm. Simulation studies and a Consumer Expenditure Survey application demonstrate bias correction and near-benchmark efficiency, confirming the practical value of incorporating callback information for distributional inequality inference.

Abstract

This paper develops semiparametric methods for estimation and inference of widely used inequality measures when survey data are subject to nonignorable nonresponse, a challenging setting in which response probabilities depend on the unobserved outcomes. Such nonresponse mechanisms are common in household surveys and invalidate standard inference procedures due to selection bias and lack of population representativeness. We address this problem by exploiting callback data from repeated contact attempts and adopting a semiparametric model that leaves the outcome distribution unspecified. We construct semiparametric full-likelihood estimators for the underlying distribution and the associated inequality measures, and establish their large-sample properties for a broad class of functionals, including quantiles, the Theil index, and the Gini index. Explicit asymptotic variance expressions are derived, enabling valid Wald-type inference under nonignorable nonresponse. To facilitate implementation, we propose a stable and computationally convenient expectation-maximization algorithm, whose steps either admit closed-form expressions or reduce to fitting a standard logistic regression model. Simulation studies demonstrate that the proposed procedures effectively correct nonresponse bias and achieve near-benchmark efficiency. An application to Consumer Expenditure Survey data illustrates the practical gains from incorporating callback information when making inference on inequality measures.
Paper Structure (19 sections, 8 theorems, 65 equations, 2 figures, 7 tables)

This paper contains 19 sections, 8 theorems, 65 equations, 2 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Semiparametric estimation and inference of inequality measures
  3. Semiparametric full-likelihood approach
  4. Semiparametric estimation of inequality measures
  5. Quantiles and their functions
  6. A class of linear functionals and its functions
  7. Gini index
  8. Asymptotic properties
  9. Asymptotic properties of $\hat{\theta}_\text{quan}^{\tau}$
  10. Asymptotic properties of $\hat{\theta}_{\boldsymbol{u},h}$
  11. Asymptotic properties of $\hat{\theta}_\text{Gini}$
  12. Semiparametric inference of inequality measures
  13. Expectation-Maximization algorithm
  14. Simulation study
  15. Performance of point estimation
  16. ...and 4 more sections

Key Result

Theorem 1

Suppose that Conditions C1--C5 in the Appendix are satisfied. Under model Alho, as $N \to \infty$, $N^{1/2}\{\hat{F}(y) - F(y)\}$ converges weakly to a zero-mean tight Gaussian process with continuous sample paths and covariance function for any $y_1, y_2 \in (0,\infty)$, where $\xi_F(Y,y) = I(Y \le y) - F(y)$ and $\boldsymbol{\Gamma}$ is defined in Gamma in the Appendix.

Figures (2)

  • Figure 1: Response rate by number of contact attempts in the CES data.
  • Figure 2: Estimated distribution function $F(y)$ based on CES data using the proposed semiparametric method (solid line) and the complete-case method (dotted line).

Theorems & Definitions (8)

  • Theorem 1
  • Lemma 1
  • Theorem 2
  • Corollary 1
  • Theorem 3
  • Corollary 2
  • Theorem 4
  • Proposition 1