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Estimation beyond Missing (Completely) at Random

Tianyi Ma, Kabir A. Verchand, Thomas B. Berrett, Tengyao Wang, Richard J. Samworth

Abstract

We study the effects of missingness on the estimation of population parameters. Moving beyond restrictive missing completely at random (MCAR) assumptions, we first formulate a missing data analogue of Huber's arbitrary $ε$-contamination model. For mean estimation with respect to squared Euclidean error loss, we show that the minimax quantiles decompose as a sum of the corresponding minimax quantiles under a heterogeneous, MCAR assumption, and a robust error term, depending on $ε$, that reflects the additional error incurred by departure from MCAR. We next introduce natural classes of realisable $ε$-contamination models, where an MCAR version of a base distribution $P$ is contaminated by an arbitrary missing not at random (MNAR) version of $P$. These classes are rich enough to capture various notions of biased sampling and sensitivity conditions, yet we show that they enjoy improved minimax performance relative to our earlier arbitrary contamination classes for both parametric and nonparametric classes of base distributions. For instance, with a univariate Gaussian base distribution, consistent mean estimation over realisable $ε$-contamination classes is possible even when $ε$ and the proportion of missingness converge (slowly) to 1. Finally, we extend our results to the setting of departures from missing at random (MAR) in normal linear regression with a realisable missing response.

Estimation beyond Missing (Completely) at Random

Abstract

We study the effects of missingness on the estimation of population parameters. Moving beyond restrictive missing completely at random (MCAR) assumptions, we first formulate a missing data analogue of Huber's arbitrary -contamination model. For mean estimation with respect to squared Euclidean error loss, we show that the minimax quantiles decompose as a sum of the corresponding minimax quantiles under a heterogeneous, MCAR assumption, and a robust error term, depending on , that reflects the additional error incurred by departure from MCAR. We next introduce natural classes of realisable -contamination models, where an MCAR version of a base distribution is contaminated by an arbitrary missing not at random (MNAR) version of . These classes are rich enough to capture various notions of biased sampling and sensitivity conditions, yet we show that they enjoy improved minimax performance relative to our earlier arbitrary contamination classes for both parametric and nonparametric classes of base distributions. For instance, with a univariate Gaussian base distribution, consistent mean estimation over realisable -contamination classes is possible even when and the proportion of missingness converge (slowly) to 1. Finally, we extend our results to the setting of departures from missing at random (MAR) in normal linear regression with a realisable missing response.

Paper Structure

This paper contains 50 sections, 47 theorems, 343 equations, 6 figures, 1 table, 5 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Notation
  4. Statistical setting
  5. The extended space X_star and classical models of missing data
  6. Models of departures from M(C)AR
  7. Identifiability issues under the missing at random assumption
  8. Huber-style models of departure from MCAR
  9. Regression with missing response
  10. Characterisation of realisability
  11. Minimax quantile framework
  12. Mean estimation under arbitrary contamination
  13. Mean estimation under realisable contamination
  14. Gaussian realisable model
  15. Univariate case
  16. ...and 35 more sections

Key Result

Theorem 1

Let $\mathcal{X}_1,\ldots,\mathcal{X}_d$ be locally compact Hausdorff spacesFor the convenience of the reader, definitions of these terms from topology are provided in Section sec:proof-general-realisable. and let $\mathcal{X} \coloneqq \prod_{j=1}^d \mathcal{X}_j$. Assume that every open set in $\m for all $f \in C_{\mathrm{b}}(\mathcal{X}_\star)$.

Figures (6)

  • Figure 1: An illustration of the arbitrary $\epsilon$-contamination model $\mathcal{P}^{\mathrm{arb}}(P, \epsilon, \pi)$, which interpolates between $\mathsf{MCAR}_{(\pi,P)}$ and $\mathcal{P}(\mathcal{X}_\star)$.
  • Figure 2: An illustration of the realisable $\epsilon$-contamination model $\mathcal{R}(P, \epsilon, \pi)$, which interpolates between $\mathsf{MCAR}_{(\pi,P)}$ and $\mathsf{MNAR}_P$.
  • Figure 3: An example of a Gaussian-realisable distribution. Let $q = 1$ and $\epsilon = 0.8$. Panel (a) plots (i) $\{q(1 - \epsilon) + \epsilon\} \cdot \phi(x)$ as a solid black curve, (ii) $q(1-\epsilon) \cdot \phi(x)$ as a dashed black curve and (iii) $\{q (1 - \epsilon) + m(x)\} \cdot \phi(x)$ as a solid red curve, for some $m:\mathbb{R} \to [0,1]$. Note that the red curve is realisable by $\mathsf{N}(0, 1)$. By contrast, panel (b) plots the red curve with no changes and uses $\phi(x - 1/2)$ in place of $\phi(x)$ for the two black curves. In this case, the red curve is not realisable by $\mathsf{N}(1/2, 1)$.
  • Figure 4: Illustration of the Kolmogorov projection onto two distinct realisable sets. The realisable sets are disjoint when $\theta_1 \neq \theta_2$, by Lemma \ref{['lemma:one-dim-kolmogorov-distance-realisable-sets']}.
  • Figure 5: Schematic diagrams of various maps defined in the proof. The fact that the maps in the right panel commute follows from the fact that $h_*(\mu)(g) = \mu(g\circ h)$ for all $g\in C_{\mathrm{b}}(\mathcal{X}_\star)$.
  • ...and 1 more figures

Theorems & Definitions (94)

  • Example 1
  • Example 2
  • Theorem 1
  • Proposition 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • ...and 84 more