Preservation of the fixed-sample-size asymptotic distribution of maximum likelihood estimators under sample size re-estimation

In confirmatory treatment-effect experiments designed with a predetermined type I error and power, the total sample size is commonly re-estimated at an interim analysis from first-stage data. When the re-estimation rule uses the estimated treatment effect, the full-sample maximum likelihood estimator (MLE) no longer has the asymptotic distribution it would have under a fixed sample size: it converges to a random mixture of truncated normal distributions. The source paper (Tarima and Flournoy, 2019) settles the single-interim case up to sufficient degeneracy conditions and explicitly defers multiple interim analyses to a subsequent manuscript. The open problem is to complete this asymptotic theory — the multiple-interim extension and, more generally, a characterization of which sample-size re-estimation (SSR) rules and model classes preserve the fixed-sample-size asymptotic distribution of the MLE; a related open direction is post-SSR estimation and inference with nominal asymptotic coverage when preservation fails.

Background

Many confirmatory experiments, notably randomized clinical trials, are designed to detect a treatment effect with a predetermined type I error rate (the probability of falsely declaring an effect, $\alpha$) and statistical power (the probability of detecting a prespecified effect, $1-\beta$). The sample size needed for these operating characteristics depends on unknown quantities, so a common design pauses the study after a first stage of $n$ observations (an interim analysis), re-estimates the required total sample size $N$ from the first-stage data, and collects $N-n$ further observations. This is sample size re-estimation (SSR). Rules differ in what they plug in: nuisance-based rules re-estimate only parameters such as the variance, while effect-based rules use the estimated treatment effect itself; to control cost and feasibility, $N$ is usually truncated to an interval $[n, n_{\max}]$, which induces truncation in the distributions of downstream statistics.

A secondary objective of such experiments is estimation of the treatment effect. The maximum likelihood estimator (MLE) is the parameter value maximizing the joint density of all observations; under standard regularity conditions with a fixed sample size, the rescaled MLE is asymptotically normal, and this asymptotic normality underlies the usual Wald confidence intervals and tests (intervals and tests formed as the estimate plus or minus a multiple of its standard error). Classical central limit theory for randomly stopped sums covers sample sizes driven by stopping rules that do not depend on the parameter being estimated, but effect-based SSR violates its premises.

Tarima and Flournoy (2019) studied what happens to this picture under SSR in regular parametric models $X\sim f(x\,|\,\theta,\eta)$ with an effect parameter $\theta$ and nuisance parameter $\eta$, under local alternatives $\theta=h/\sqrt{n}$ that approach the null as the first-stage size grows. Their Theorem 1 shows that the standardized full-sample MLE $\sqrt{N}(\tilde\theta-\theta)/\sigma_\theta$ converges to a random mixture of truncated normal distributions whose mixing is governed by the random ratio $N/n$ and by the boundary probabilities that the recalculated size falls at $n$ or $n_{\max}$. Normality is preserved only in degenerate situations: when $N/n$ converges to a constant (as happens when the rule uses only a consistently estimated nuisance parameter), or when the asymptotic probability mass of $N$ concentrates near $n$ or far above $n$. Rules that plug the first-stage effect estimate into the sample-size formula generically produce non-normal mixtures, and intervals based on the usual central-limit approximation can severely under-cover — in the authors' real-data example, 77.5% empirical coverage at a nominal 95% level, improved only to 89.7% by their local-alternative calibration.

This phenomenon sits next to the classical local asymptotic normality (LAN) framework — the property that log-likelihood ratios of a regular model are asymptotically quadratic — and its mixed-normal extension (LAMN), in which a random information quantity appears in the limit law; the SSR-induced limits are more singular, being mixtures of truncated laws rather than mixed normals. Parallel results were obtained by Flournoy, May and Tommasi (2021), who proved for nonlinear regression with Gaussian errors and a two-stage adaptive design that the MLE is consistent with a Gaussian-mixture limit obtained via stable convergence (a mode of convergence strong enough to preserve conditioning on the first-stage data), with an asymptotic covariance that is not the inverse Fisher information; their technique requires normal errors. Tarima and Flournoy (2024) quantified the information cost of informative adaptation by decomposing the total Fisher information (the quantity that in fixed-sample-size theory determines the MLE's asymptotic variance) into a component consumed by the design and a remainder, deriving a new lower bound for the mean squared error of post-adaptation estimators and showing that the classical Cramér–Rao and Wolfowitz sequential bounds (classical lower bounds on, respectively, the variance of unbiased estimators at a fixed sample size and the mean squared error of sequentially stopped estimators) do not apply after informative adaptation. Lin, Flournoy and Rosenberger (2020) showed that random norming — rescaling the MLE by observed rather than expected information — better normalizes the estimator in adaptive nonlinear regression, though without a coverage guarantee for effect-based SSR.

What the source explicitly defers is the multiple-interim extension: "to elicit the asymptotic distribution for multiple interim analyses, simply apply Theorem 2 recursively. These results will be studied further in a subsequent manuscript." More generally, what remains unresolved in the literature is a necessary-and-sufficient characterization, across SSR rule classes and model choices, of when the fixed-sample-size asymptotic distribution survives the interim adaptation, together with a post-SSR estimation and inference methodology whose operating characteristics are provably nominal when it does not.

Problem Statement

Setting (source-faithful): observations $X_1,\dots,X_N$ come from a regular parametric model $f(x\,|\,\theta,\eta)$ with effect parameter $\theta$ and nuisance parameter $\eta$. After a first stage of $n$ observations, the maximum likelihood estimates $(\hat\theta_1,\hat\eta_1)$ are computed and the total sample size is re-estimated as $N=n\,g(\hat\theta_1,\hat\eta_1)$, truncated to $[n,n_{\max}]$, by a rule $g$ chosen to secure type I error $\alpha$ and power $1-\beta$ at a minimally clinically relevant effect; $N-n$ further observations are then collected, and $\tilde\theta$ denotes the full-sample MLE. Without SSR, the standardized MLE is asymptotically standard normal (or, in non-regular models, has a known non-normal limit law). The open questions below concern extensions beyond the source's single-interim setting: multiple interim analyses, multivariate parameters, non-Gaussian error distributions, and models with known non-normal fixed-sample-size limits, for which the source's Theorem 2 already supplies the post-SSR limit representation.

Open questions:

1. Preservation characterization. Determine necessary and sufficient conditions on the SSR rule $g$ and on the model under which the standardized full-sample MLE retains the asymptotic distribution it would have under a fixed sample size. The single-interim case is settled only for rules under which the random ratio $N/n$ converges in distribution — the working assumption of the source's Theorem 1, whose Corollary 1 specializes to rules that are functions of the estimated coefficient of variation (the ratio of the standard deviation of the first-stage effect estimate to its expectation): preservation holds when the mixing degenerates ($N/n$ converging to a constant, or asymptotic mass of $N$ concentrating near $n$ or far above $n$); otherwise a mixture of truncated laws results, and the source states that its asymptotic results apply as well to rules without explicit formulae such as conditional-power-based re-estimation (rules that set the total sample size so that the probability of a significant final result, computed under the interim effect estimate, reaches a target). The characterization remains open for multiple interim analyses — the extension the source explicitly defers, noting that its Theorem 2 (which gives the post-SSR limit of any statistic with a stationary non-degenerate asymptotic distribution, including non-normal fixed-sample-size limits) can be applied recursively — for multivariate parameters, and for non-Gaussian error distributions. These open directions are independently sufficient rather than jointly mandatory: establishing the necessary-and-sufficient characterization in any one of them — single-interim rules for which $N/n$ does not converge in distribution, the multiple-interim setting, multivariate parameters, non-Gaussian errors, or models with known non-normal fixed-sample-size limits — completes question 1, while the general characterization over all SSR rule classes and model classes is the broader target that subsumes them.

2. Post-SSR inference. A related open direction, not a requirement conjoined with question 1: for each class of SSR rules and models in which preservation fails, construct estimators, confidence intervals, and tests whose asymptotic coverage (for intervals) and type I error (for tests) equal their nominal levels under the resulting mixture limit — closing the coverage gap left by current calibration approaches, which reached 89.7% versus nominal 95% coverage in the source paper's illustrative data example.

References

1. Sergey Tarima, Nancy Flournoy, 'Asymptotic properties of maximum likelihood estimators with sample size recalculation', Statistical Papers 60(2):373-394 (2019), DOI 10.1007/s00362-019-01095-x, https://doi.org/10.1007/s00362-019-01095-x (source paper; full text open-access at PMC6905624)
2. Nancy Flournoy, Caterina May, Chiara Tommasi, 'The effects of adaptation on maximum likelihood inference for nonlinear models with normal errors', Journal of Statistical Planning and Inference 214:139-150 (2021), DOI 10.1016/j.jspi.2021.02.002, https://doi.org/10.1016/j.jspi.2021.02.002 (preprint arXiv:1812.03970, posted under the earlier title 'The Effects of Adaptation on Inference for Non-Linear Regression Models with Normal Errors', https://arxiv.org/abs/1812.03970)
3. Sergey Tarima, Nancy Flournoy, 'The cost of sequential adaptation and the lower bound for mean squared error', Statistical Papers 65(9):5529-5553 (2024), DOI 10.1007/s00362-024-01565-x, https://doi.org/10.1007/s00362-024-01565-x (preprint arXiv:2209.02436)
4. Sergey Tarima, Nancy Flournoy, 'Distribution theory following blinded and unblinded sample size re-estimation under parametric models', Communications in Statistics - Simulation and Computation 51(4):2053-2064 (2022), DOI 10.1080/03610918.2019.1692030, https://doi.org/10.1080/03610918.2019.1692030
5. Sergey Tarima, Nancy Flournoy, 'Effect of interim adaptations in group sequential designs' (2019), preprint arXiv:1908.01411, https://arxiv.org/abs/1908.01411
6. Sergey Tarima, Nancy Flournoy, 'Most powerful test sequences with early stopping options', Metrika 85(4):491-513 (2022), DOI 10.1007/s00184-021-00839-w, https://doi.org/10.1007/s00184-021-00839-w (preprint arXiv:2101.11595)
7. Zhantao Lin, Nancy Flournoy, William F. Rosenberger, 'Random norming aids analysis of non-linear regression models with sequential informative dose selection', Journal of Statistical Planning and Inference 206:29-42 (2020), DOI 10.1016/j.jspi.2019.09.003, https://doi.org/10.1016/j.jspi.2019.09.003
8. Sergey Tarima, Nancy Flournoy, 'Choosing interim sample sizes in group sequential designs', Studies in Health Technology and Informatics, German Medical Data Sciences: Bringing Data to Life (2021), DOI 10.3233/SHTI210043, https://doi.org/10.3233/SHTI210043
9. Sergey Tarima, Nancy Flournoy, 'Group sequential tests: beyond exponential family models', Statistical Papers 64(4):1361-1372 (2023), DOI 10.1007/s00362-023-01432-1, https://doi.org/10.1007/s00362-023-01432-1
10. Peijin Wang, Shein-Chung Chow, 'Sample size re-estimation in clinical trials', Statistics in Medicine 40(27):6133-6149 (2021), DOI 10.1002/sim.9175, https://doi.org/10.1002/sim.9175
11. Nancy Flournoy, Sergey Tarima, 'Discussion on "Adaptive enrichment designs with a continuous biomarker" by Nigel Stallard', Biometrics 79(1):31-35 (2023), DOI 10.1111/biom.13641, https://doi.org/10.1111/biom.13641
12. Giacomo Aletti, Nancy Flournoy, Caterina May, Chiara Tommasi, 'Maximum likelihood estimation under the Emax model: existence, geometry and efficiency', Statistical Papers 66(5):article 106 (2025), DOI 10.1007/s00362-025-01673-2, https://doi.org/10.1007/s00362-025-01673-2