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Deep non-parametric logistic model with case-control data and external summary information

Hengchao Shi, Ming Zheng, Wen Yu

TL;DR

This work considers the estimation of a non-parametric logistic model with the case-control data supplemented by external summary information and derives the non-asymptotic error bound of the proposed estimator.

Abstract

The case-control sampling design serves as a pivotal strategy in mitigating the imbalanced structure observed in binary data. We consider the estimation of a non-parametric logistic model with the case-control data supplemented by external summary information. The incorporation of external summary information ensures the identifiability of the model. We propose a two-step estimation procedure. In the first step, the external information is utilized to estimate the marginal case proportion. In the second step, the estimated proportion is used to construct a weighted objective function for parameter training. A deep neural network architecture is employed for functional approximation. We further derive the non-asymptotic error bound of the proposed estimator. Following this the convergence rate is obtained and is shown to reach the optimal speed of the non-parametric regression estimation. Simulation studies are conducted to evaluate the theoretical findings of the proposed method. A real data example is analyzed for illustration.

Deep non-parametric logistic model with case-control data and external summary information

TL;DR

This work considers the estimation of a non-parametric logistic model with the case-control data supplemented by external summary information and derives the non-asymptotic error bound of the proposed estimator.

Abstract

The case-control sampling design serves as a pivotal strategy in mitigating the imbalanced structure observed in binary data. We consider the estimation of a non-parametric logistic model with the case-control data supplemented by external summary information. The incorporation of external summary information ensures the identifiability of the model. We propose a two-step estimation procedure. In the first step, the external information is utilized to estimate the marginal case proportion. In the second step, the estimated proportion is used to construct a weighted objective function for parameter training. A deep neural network architecture is employed for functional approximation. We further derive the non-asymptotic error bound of the proposed estimator. Following this the convergence rate is obtained and is shown to reach the optimal speed of the non-parametric regression estimation. Simulation studies are conducted to evaluate the theoretical findings of the proposed method. A real data example is analyzed for illustration.
Paper Structure (12 sections, 9 theorems, 69 equations, 2 figures, 3 tables)

This paper contains 12 sections, 9 theorems, 69 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Method
  3. Model specification and identifiability
  4. Functional approximation
  5. Model estimation
  6. Theoretical results
  7. Numerical results
  8. Simulation studies
  9. Real data example
  10. Concluding remarks
  11. Appendix
  12. Proof of theorems and corollary

Key Result

Theorem 1

Under Assumption 4 and Assumption 5, we have that i) $\hat{P}_1\xrightarrow{p}P_1$ as $n\to\infty$, where $\xrightarrow{p}$ stands for converging in probability, and ii) $\sqrt{n}(\hat{P}_1 - P_1) \xrightarrow{d} N(0,a_1^2 V_1 + a_2^2 V_0 + a_3^2 V)$ as $n\to\infty$, where $a_1, a_2, a_3, V_0$, an

Figures (2)

  • Figure 1: The average of $\widehat{g}_n(x)$ and $\widetilde{g}_n(x)$. The black line corresponds to the truce value of $g(x)$; the red dotted line corresponds to the average of $\widehat{g}_n(x)$; the blue dotted line corresponds to the average of $\widetilde{g}_n(x)$
  • Figure 2: The scatter plots of the case-control data estimator against full data estimator. The left panel is for $\widehat{g}_n(x)$ and the right panel is for $\widetilde{g}_n(x)$. The three rows correspond to three case and control sizes.

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6