Dependence-Robust Resampling and Moment Tests
Suppose a mean is root-\(n\) consistent, but estimating its long-run variance is difficult. (Leung 2022) adds random draws whose variance can be calculated conditional on the data. The resulting test works when the original sample’s error is small relative to this artificial fluctuation. As in ordering-free resampling, this requirement limits how many draws can be used.
1. The main limit theorem
Let \(X_i\in\mathbb R^m\), with sample mean \(\overline X\) and sample covariance \(\widehat\Sigma=n^{-1}\sum_i(X_i-\overline X)(X_i-\overline X)'\). Draw independent uniform indices \(I_r\) and independent ordered distinct pairs \((J_r,K_r)\).
The mean statistic averages randomly selected observations. The pair statistic instead averages products from distinct pairs:
Theorem 1. Suppose \(\sqrt n(\overline X-\mu_0)=O_P(1)\), \(\widehat\Sigma\) converges to a positive definite matrix, and empirical \(2+\delta\) moments are bounded in probability. As \(R\to\infty\), \(Z_M(\mu_0)\Rightarrow \mathcal{N}(0,I_m)\) if \(R/n\to0\), while \(T_U(\mu_0)\Rightarrow \mathcal{N}(0,1)\) if \(\sqrt R/n\to0\).
2. The mean statistic in one line
To locate the original-sample error, add and subtract the sample mean inside every draw:
The centered draws in the first term have conditional covariance equal to the identity and obey a conditional i.i.d. CLT. The remaining drift is \(O_P(\sqrt{R/n})\), which vanishes under the theorem’s repetition restriction. Thus the sample covariance standardizes the artificial draws; it is not being used to estimate the original mean’s long-run variance.
3. Why pairs allow more repetitions
Put \(a=\overline X-\mu\). Summing over distinct pairs gives the exact conditional expectation
The linear terms cancel because centered observations sum to zero. Under the null, both remaining terms are order \(1/n\), so aggregation creates a drift of order \(\sqrt R/n\). This explains the weaker restriction on \(R\).
The mean statistic detects shifts of order \(R^{-1/2}\), and the pair statistic detects shifts of order \(R^{-1/4}\). Balancing the paper’s leading error terms suggests \(R_M\asymp n^{1/2}\) and \(R_U\asymp n^{4/3}\), giving detectable scales \(n^{-1/4}\) and \(n^{-1/3}\). These are tuning heuristics, not universally optimal choices.
4. Extensions and limits
For a one-sided moment inequality, squaring creates an additional problem: large deviations on the null side also produce a large statistic. The paper subtracts an estimated null-side drift and proves uniform size control under uniform versions of the assumptions. Estimated nuisance scores need additional consistency and smoothness conditions.
The initial root-\(n\) consistency assumption already restricts dependence. If the known convergence rate is slower, the allowable \(R\) must shrink accordingly. The artificial draws provide a tractable reference law while retaining the information limitation of the original observations.
5. Whiten the empirical distribution first
Define \(Z_i=\widehat\Sigma^{-1/2}(X_i-\overline X)\). Exactly, conditional on the original data,
A uniform draw \(I\) therefore has \(\mathbb{E}[Z_I\mid\mathbf X]=0\) and covariance \(I_m\). With empirical \(2+\delta\) moments bounded and a nonsingular limiting covariance, the conditional Lyapunov bound for \(R\) independent draws is of order \(R^{-\delta/2}\) times a bounded empirical moment. It vanishes as \(R\to\infty\).
This proves the normal limit of the centered artificial mean. To pass from conditional to unconditional characteristic functions, the conditional characteristic function approaches the deterministic Gaussian one in probability and is bounded by one; taking expectations preserves the limit. The remaining drift is \(\sqrt R\widehat\Sigma^{-1/2}(\overline X-\mu_0)\), which is negligible under \(R/n\to0\) and the root-\(n\) consistency assumption.
6. The distinct-pair second moment
For a uniformly chosen ordered pair \(J\ne K\), let \(H=Z_J'Z_K\). Since \(\sum_iZ_i=0\),
so \(\mathbb{E}H=-m/(n-1)\). For the second moment, use the trace identity
The trace identity includes all pairs. Subtract the squared diagonal contributions and divide by the number of ordered distinct pairs:
The empirical \(2+\delta\) moment condition implies \(\max_j\|Z_j\|^2/n=o_p(1)\). Since \(\sum_j\|Z_j\|^2=nm\),
Thus the conditional variance of \(H\) converges to \(m\), explaining the factor \(\sqrt{mR}\) in \(T_U\). Bounds on pair moments follow from \(|Z_j'Z_k|\leq\|Z_j\|\|Z_k\|\), giving the conditional CLT for the centered pair average. The excluded diagonal produces a small negative mean, not an exactly mean-zero statistic.
7. Add back a candidate mean
Put \(b=\widehat\Sigma^{-1/2}(\overline X-\mu)\). A pair contribution at \(\mu\) is
Each centered observation has zero conditional average, so the linear terms disappear on taking expectation. The remaining terms give
Under the null, \(b=O_p(n^{-1/2})\), so this drift is \(O_p(\sqrt R/n)\). The random linear terms have vanishing conditional variance when \(b=o_p(1)\), while the centered quadratic term retains variance one. These are the separate mean and variance checks behind the pair limit.
For a local alternative \(\mu_0-\mu=hR^{-1/4}\), and with original-sample error negligible at that scale, the positive drift is \(h'\Sigma^{-1}h/\sqrt m\) to first order. The mean statistic instead shifts linearly at scale \(R^{-1/2}\). As in the Song comparison, their useful repetition counts differ because the mean and quadratic sampling drifts have different orders.
The variance of the random linear term can be computed exactly as well. Since \(J\ne K\),
Hence the linear part of \(T_U(\mu)\), after aggregation over \(R\) independent pairs, has conditional variance
Its size does not grow with \(R\), because it is a centered sum divided by \(\sqrt R\). The deterministic quadratic drift does grow with \(\sqrt R\). This separates the two reasons for requiring a candidate mean close to the population mean. Consistency makes the random linear part negligible; the stronger repetition restriction controls the accumulated squared error.
The cross-covariance with the centered quadratic term is harmless once the latter’s variance is bounded: Cauchy–Schwarz bounds it by a constant times \(\|b\|\). Thus no symmetry assumption on the empirical distribution is needed to remove this cross-term asymptotically.
Whitening works because the artificial draws are independent conditional on the data. Their covariance can differ from the original sample mean’s long-run covariance. The method avoids estimating the latter by making the artificial fluctuation large enough to dominate the original estimation error. A persistent common shock prevents that domination.
8. A scalar numerical identity
For centered scalar observations with empirical variance one, \(\sum_iZ_i^2=n\). With two observations they must be opposite in sign, and every distinct-pair product is \(-1\). The pair statistic then has no conditional variance. This does not contradict the asymptotic theorem: the diagonal correction is substantial at such a small \(n\), and the no-dominant-observation argument has not taken effect.
As \(n\) grows under the moment assumptions, the discarded diagonal share \(\sum_iZ_i^4/n^2\) vanishes and the pair variance approaches one. The exact formula provides a useful finite-sample diagnostic for unusually concentrated observations, even when an asymptotic normal approximation is ultimately used.
9. What slower original-sample rates would require
Suppose instead that \(\overline X-\mu_0=O_p(a_n^{-1})\) for a known diverging rate \(a_n\). The mean drift is negligible if \(\sqrt R/a_n\to0\). The quadratic mean component is negligible if \(\sqrt R/a_n^2\to0\), and the distinct-pair correction additionally requires \(\sqrt R/n\to0\).
These bounds check the drifts in a possible slower-rate extension. Covariance convergence, empirical moments, and the conditional approximation remain to be proved. Random sampling cannot correct a persistent original-sample bias or identify a mean that the original model does not identify.
The moment bound needed for the pair CLT
The pair argument does not require an empirical fourth moment to remain bounded. For \(r=2+\delta\) and \(M_{r,n}=n^{-1}\sum_i\|Z_i\|^r=O_p(1)\),
Centering preserves this moment order by \(|x-y|^r\le2^{r-1}(|x|^r+|y|^r)\). With conditional pair variance tending to \(m>0\), the Lyapunov ratio for \(R\) independent pairs is therefore \(O_p(R^{-\delta/2})\). It tends to zero and proves the conditional centered-pair CLT used above.
The maximum bound in Section 6 is also explicit:
Thus the same \(2+\delta\) moment assumption both eliminates the discarded diagonal share and supplies the Lyapunov bound. No independence of the original observations is used in these conditional algebraic steps; their original dependence is restricted through the sample-mean and covariance premises.