Bootstrap ratio bias with unequal cluster sizes

Bootstrap ratio bias with unequal cluster sizes arises because the pooled mean in a cluster bootstrap is a ratio with a random denominator. Let the empirical distribution over the $K$ observed groups assign mass $1/K$ to each pair $(n_j,T_j)$ of group size and response sum. One bootstrap replicate draws $K$ such pairs and computes ${\overline{T}}^{\ast }/{\overline{n}}^{\ast }$. Its conditional expectation need not equal $r=\overline{T}/\overline{n}$, even though the numerator and denominator averages are individually unbiased for their empirical means.

A second-order Taylor expansion of $f(t,n)=t/n$ explains the leading effect. Write $v_n$ and $c_{Tn}$ for the empirical variance of one group size and empirical covariance of one group size with its response sum, using denominator $K$. The conditional bootstrap bias is approximatelyThe second derivatives are $f_{nn}=2\overline{T}/{\overline{n}}^3$ and $f_{tn}=-1/{\overline{n}}^2$; the variances/covariance of the averages bring in $1/K$. This delta method approximation requires a denominator sufficiently concentrated away from zero and control of the remainder. It is not an exact identity for a few highly unequal groups.

For an original three-group example, take $(n,T)=(1,0),(2,2),(4,16)$. Then $r=18/7\approx 2.57143$, $\overline{n}=7/3$, $v_n=14/9$, and $c_{Tn}=26/3$. The second-order bias estimate is $-2/7\approx -0.28571$. Enumerating all $3^3=27$ ordered bootstrap draws instead gives mean about $2.24797$, hence exact conditional bias about $-0.32346$. The approximation identifies the direction and mechanism but misses a visible remainder at this small group count.

Equal group sizes remove the random denominator and this particular ratio bias. If every group has the same response mean, $T_j=r{n}_j$, every replicate ratio is exactly $r$ regardless of size. Neither case justifies replacing a row-weighted estimand with an equally weighted group mean in general; see unequal-cluster weighting in a cluster bootstrap. The primary Stata specification samples entire clusters and allows replicate row counts to vary.^[ref]