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Asymptotic properties of adaptive designs through differentiability in quadratic mean

Dennis Christensen, Emil Aas Stoltenberg, Nils Lid Hjort

TL;DR

The paper addresses the challenge of establishing asymptotic normality for the maximum likelihood estimator under adaptive designs where the covariate sequence $X_i$ depends on past data, circumventing classical regularity conditions on derivatives. It introduces summable differentiability in quadratic mean (S-DQM) as a weaker regularity framework and proves a local asymptotic normality expansion using a martingale structure for the score process, plus conditions ensuring the observed information converges to a nondegenerate limit $J$. The authors develop practical criteria to verify S-DQM, including domination by a measure and smoothness of $\sqrt{f_{\theta}(y|x)}$, and provide compact-domain results to handle edge cases. They then apply the theory to two classical adaptive designs, Bruceton and Robbins–Monro, and analyze a Markovian Langlie design, showing S-DQM holds and $J_n \to J$, yielding $\sqrt{n}(\widehat{\theta}_n-\theta_0) \xrightarrow{d} N(0, J^{-1})$. The work broadens inferential validity for adaptive experiments by reducing regularity requirements and extending LAN to dependent data, with direct implications for confidence intervals and hypothesis testing in binary regression settings.

Abstract

There exist multiple regression applications in engineering, industry and medicine where the outcomes follow an adaptive experimental design in which the next measurement depends on the previous observations, so that the observations are not conditionally independent given the covariates. In the existing literature on such adaptive designs, results asserting asymptotic normality of the maximum likelihood estimator require regularity conditions involving the second or third derivatives of the log-likelihood. Here we instead extend the theory of differentiability in quadratic mean (DQM) to the setting of adaptive designs, which requires strictly fewer regularity assumptions than the classical theory. In doing so, we discover a new DQM assumption, which we call summable differentiability in quadratic mean (S-DQM). As applications, we first verify asymptotic normality for two classical adaptive designs, namely the Bruceton 'up-and-down' design and the Robbins-Monro design. Next, we consider a more complicated problem, namely a Markovian version of the Langlie design.

Asymptotic properties of adaptive designs through differentiability in quadratic mean

TL;DR

The paper addresses the challenge of establishing asymptotic normality for the maximum likelihood estimator under adaptive designs where the covariate sequence depends on past data, circumventing classical regularity conditions on derivatives. It introduces summable differentiability in quadratic mean (S-DQM) as a weaker regularity framework and proves a local asymptotic normality expansion using a martingale structure for the score process, plus conditions ensuring the observed information converges to a nondegenerate limit . The authors develop practical criteria to verify S-DQM, including domination by a measure and smoothness of , and provide compact-domain results to handle edge cases. They then apply the theory to two classical adaptive designs, Bruceton and Robbins–Monro, and analyze a Markovian Langlie design, showing S-DQM holds and , yielding . The work broadens inferential validity for adaptive experiments by reducing regularity requirements and extending LAN to dependent data, with direct implications for confidence intervals and hypothesis testing in binary regression settings.

Abstract

There exist multiple regression applications in engineering, industry and medicine where the outcomes follow an adaptive experimental design in which the next measurement depends on the previous observations, so that the observations are not conditionally independent given the covariates. In the existing literature on such adaptive designs, results asserting asymptotic normality of the maximum likelihood estimator require regularity conditions involving the second or third derivatives of the log-likelihood. Here we instead extend the theory of differentiability in quadratic mean (DQM) to the setting of adaptive designs, which requires strictly fewer regularity assumptions than the classical theory. In doing so, we discover a new DQM assumption, which we call summable differentiability in quadratic mean (S-DQM). As applications, we first verify asymptotic normality for two classical adaptive designs, namely the Bruceton 'up-and-down' design and the Robbins-Monro design. Next, we consider a more complicated problem, namely a Markovian version of the Langlie design.
Paper Structure (9 sections, 10 theorems, 65 equations)

This paper contains 9 sections, 10 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Local asymptotic normality for adaptive designs
  3. Summable differentiability in quadratic mean (S-DQM)
  4. Techniques for establishing S-DQM
  5. From S-DQM to asymptotic normality
  6. Illustration: The Bruceton and Robbins--Monro designs
  7. Bruceton
  8. Robbins--Monro
  9. A Markovian Langlie design

Key Result

Proposition 2.1

Suppose that the conditional density $f_{\theta}(y \,|\, x)$ is S-DQM at $\theta_0$; that $i \mapsto \mathbb{E}_{\theta_0}\,\lVert u_{\theta_0}(Y_i \,|\, X_i)\rVert^4$ is uniformly bounded; and that $\langle U_{n,\cdot},U_{n,\cdot}\rangle_n$ is bounded in probability. Then $(U_{n,j})_{1\leq j \leq,

Theorems & Definitions (26)

  • Definition 2.1
  • Proposition 2.1
  • proof
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Theorem 2.1
  • proof
  • Lemma 3.1
  • ...and 16 more