= 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 $\overline T^*/\overline n^*$. 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 approximately
$$
E_*\!\left(\frac{\overline T^*}{\overline n^*}\right)-r
\approx\frac{r v_n-c_{Tn}}{K\overline n^2}.
$$
The 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 $r=18/7\approx2.57143$, $\overline n=7/3$, $v_n=14/9$, 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.<Bootstrap sampling and estimation in the Stata documentation>[Printed pp. 3–5, cluster() and variable replicate size]{ref}
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