Resampled Statistics: Size and Local Power
Repeated random subsamples create a reference distribution conditional on the data. Taking more draws stabilizes that distribution, but also magnifies the error already present in the original sample. The results of (Song 2018) determine how subsample size and repetition count can balance these effects, and what local power remains when dependence ordering is unknown.
1. Two kinds of signal
Let \(b_n\) be the size of each random subsample, \(R_n\) the number of repetitions, and \(d\) the dimension of the tested mean. An \(M\)-type statistic aggregates standardized subsample means, producing a squared Gaussian norm after its bias correction. A \(U\)-type statistic instead aggregates off-diagonal quadratic terms, with a normal calibration after the paper’s normalization.
Theorem 1. Under the paper’s uniform moment, covariance-nondegeneracy, and ordering-free dependence conditions, the randomized tests have asymptotic size \(\alpha\). The \(M\) statistic has the corresponding \(\chi_d^2/d\) null limit; the standardized \(U\) statistic has a Gaussian null limit. Local mean shifts of order \((R_nb_n)^{-1/2}\) affect the \(M\) limit, while shifts of order \(R_n^{-1/4}b_n^{-1/2}\) affect the \(U\) limit. Under the additional higher-order conditions used for the comparison, the \(U\) procedure can detect smaller shifts at a matched rate of size error.
The power comparison holds size accuracy fixed. Since the procedures accumulate original-sample error at different rates, equal size accuracy may permit different repetition counts.
2. Derive the local scales
For the mean statistic, a population shift \(\Delta_n\) contributes a drift proportional to \(\sqrt{R_nb_n}\,\Sigma^{-1/2}\Delta_n\). A nonvanishing drift therefore appears at the first displayed rate. Squaring the Gaussian vector yields the noncentral chi-square form.
For the quadratic statistic, the leading shift is proportional to
Making this order one gives \(\|\Delta_n\|\asymp R_n^{-1/4}b_n^{-1/2}\). The signal is quadratic, so this test detects deviations in either direction.
3. Dependence limits how much to aggregate
The \(M\) conditions include
The \(U\) conditions instead control
together with bounds on higher-order dependence coefficients. The graph-counting result supplies primitive sufficient bounds in dependency-graph examples.
The restrictions bound the conditional drift and higher moments left by the original data. More repetitions reduce Monte Carlo error relative to that drift. Larger subsamples also increase the chance of drawing dependent observations together. Both tuning parameters therefore enter the size calculation.
4. How to read the result in practice
The procedure changes the statistic itself by introducing random subsampling. Its detectable-alternative rate describes the resulting test, and can be slower than the rate of a full-sample test whose sampling variance is known.
Centered resampling can approximate critical values when the simulation count grows at an admissible rate. Combining repeated tests requires the paper’s confidence-function construction or another justified rule. Choosing a seed after seeing its answer changes the rejection event and hence the test’s size.
5. Conditional randomness and full-sample error enter separately
For a schematic whitened subsample-mean vector, write the aggregate as
Conditional on \(\mathcal X_n\), the draws in the first term are independent, so their covariance and higher moments can be calculated from the permutation experiment. The second term is fixed by the original sample. Repeating the experiment cannot average that term away.
For a mean-based draw, its conditional mean is proportional to \(\sqrt{b_n}(\overline X_n-\mu_0)\). Under an alternative \(\mathbb{E}\overline X_n=\mu_0+\Delta_n\), separate
The same multiplier acts on the population shift and the original sample’s error. A useful test keeps the former visible while making the latter negligible relative to the artificial reference fluctuation. This imposes an upper restriction on \(R_n\), even though more repetitions would improve Monte Carlo precision alone.
6. The noncentral limits explain the detection scales
Suppose the mean alternative is \(\Delta_n=h/\sqrt{R_nb_n}\). In coordinates where the limiting randomized covariance is identity, the mean vector approaches \(\Sigma^{-1/2}h\). If the remainder and bias-correction conditions continue to hold along this alternative, the squared norm has limit
This is a scaled noncentral chi-square law with noncentrality \(h'\Sigma^{-1}h\). A shift smaller than \((R_nb_n)^{-1/2}\) disappears from this limit; a shift at that scale changes the rejection probability by an order-one amount.
For the quadratic statistic, take \(\Delta_n=h/(R_n^{1/4}\sqrt{b_n})\). Then
The quadratic shift is nonnegative after the stated variance normalization. It therefore raises the rejection probability of an upper-tail test for mean departures of either sign. Its multiplier must be calculated from the chosen normalization; an unnormalized quadratic sum need not have variance one.
At identical \(R_n,b_n\), the mean statistic’s detection scale is smaller. The paper’s comparison allowing better local power for the quadratic procedure uses different admissible repetition counts at matched size accuracy. Omitting that matching would make the comparison appear contradictory.
7. Reading the tuning restrictions as a budget
Write \(L_{2,n}=\overline\lambda_n(2)+n^{-1}\) and
The displayed restrictions become \(R_nb_nL_{2,n}\to0\) and \(R_nb_n^2L_{4,n}\to0\). If one informally substitutes \(b_n=n^\beta\), \(R_n=n^\gamma\), and \(L_{2,n}\asymp n^{-a_2}\), the mean condition is
Likewise, \(L_{4,n}\asymp n^{-a_4}\) yields \(\gamma+2\beta<a_4\). Increasing either tuning exponent uses some of the room allowed by the decay of dependence. These are drift restrictions only; the conditional CLT and higher-order size approximation add further conditions.
Even if the original observations are independent, diagonal and repeated-index terms produce the explicit powers of \(n^{-1}\) and \(n^{-2}\). In that benchmark, the two displayed restrictions reduce to \(R_nb_n/n\to0\) and \(R_nb_n^2/n^2\to0\). Treating every dependence coefficient as zero and then allowing arbitrarily many repetitions would overlook these finite-sample contributions.
The size restriction and the local-power scale can be compared without choosing numerical tuning constants. In the independent-data benchmark, write the mean restriction as \(R_nb_n=n\epsilon_n\), with \(\epsilon_n\to0\). Then its detectable mean scale is
For the quadratic restriction, write \(\sqrt{R_n}b_n=n\delta_n\), with \(\delta_n\to0\). Its detectable scale is
Both are larger than \(n^{-1/2}\) by a diverging factor under these sufficient drift restrictions. This explains the price of making the unknown original-sample error negligible relative to the artificial noise, even when the data happen to be independent. A conventional efficient mean test can use root-\(n\) alternatives when its sampling variance is estimable.
These identities do not determine which randomized statistic is better: that requires relating \(\epsilon_n\) and \(\delta_n\) to the same desired size error, including the higher-order terms. They do prevent two misleading comparisons. Holding \(R_n,b_n\) equal favors the mean statistic’s linear signal; separately maximizing each repetition count without comparing size accuracy can favor a poorly calibrated procedure. The paper’s refinement compares the two after controlling that calibration cost.
Under a local quadratic shift with limiting drift \(\kappa(h)\geq0\), the upper-tail power is \(1-\Phi(z_{1-\alpha}-\kappa(h))\). Under the mean statistic it is the upper tail of a noncentral chi-square distribution. The two shapes reflect linear versus squared accumulation of the same underlying mean discrepancy, rather than different null hypotheses.
8. Why centered resampling can estimate a critical value
Subtracting the sample mean removes empirical drift from each artificial draw used for the reference distribution. A conditional CLT gives a limiting law. Uniform convergence of its cdf, together with continuity and a regular critical quantile, then justifies convergence of the simulated critical value.
The statistic being tested still contains the original sample-mean error. The size proof must show that this drift is negligible relative to the centered reference law. Centering the simulations establishes only the reference side of that comparison.
Repeating the whole randomized test produces a conditional acceptance probability. It does not authorize choosing the most favorable seed or averaging arbitrary p-values. A rule combining repetitions must be analyzed as a new test, as in the paper’s confidence-function construction. The moment calculation for ordering-free resampling supplies the conditional means and variances that make these distinctions explicit.