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arxiv: 2604.21735 · v1 · submitted 2026-04-23 · 📊 stat.ME

A semiparametric two-sample homogeneity test with nonignorable nonresponse using callback data

Xinyu Wang , Tao Yu , Chunlin Wang , Pengfei Li This is my paper

Pith reviewed 2026-05-09 20:52 UTC · model grok-4.3

classification 📊 stat.ME
keywords semiparametric testnonignorable nonresponsecallback dataempirical likelihood ratiodensity ratio modeltwo-sample homogeneitymissing data analysis
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The pith

Callback data enable a semiparametric empirical likelihood test for distributional homogeneity despite nonignorable nonresponse.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The authors develop a framework for testing whether two distributions are the same when data are missing in a nonignorable manner. They use callback records of contact attempts to model the response process semiparametrically and connect the populations with a density ratio model. An empirical likelihood ratio statistic is then proposed for the homogeneity hypothesis. This statistic is shown to follow a chi-square distribution under the null. Simulations indicate that the procedure maintains the correct type I error rate while delivering greater power than approaches that ignore the missingness mechanism.

Core claim

The paper proposes an empirical likelihood ratio test for the homogeneity of two distributions within a semiparametric framework that incorporates callback data through a flexible response model and links the distributions via a density ratio model. Under the null hypothesis of identical distributions, the test statistic converges to a Wilks-type chi-square limit. An expectation-maximization algorithm is developed to compute the test, and simulation studies demonstrate reliable type I error control along with substantially improved power compared to methods that do not account for nonignorable nonresponse.

What carries the argument

Empirical likelihood ratio test statistic arising from the semiparametric callback response model combined with the density ratio model linking the two population distributions.

Load-bearing premise

The semiparametric model for the callback response mechanism accurately describes how nonresponse depends on the unobserved outcomes, and the density ratio model correctly relates the two population distributions.

What would settle it

Simulating data from two identical distributions under the specified callback response mechanism and verifying that the empirical likelihood ratio statistic's finite-sample distribution approaches the chi-square limit as sample size grows would confirm the claim; consistent over-rejection or under-rejection would falsify the asymptotic result.

read the original abstract

Testing the homogeneity of two distributions is fundamental in statistics, but classical procedures may fail under nonignorable nonresponse. In many surveys, callback data record repeated contact attempts and provide auxiliary information about the response mechanism. We develop a semiparametric framework for two-sample homogeneity testing that explicitly incorporates such information. The response mechanism is modeled by a flexible semiparametric callback model, while the two population distributions are linked through a density ratio model. Within this unified framework, we propose an empirical likelihood ratio test for distributional homogeneity and show that, under the null hypothesis, it has a Wilks-type chi-square limit. To facilitate computation, we develop an efficient expectation-maximization-type algorithm. Simulation results show that the proposed method controls type I error well and achieves substantially higher power than existing methods that ignore nonignorable missingness. An application to real survey income data illustrates its practical value.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a semiparametric framework for testing homogeneity between two distributions in the presence of nonignorable nonresponse, using callback data to model the response mechanism semiparametrically and a density-ratio model to link the two populations. Within this setup it proposes an empirical likelihood ratio test statistic for the homogeneity hypothesis and claims that the statistic converges to a Wilks-type chi-square limit under the null; an EM-type algorithm is derived for computation. Simulations are reported to show type-I error control and substantially higher power than methods that ignore nonignorable missingness, and the method is illustrated on real survey income data.

Significance. If the claimed asymptotic result holds, the work supplies a practically relevant tool for two-sample testing when callback information is available to mitigate nonignorable nonresponse bias. The combination of a flexible semiparametric response model with an empirical-likelihood approach is a methodological strength, and the reported simulation gains in power are potentially useful for applied survey analysis.

major comments (2)
  1. [Abstract / §3] Abstract and the statement of the main theorem (presumably §3): the claim that the profiled empirical likelihood ratio converges to a chi-square whose degrees of freedom equal only the dimension of the homogeneity parameter rests on an unverified orthogonality condition. The semiparametric callback model introduces a nonparametric or high-dimensional nuisance whose tangent space may have a non-zero projection onto the density-ratio score; without an explicit demonstration that this projection is o_p(1) or that the efficient information matrix is block-diagonal after EM profiling, the limiting distribution is generally a weighted sum of chi-squares rather than the asserted Wilks form.
  2. [§4 / Simulation section] §4 (algorithm) and the simulation design: the EM procedure is said to maximize the profiled likelihood, yet no verification is given that the Lagrange multipliers or the M-step updates enforce the necessary orthogonality between the homogeneity parameter and the callback nuisance. If the simulation data-generating process uses the same semiparametric callback specification as the estimator, the reported power advantage may be partly an artifact of correct specification rather than robustness to the nonignorable mechanism.
minor comments (2)
  1. The notation distinguishing the callback probability model from the density-ratio tilt parameter could be made more explicit (e.g., by adding a short table of symbols).
  2. A few simulation tables report power at fixed sample sizes; adding a brief sensitivity check under misspecified callback models would strengthen the practical claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. The comments raise important points about the rigor of the asymptotic justification and the simulation design. We address each major comment below and outline the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract / §3] Abstract and the statement of the main theorem (presumably §3): the claim that the profiled empirical likelihood ratio converges to a chi-square whose degrees of freedom equal only the dimension of the homogeneity parameter rests on an unverified orthogonality condition. The semiparametric callback model introduces a nonparametric or high-dimensional nuisance whose tangent space may have a non-zero projection onto the density-ratio score; without an explicit demonstration that this projection is o_p(1) or that the efficient information matrix is block-diagonal after EM profiling, the limiting distribution is generally a weighted sum of chi-squares rather than the asserted Wilks form.

    Authors: We appreciate the referee drawing attention to the need for explicit verification of the orthogonality condition. In the proof of the main theorem (Theorem 3.1 and its supporting lemmas in the appendix), we derive the efficient score for the homogeneity parameter after profiling out the semiparametric callback nuisance. The density-ratio model structure ensures that the projection of the homogeneity score onto the tangent space of the callback model is exactly zero, yielding a block-diagonal efficient information matrix and the standard Wilks chi-square limit with degrees of freedom equal to the dimension of the homogeneity parameter. However, we acknowledge that this step could be presented more transparently. In the revised manuscript we will add a dedicated paragraph immediately following the statement of the theorem that explicitly computes the projection and confirms it is zero under the model assumptions. revision: yes

  2. Referee: [§4 / Simulation section] §4 (algorithm) and the simulation design: the EM procedure is said to maximize the profiled likelihood, yet no verification is given that the Lagrange multipliers or the M-step updates enforce the necessary orthogonality between the homogeneity parameter and the callback nuisance. If the simulation data-generating process uses the same semiparametric callback specification as the estimator, the reported power advantage may be partly an artifact of correct specification rather than robustness to the nonignorable mechanism.

    Authors: We agree that the algorithm section would benefit from an explicit statement on how the EM updates preserve orthogonality. The Lagrange multipliers in the E-step and the closed-form M-step updates for the homogeneity parameters are constructed precisely so that the profiled score remains orthogonal to the nuisance tangent space at convergence; we will insert a short verification of this property in the revised Section 4. On the simulation design, the primary data-generating process matches the assumed semiparametric callback model to isolate the effect of correctly accounting for nonignorable nonresponse. To address potential concerns about specification, the supplementary material already contains additional simulation results under misspecified callback models (both parametric and nonparametric perturbations). We will move a concise summary of these robustness checks into the main simulation section and report the corresponding type-I error and power figures. revision: partial

Circularity Check

0 steps flagged

No circularity: Wilks limit follows from standard EL asymptotics on profiled semiparametric model

full rationale

The paper constructs an empirical likelihood ratio statistic for homogeneity under a density-ratio link and semiparametric callback response model, then invokes the classical Wilks theorem for the profiled EL ratio under the null. This is a direct application of established EL large-sample theory to the composite estimating equations after nuisance profiling; the limiting chi-square degrees of freedom are determined by the dimension of the homogeneity parameter alone once the callback and tilt parameters are concentrated out. No step reduces a claimed prediction to a fitted quantity by construction, no self-citation supplies the core limit theorem, and the EM algorithm is presented only as a computational device rather than as the source of the distributional result. The derivation chain is therefore self-contained against external EL theory and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The approach relies on standard asymptotic regularity conditions for empirical likelihood and semiparametric estimation, plus modeling assumptions for the callback response mechanism and density ratio that are not independently verified in the provided abstract.

free parameters (1)
  • parameters in semiparametric callback model
    The flexible callback model for response mechanism requires estimation of unspecified components from data.
axioms (2)
  • standard math Regularity conditions ensuring Wilks-type chi-square limit for the empirical likelihood ratio test
    Invoked to establish the asymptotic distribution under the null hypothesis.
  • domain assumption The density ratio model correctly links the two population distributions
    Central modeling choice for the two-sample setup.

pith-pipeline@v0.9.0 · 5460 in / 1307 out tokens · 23524 ms · 2026-05-09T20:52:27.883930+00:00 · methodology

discussion (0)

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