Extending nonparametric two-stage fixed-width confidence intervals to heavy-tailed, weaker-moment, and dependent data regimes

Steland and Chang (2021) showed that a fully nonparametric two-stage procedure for a fixed-width confidence interval is consistent, asymptotically correct, and first- and second-order efficient — under both the classical high-accuracy asymptotics (interval width $d\downarrow 0$) and their novel high-confidence asymptotics (confidence level tending to 1) — whenever the point estimator satisfies a random central limit theorem (RCLT) and its asymptotic variance admits a consistent, asymptotically unbiased estimator. All of their theory assumes i.i.d. observations and high-order moments ($E(Y_1^8)<\infty$ for the one-sample second-order bound, $E|X_{i1}|^{12}<\infty$ for the common-mean RCLT), and the pilot-sample tuning constants are chosen heuristically. The open question is whether these guarantees — asymptotic coverage, consistency and asymptotic unbiasedness of the data-dependent final sample size, and first- and second-order efficiency — can be established, or shown to fail, when these sufficient conditions are weakened: in heavy-tailed regimes where the RCLT does not hold, under lighter moment assumptions, and for dependent (time-series, spatial, or networked) observations.

Background

A fixed-width confidence interval for a parameter $\theta$ is an interval of prespecified half-width $d$ whose coverage probability is required to approach a target level $1-\alpha$. Since the sample size needed to achieve width $d$ depends on the unknown asymptotic variance, it must itself be estimated from the data. Chow and Robbins (1965) introduced the purely sequential solution, in which sampling continues until a data-dependent stopping rule is met; Stein (1945) and Mukhopadhyay (1980) developed two-stage alternatives, in which a pilot sample first estimates the variance and determines the final sample size, which is then sampled in one additional batch. Two-stage procedures trade some statistical efficiency for operational simplicity, which matters when each data query is expensive, as in distributed big-data analysis — the motivating setting of the source paper. Two performance notions are central: a procedure is first-order efficient if the ratio of its expected final sample size to the optimal fixed sample size $N_{\mathrm{opt}}$ (known only under the true variance) tends to 1, and second-order efficient if the difference of the expected sample size and $N_{\mathrm{opt}}$ stays bounded (or grows slower than the leading term) — a strictly sharper property that typically requires high-order moment control. The key probabilistic tool is Anscombe's random central limit theorem (RCLT): a statistic $\hat{\mu}_n$ satisfies the RCLT if $\sqrt{n}(\hat{\mu}_n-\mu)/\sigma_\mu$ not only converges in distribution to a standard normal along deterministic sample sizes but remains asymptotically standard normal when evaluated at a random sample size $N$ with $N/n\to 1$ — the property that makes inference at data-dependent sample sizes valid. Steland and Chang (2021) distilled the guarantees of a two-stage fixed-width procedure into two sufficient conditions: Assumption (A), the RCLT for the estimator, and Assumption (E), the existence of a consistent and asymptotically unbiased estimator of the asymptotic variance $\sigma_\mu^2$. Under these they proved consistency and asymptotic unbiasedness of the random sample size for the optimal fixed size, asymptotic coverage $1-\alpha$, and first- and second-order efficiency, both when $d\downarrow 0$ at fixed level (high-accuracy asymptotics) and — a framework they introduce — when $d$ is fixed and the confidence level tends to 1 (high-confidence asymptotics). They instantiated the theory for three problems: the mean of univariate data, the mean of a projection of high-dimensional data (extending the two-stage high-dimensional program of Aoshima and Yata, 2011), and the common mean of two samples with ordered variances, where the nonlinear common-mean estimators (convex combinations of the two sample means with weights depending on sample variances) required a dedicated RCLT analysis based on jackknife variance estimation — the jackknife re-evaluates the statistic leaving out one observation at a time and estimates its variance from the spread of the resulting values — the latter built on Steland and Chang (2019) under a twelfth-moment condition. The first-stage sample size follows Mukhopadhyay's modified rule with tuning constants ($\bar{N}_0$, a scale constant $f$, and an exponent $\gamma$), or the paper's three-observation rule, which estimates $\sigma_\mu$ from only three pilot observations; no systematic calibration theory for these constants is given. Three boundary facts delimit what the paper does not resolve. First, the theory is silent when the RCLT fails — e.g. for heavy-tailed data with infinite variance, where even the target $N_{\mathrm{opt}}=O(\sigma^2/d^2)$ underlying the two-stage rule is undefined and different limit laws (non-Gaussian stable limits, or self-normalized statistics that divide by a random scale statistic) govern the estimator. For the simple mean, distribution-free sequential intervals under no moment conditions have long existed via rank tests (Geertsema, 1970), and sequential fixed-width intervals are valid for dependent steady-state simulation output (Glynn and Whitt, 1992), but neither strand addresses this two-stage framework. Second, the second-order results genuinely use the imposed moments: $E(Y_1^8)<\infty$ for the one-sample second-order bound and $E|X_{i1}|^{12}<\infty$ for the common-mean RCLT, and the paper does not reduce these orders. Third, all theory assumes i.i.d. observations within and between stages; simulations with $t(5)$ errors probe heavier tails only empirically.

Problem Statement

The Steland–Chang framework reduces all guarantees of a nonparametric two-stage fixed-width confidence interval — asymptotic coverage $1-\alpha$ in both the high-accuracy asymptotics ($d\downarrow 0$) and the high-confidence asymptotics (confidence level tending to 1), consistency and asymptotic unbiasedness of the data-dependent final sample size for the optimal fixed sample size, and first- and second-order efficiency — to two sufficient conditions: the random central limit theorem (RCLT; Assumption (A)) — asymptotic normality that survives evaluation at random rather than deterministic sample sizes — and a consistent, asymptotically unbiased variance estimator (Assumption (E)), proved under i.i.d. sampling with high-order moment conditions. The open question is to determine how far these sufficient conditions can be weakened, and where the framework breaks. Concretely, for the two-stage procedure (and, where the variance-based sample-size target is undefined, for a principled modification of the two-stage sample-size rule that retains its pilot-sample-plus-one-batch structure; replacing the two-stage design by a purely sequential procedure does not resolve the regime): (i) establish or refute asymptotic validity and efficiency in regimes where the RCLT fails, such as heavy-tailed data with infinite variance governed by non-normal limit laws or self-normalization; (ii) prove the second-order efficiency bounds and the common-mean RCLT under moment assumptions weaker than $E(Y_1^8)<\infty$ and $E|X_{i1}|^{12}<\infty$, identifying the minimal moment order, if any, at which each property survives; and (iii) extend the guarantees to dependent observations — stationary time series, spatial, or networked data — including verification of an Anscombe-type RCLT and construction of consistent, asymptotically unbiased variance estimators for the two-stage rule. Items (i)–(iii) share one scientific objective: remove or weaken the regularity conditions so that the same two-stage fixed-width guarantees hold over a strictly larger class of data-generating mechanisms, for the mean, the high-dimensional projection, and the common-mean applications alike; a negative answer for any regime — a proof that coverage or second-order efficiency must fail, or that no asymptotically unbiased variance estimator exists in a stated class — resolves that regime in the opposite direction. Each of (i)–(iii) is an independently resolvable sub-question, and each may be settled for any one of the three applications on its own.

References

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