Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Learning Centre Partitions from Summaries

Zinsou Max Debaly, Jean-Francois Ethier, Michael H. Neumann, Félix Camirand-Lemyre

TL;DR

To improve finite-sample integration, a multi-round bootstrap CoC that re-evaluates merges across independently resampled summary sets is introduced and it is proved a \emph{golden-partition recovery} result: as the number of rounds grows with $n$, the true partition is recovered with probability tending to one.

Abstract

Multi-centre studies increasingly rely on distributed inference, where sites share only centre-level summaries. Homogeneity of parameters across centres is often violated, motivating methods that both \emph{test} for equality and \emph{learn} centre groupings before estimation. We develop multivariate Cochran-type tests that operate on summary statistics and embed them in a sequential, test-driven \emph{Clusters-of-Centres (CoC)} algorithm that merges centres (or blocks) only when equality is not rejected. We derive the asymptotic $χ^2$-mixture distributions of the test statistics and provide plug-in estimators for implementation. To improve finite-sample integration, we introduce a multi-round bootstrap CoC that re-evaluates merges across independently resampled summary sets; under mild regularity and a separation condition, we prove a \emph{golden-partition recovery} result: as the number of rounds grows with $n$, the true partition is recovered with probability tending to one. We also give simple numerical guidelines, including a plateau-based stopping rule, to make the multi-round procedure reproducible. Simulations and a real-data analysis of U.S.\ airline on-time performance (2007) show accurate heterogeneity detection and partitions that change little with the choice of resampling scheme.

Learning Centre Partitions from Summaries

TL;DR

To improve finite-sample integration, a multi-round bootstrap CoC that re-evaluates merges across independently resampled summary sets is introduced and it is proved a \emph{golden-partition recovery} result: as the number of rounds grows with , the true partition is recovered with probability tending to one.

Abstract

Multi-centre studies increasingly rely on distributed inference, where sites share only centre-level summaries. Homogeneity of parameters across centres is often violated, motivating methods that both \emph{test} for equality and \emph{learn} centre groupings before estimation. We develop multivariate Cochran-type tests that operate on summary statistics and embed them in a sequential, test-driven \emph{Clusters-of-Centres (CoC)} algorithm that merges centres (or blocks) only when equality is not rejected. We derive the asymptotic -mixture distributions of the test statistics and provide plug-in estimators for implementation. To improve finite-sample integration, we introduce a multi-round bootstrap CoC that re-evaluates merges across independently resampled summary sets; under mild regularity and a separation condition, we prove a \emph{golden-partition recovery} result: as the number of rounds grows with , the true partition is recovered with probability tending to one. We also give simple numerical guidelines, including a plateau-based stopping rule, to make the multi-round procedure reproducible. Simulations and a real-data analysis of U.S.\ airline on-time performance (2007) show accurate heterogeneity detection and partitions that change little with the choice of resampling scheme.

Paper Structure

This paper contains 38 sections, 15 theorems, 172 equations, 2 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. On the Homogeneity Across Data centres
  4. Cochran-Type Tests for Homogeneity
  5. Local alternatives analysis
  6. Cochran-Type Tests for the Fusion of two sets of Local Estimators
  7. Clusters of Centres Algorithm
  8. Golden-partition recovery property of multi-round bootstrap CoC-algorithm
  9. Control of the type-I and type-II errors
  10. CoC-algorithm with shrinkage rejection region
  11. Simulation study
  12. Real Data Application
  13. Arrival Delays Across Major U.S. Destination Airports (2007)
  14. Concluding Remarks
  15. Supplementary material
  16. ...and 23 more sections

Key Result

Lemma 1

Consider the centre-specific asymptotic decomposition eq::asymptoticDecomposition and suppose that the assumptions (A1) to (A4) hold. Then, under the null hypothesis $H_0 : \theta_{0, 1} = \theta_{0, 2} = \cdots = \theta_{0, K} = \theta_0,$ we have where $\{\lambda_\ell\}_{\ell=1}^{Kp}$ are the nonnegative eigenvalues of $\overline{Q}^{1/2}\, H^\top H\, \overline{Q}^{1/2}$ and $\{\chi_\ell^2\}_{\

Figures (2)

  • Figure 1: Performance curves versus $n$ (solid: $R=50$, dashed: $R=100$), for each $(\delta,u_n)$ setting for $L= 4$ and $K= 20$.
  • Figure 2: Performance curves versus $n$ (solid: $R=50$, dashed: $R=100$), for each $(\delta,u_n)$ setting for $L= 6$ and $K= 40$.

Theorems & Definitions (15)

  • Lemma 1
  • Proposition 1
  • Lemma 2
  • Proposition 2
  • Lemma 3
  • Theorem 1
  • Lemma 4
  • Proposition 3
  • Theorem 2
  • Theorem 3
  • ...and 5 more