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## SUPPLEMENT TO: “Inference for Dependent Data with Learned Clusters”

Authors: Jianfei Cao, Christian Hansen, Damian Kozbur, Lucciano Villacorta

This supplement contains proofs as well as additional simulation results.

###  S.1 Proofs of Theorems 1–7

Before proving Theorems 1–7, the basic structural results about sequences of
uniformly Ahlfors spaces are gathered. The first preliminary result is due to
Assoud, (1977) and ensures the existence of certain embedding of metric spaces
of bounded doubling into Euclidean spaces.

######  Theorem 8 (Assoud).

Let $(\mathsf{X},\mathrm{d})$ be an arbitrary (not necessarily finite) metric
space with finite doubling dimension
$\text{dim}_{2\times}(\mathsf{X})<\infty$. Let $\varepsilon\in(0,1)$. Then
there exists an $L$-bi-Lipschitz map
$(\mathsf{X},\mathrm{d}^{\nu})\rightarrow\mathbb{R}^{r}$ for some $L,\nu$
which depend only on $\varepsilon$ and $\text{dim}_{2\times}(\mathsf{X})$.

Ahlfors regularity of $\mathsf{X}$ implies constant doubling as a result of
standard arguments involving covering numbers of metric spaces. Thus the next
proposition follows.

######  Proposition 1.

Suppose that $\mathsf{X}\in\mathbf{Ahlf}_{C,\delta}$. Then the doubling
dimension of $\mathsf{X}$ is bounded by
$\text{dim}_{2\times}(\mathsf{X})\leqslant\delta\log_{2}(3C^{2})$. Therefore,
there exists an $L$-bi-Lipschitz map
$(\mathsf{X},\mathrm{d}^{3/4})\rightarrow\mathbb{R}^{\nu}$ where $\nu$ and the
Lipschitz constant $L$ depend only on $C,\delta$.

####  S.1.1 Proof of Theorem 1

###### Proof.

In the case that $\mathsf{X}_{n}$ do not embed isometrically, let $L$ be the
bi-Lipschitz constant from the maps
$(\mathsf{X}_{n},\mathrm{d}_{n}^{1-{1/4}})\rightarrow\tilde{\mathsf{X}}_{n}\subseteq\mathbb{R}^{\nu}$.
The array $\\{\\{\zeta_{i}\\}_{i\in\mathsf{X}_{n}}\\}_{n=1}^{\infty}$ indexed
on $\mathsf{X}_{n}$ yields an array
$\\{\\{\tilde{\zeta}_{i}\\}_{i\in\tilde{\mathsf{X}}_{n}}\\}_{n=1}^{\infty}$
indexed on $\tilde{\mathsf{X}}_{n}$. It is sufficient to check the conditions
of Corollary 1 in Jenish and Prucha, (2009) for this new process. Apply the
same set array of constants $c_{i}$ to $\tilde{\zeta}_{i}$. Assumption 1 in
Jenish and Prucha, (2009) is satisfied by the fact that distances are at least
$\rho_{0}$ in $\mathsf{X}_{n}$ for some $\rho_{0}>0$ by Ahlfors regularity.
Note that $L$ depends only on $C,\delta$ and in particular, does not change
with $n$. Then $\forall i,j,$ $\tilde{\mathrm{d}}_{n}(i,j)$ is also bounded
away from zero by a constant which does not depend on $n$. Condition 2(ii) is
identical to Equation 3 in Jenish and Prucha, (2009).

The next conditions in Jenish and Prucha, (2009) are mixing conditions. To
verify these, let $\tilde{\alpha}^{\mathrm{mix}}_{g,l,n}(r)$ and
$\bar{\tilde{\alpha}}^{\mathrm{mix}}_{g,l,n}(r)$ denote the corresponding
mixing coefficients for $\tilde{\zeta}_{i}$ over $\tilde{\mathsf{X}}_{n}$.
Note that $\tilde{\mathrm{d}}(\mathsf{U},\mathsf{V})\geqslant
r\Rightarrow\mathrm{d}(\mathsf{U},\mathsf{V})\geqslant
L^{-1}\nu^{-1/2}r^{\frac{3}{4}}$. Let $c=L^{-1}\nu^{-1/2}$. Then
$\bar{\tilde{\alpha}}^{\mathrm{mix}}_{g,l}(r)\leqslant\bar{\alpha}^{\mathrm{mix}}_{g,l}(cr^{3/4})$.
To verify Equation 4 in Jenish and Prucha, (2009), it is sufficient to show
that
$\sum_{m=1}^{\infty}\bar{\tilde{\alpha}}^{\mathrm{mix}}_{1,1}(m)m^{\nu\times\frac{\mu+2}{\mu}-1}<\infty.$
Note that $\bar{\tilde{\alpha}}^{\mathrm{mix}}_{1,1}(m)$ is nonincreasing and
defined for nonnegative real $m$ and $m^{\nu\times\frac{\mu+2}{\mu}-1}$ is a
polynomial in $m$. Thus, the above summation is bounded up to a constant by
the corresponding integral
$\int_{m=0}^{\infty}\bar{\tilde{\alpha}}^{\mathrm{mix}}_{1,1}(m)m^{\nu\times\frac{\mu+2}{\mu}-1}dm$.
Substituting
$\bar{\tilde{\alpha}}^{\mathrm{mix}}_{1,1}(m)\leqslant\bar{\alpha}^{\mathrm{mix}}_{1,1}(cm^{3/4})$
and a standard calculus change of variables $m^{\prime}=cm^{3/4}$,
$dm^{\prime}=\frac{3}{4}cm^{-1/4}dm$ shows that it is sufficient to verify
$\int_{m^{\prime}=0}^{\infty}\bar{\alpha}^{\mathrm{mix}}_{1,1}(m^{\prime}){m^{\prime}}^{\frac{4}{3}(\nu\times\frac{\mu+2}{\mu}-1)}{m^{\prime}}^{1/3}dm^{\prime}<\infty.$
This integral is in turn bounded by a constant times the summation

$\sum_{m^{\prime}=1}^{\infty}\bar{\alpha}^{\mathrm{mix}}_{1,1}(m^{\prime}){m^{\prime}}^{\frac{4}{3}\nu\times\frac{\mu+2}{\mu}-1}=\sum_{m^{\prime}=1}^{\infty}\bar{\alpha}_{1,1}^{\mathrm{mix}}(m^{\prime}){m^{\prime}}^{\nu\times\frac{\mu+2}{\mu}-1}<\infty$

which is assumed to be finite under Condition 2(iii), thus verifying Equation
4 in Jenish and Prucha, (2009). Using similar arguments, Condition 2(iv)
implies Assumption 4(2) in Jenish and Prucha, (2009). Next, Condition 2(v)
implies that

$\bar{\tilde{\alpha}}^{\mathrm{mix}}_{1,\infty}(m)\leqslant\bar{\alpha}^{\mathrm{mix}}_{1,\infty}(cm^{3/4})=O((cm^{3/4})^{\frac{4}{3}(\nu-\mu)})=O(m^{\nu-\mu})$

thus verifying Assumption 4(3) in Jenish and Prucha, (2009). Finally,
Condition 2(vi) implies Assumption 5 in Jenish and Prucha, (2009). This
verifies the assumptions of Corollary 1 in Jenish and Prucha, (2009).

∎

####  S.1.2 Proof of Theorem 2

###### Proof.

Theorem 2 is proven in the case $|\mathcal{C}|=2$. In this case for
$\mathsf{C}\subseteq\mathsf{X}_{n}$, take
$\mathsf{D}=\mathsf{X}_{n}\setminus\mathsf{C}$. The general case follows by
applying the same arguments, and by retracing boundaries when necessary. By
arguments in Bester et al., (2011) which appeared previously in Jenish and
Prucha, (2009) and Bolthausen, (1982),
$\sigma({\mathsf{C}})^{-1}\sigma({\mathsf{D}})^{-1}n=O(1)$ and

$\Big{|}\text{cov}\Big{(}\sigma({\mathsf{C}})^{-1}\sum_{i\in\mathsf{C}}\zeta_{j},\sigma({\mathsf{D}})^{-1}\sum_{j\in\mathsf{D}}\zeta_{j}\Big{)}\Big{|}\leqslant\sigma({\mathsf{C}})^{-1}\sigma({\mathsf{D}})^{-1}\sum_{(i,j)\in\mathsf{C}\times\mathsf{D}}\bar{\alpha}_{1,1}(\lceil\mathrm{d}_{n}(i,j)\rceil)^{\mu/(2+\mu)}$
$=\sigma({\mathsf{C}})^{-1}\sigma({\mathsf{D}})^{-1}\sum_{g=1}^{\infty}|\\{(i,j)\in\mathsf{C}\times\mathsf{D}:g-1\leqslant\mathrm{d}_{n}(i,j)<g\\}|\bar{\alpha}_{1,1}(g)^{\mu/(2+\mu)}.$

Next, to proceed in the current setting, note that by Condition 1,

$|\\{(i,j)\in\mathsf{C}\times\mathsf{D}:g-1\leqslant\mathrm{d}_{n}(i,j)<g\\}|\leqslant|\\{(i,j)\in\mathsf{X}\times\mathsf{X}:g-1\leqslant\mathrm{d}_{n}(i,j)<g\\}|\leqslant
nCg^{\delta}.$

By Condition 3, it also follows that

$\max_{g\leqslant
r}|\\{(i,j)\in\mathsf{C}\times\mathsf{D}:g-1\leqslant\mathrm{d}_{n}(i,j)<g\\}|\leqslant
o(n).$

Then the original covariance is bounded by

$\sigma({\mathsf{C}})^{-1}\sigma({\mathsf{D}})^{-1}\left[\sum_{g=1}^{r}o(n)\bar{\alpha}_{1,1}(g)+\sum_{m=r}^{\infty}nCg^{\delta}\bar{\alpha}_{1,1}(g)\right]$

The first term satisfies
$\sigma({\mathsf{C}})^{-1}\sigma_{\mathsf{D}}^{-1}\sum_{g=1}^{r}o(n)\bar{\alpha}_{1,1}(g)\rightarrow
0$ by
$\sum_{g=1}^{r}o(n)\bar{\alpha}_{1,1}(g)\leqslant\sum_{g=1}^{\infty}o(n)\bar{\alpha}_{1,1}(g)<\infty$
and $\sigma_{\mathsf{C}}^{-1}\sigma({\mathsf{D}})^{-1}o(n)\rightarrow 0$. The
second term satisfies
$\sigma_{\mathsf{C}}^{-1}\sigma_{\mathsf{D}}^{-1}\sum_{m=r}^{\infty}nCg^{\delta}\bar{\alpha}_{1,1}(g)\rightarrow
0$ by $n\sigma({\mathsf{C}})^{-1}\sigma({\mathsf{D}})^{-1}=O(1)$,
$\sum_{m=1}^{\infty}nCg^{\delta}\bar{\alpha}_{1,1}(g)<\infty$ and
$r\rightarrow\infty\implies\sum_{m=r}^{\infty}Cg^{\delta}\bar{\alpha}_{1,1}(g)\rightarrow
0$. The proposition follows.

∎

####  S.1.3 Proof of Theorem 3

###### Proof.

Let $\bar{r}_{n}=\log n$. From this point on in this proof, $n$ is excluded
from notation when used as an index in a subscript and when it provides no
added clarity. Consider two arbitrary (sequences of) points
$i,j\in\mathsf{X}$. Let
$M=\\{l:|\mathrm{d}(i,l)-\mathrm{d}(j,l)|\leqslant\bar{r}\\}.$ Let $M_{0}$ be
an $\bar{r}^{2}$-net of $M$. Note that by construction, if $|M|\neq o(n)$ then
$|M_{0}|\neq o(n/\bar{r}^{2\delta})$. Let $A\subseteq[\frac{1}{2},\hskip
1.42262pt\frac{9}{10}]$ satisfy
$|a-a^{\prime}|\geqslant\bar{r}^{3}/\mathrm{d}(i,j)$ for each $a,a^{\prime}\in
A$. Take $|A|\geqslant\frac{4}{10}\frac{\mathrm{d}(i,j)}{2}/\bar{r}^{3}$. For
$a\in A,$ let $M_{a}$ consist of interpolants $l_{a}$ such that
$|\mathrm{d}(i,l_{a})-a\mathrm{d}(i,l)|\leqslant K$ and
$|\mathrm{d}(l_{a},l)-(1-a)\mathrm{d}(i,l)|\leqslant K$ for each $l\in M_{0}$.
Then by trigonometry, $\cup_{a\in A}M_{a}$ is $3\bar{r}$-separated for $n$
sufficiently large and contains $|A|\times|M_{0}|$ elements. This shown by
constructing the line segments $\overline{\iota(i)\iota(l)}$ and
$\overline{\iota(i)\iota(l^{\prime})}$ where $\iota$ is the coarse isometry to
Euclidean space $\mathsf{E}$, which gives points $u,u^{\prime}$ belonging to
the above constructed line segments with distances
$d_{\mathsf{E}}(\iota(l_{a}),u)$,
$d_{\mathsf{E}}(\iota(l_{a}^{\prime}),u^{\prime})$ bounded, implying that
$u,u^{\prime}$ are sufficiently separated to yield the claim. By
$3\bar{r}$-separation and $|\textsf{B}_{\bar{r}}(l_{a})|\geqslant
C^{-1}\bar{r}^{\delta}$, it follows that

$\left|\bigcup_{a\in A,l_{a}\in
M_{a}}\textsf{B}_{\bar{r}}(l_{a})\right|\geqslant|A|\times|M_{0}|C^{-1}\bar{r}^{\delta}\geqslant\frac{4}{20}\frac{\mathrm{d}(i,j)}{\bar{r}^{3}}|M_{0}|C^{-1}\bar{r}^{\delta}.$

As the left hand side above is $\leqslant n$, it follows that the two
statements (1.) $d(i,j)\geqslant\bar{r}^{4+\delta}$ for $n$ sufficiently
large, and (2.) $|M_{0}|\neq o(n/{\bar{r}}^{2\delta})$ cannot be true
simultaneously.

Next, it is shown that $k$-medoids terminates with medoids $i_{1},...,i_{k}$
such that $\mathrm{d}(i_{g},i_{l})\geqslant(\log n)^{4+\delta}$ for $n$
sufficiently large. This implies the small boundaries statement of Theorem 3,
as a violation of this statement would require that $M$ corresponding to some
pair $i_{g},i_{l}$ satisfies $M\neq o(n)$, subsequently implying $M_{0}\neq
o(n/\bar{r}^{2\delta})$, violating the above dicotomy.

Again for contradiction, suppose there is a sequence $\ell\rightarrow 0$ such
that for infinitely many $n$, there are two clusters
$\mathsf{C}_{1},\mathsf{C}_{2}$ with medoids $i_{1},i_{2}$ satisfying
$\mathrm{d}(i_{1},i_{2})<\ell n^{1/\delta}$. By the pigeonhole principal,
there must be a cluster $\mathsf{C}_{3}$ with $n/k$ members. Then
$diam(\mathsf{C}_{3})$ must be at least $C^{-1}(n/k)^{1/\delta}$. Let $x_{3}$
be the corresponding medoid. Then there must be
$i_{3}^{\prime}\in\mathsf{C}_{3}$ such that
$\mathrm{d}(i_{3},i_{3}^{\prime})\geqslant\frac{1}{4}C^{-1}(n/k)^{1/\delta}$
and
$\mathrm{d}(i_{3}^{\prime},i_{g})\geqslant\frac{1}{4}C^{-1}(n/k)^{1/\delta}$
for any other medoid $i_{g}$. Then consider the update in the partitioned
medoid algorithm given by $i_{2}\leftarrow i_{3}^{\prime}$. This update is
cost reducing for $n$ sufficiently large. To see this, note that for elements,
$i\in\textsf{B}_{\frac{1}{4}C^{-1}(n/k)^{1/\delta}}(i_{3}^{\prime})$ the total
cost reduction from being reassigned from a medoid centered around $i_{2}$ to
a medoid centered around $i_{3}^{\prime}$ is at least
$|\textsf{B}_{\frac{1}{4}C^{-1}(n/k)^{1/\delta}}(i_{3}^{\prime})|{\frac{1}{4}C^{-1}(n/k)^{1/\delta}}\geqslant{\frac{1}{4}C^{-1}(n/k)^{1/\delta}}C^{-1}({\frac{1}{4}C^{-1}(n/k)^{1/\delta}})^{\delta}$
. The total cost increase from reassigning elements in $\mathsf{C}_{2}$ to
$\mathsf{C}_{1}$ is at most $\ell n^{1/\delta}|\mathsf{C}_{2}|\leqslant\ell
n^{1/\delta}n$. The difference between the above two quantities is a lower
bound on the cost reduction for the update. Comparing the above to quantities
for $n$ sufficiently large, the $k$-medoids algorithm could not have stopped
at a step with $\mathrm{d}(i_{1},i_{2})<\ell n^{1/\delta}$ giving the desired
contradiction.

Finally, note that $\mathrm{d}(i_{g},i_{l})\geqslant\ell n^{1/\delta}$ for all
medoids $i_{g},i_{l}$, some $\ell$ bounded uniformly away from $0$, and for
$n$ sufficiently large implies the balanced clusters condition after applying
Ahlfors regularity. ∎

####  S.1.4 Proof of Theorem 4

###### Proof.

Fix $k\leqslant k_{\max}$ and consider the subsequence of
$\mathcal{C}_{\shortrightarrow}$ for which $|\mathcal{C}_{n}|=k$. Let $V_{k}$
be a $k$-dimensional real vector space with distinguished basis indexed by
$\mathcal{C}_{\infty,k}$. Choose bijections
$\mathcal{C}_{n}\rightarrow\mathcal{C}_{\infty,k}$ and induce linear maps
$\ell_{n}:\mathbb{R}^{\mathcal{C}_{n}}\rightarrow V\rightarrow$ such that the
image of $S_{n}$ in the direction of any vector corresponding to an element of
$\mathcal{C}_{\infty}$ has variance 1. Applying Theorems 1 and 2 to $S_{n}$,
the Cramèr-Wold device to $\ell(S_{n})$ and the Almost-Sure Representation
theorem (Theorem 2.19 in van der Vaart, (1998)) then gives the
$\breve{S}^{*}_{\infty,k}$ and $\breve{S}_{\mathcal{C}_{n}}$ such that
$\|\breve{S}^{*}_{\infty,k}\|_{2}\rightarrow 0$, $\tilde{\mathrm{P}}_{k}$\-
a.s. for some $\tilde{\mathrm{P}}_{k}$ for all $n$ in the subsequence. Set for
$n$ in the subsequence
$\tilde{S}_{\mathcal{C}_{n}}^{*}=\ell_{n}^{-1}\tilde{S}^{*}_{\infty,k}$ and
$\tilde{S}_{\mathcal{C}_{n}}=\ell_{n}^{-1}\breve{S}_{\mathcal{C}_{n}}$. Repeat
this construction for all $k$ and take all $\tilde{\mathrm{P}}_{k}$ to be a
common $\tilde{\mathrm{P}}$. ∎

####  S.1.5 Proof of Theorem 5

###### Proof.

The proof follows directly from the definition of
$\widehat{\theta}_{\mathsf{C}}$ along with standard arguments and the
application of Theorems 1–4. ∎

###  S.2 Proof of Theorem 6

###### Proof.

Define the function $\psi$ by
$\psi(S)=\textbf{1}_{|t(S)|>t_{1-\alpha/2,k-1}}$, where $S$ has $k$
components. Let $\tilde{\mathrm{E}}$ be expectation with respect to the
measure $\tilde{\mathrm{P}}$. Note that for a partition $\mathcal{C}$,
$T_{\mathrm{IM}(\alpha),\mathcal{C}}=\psi(S_{\mathcal{C}})$. By construction
of $\tilde{S}_{\mathcal{C}}^{*}$ and Theorem 1 of Ibragimov and Müller,
(2010), for any $n$ and any $\mathcal{C}\in\mathscr{C}_{n}$,
$\tilde{\mathrm{E}}[\psi(\tilde{S}_{\mathcal{C}}^{*})]\leq\alpha.$

Next, show
$\sup_{\mathcal{C}\in\mathscr{C}_{n}}|\tilde{\mathrm{E}}[\psi(\tilde{S}_{\mathcal{C}})]-\tilde{\mathrm{E}}[\psi(\tilde{S}_{\mathcal{C}}^{*})]|\rightarrow
0.$ Define the events
$\mathcal{E}_{1,\mathcal{C}}=\\{|t(\tilde{S}^{*}_{\mathcal{C}})-t_{1-\alpha/2,|\mathcal{C}|-1}|>b_{\mathcal{C}}\\},\
\
\mathcal{E}_{2,\mathcal{C}}=\\{|\mathrm{sd}(\tilde{S}^{*}_{\mathcal{C}})|>b_{\mathcal{C}}^{\prime}\\}$
for $b_{\mathcal{C}},b_{\mathcal{C}}^{\prime}\in\mathbb{R}$ specified below.
Note that by the upper and lower bounds on the variances of
$\tilde{S}^{*}_{\mathcal{C}}$ in Condition 7, and continuity properties of
$t(\cdot)$ and $\mathrm{sd}(\cdot)$ one can select a common positive function,
denoted $M$ which is independent of $\mathcal{C}$, such that
$\lim_{b\rightarrow 0}M(b,b^{\prime})=\lim_{b^{\prime}\rightarrow
0}M(b,b^{\prime})=0$ and such that
$\tilde{\mathrm{P}}(\mathcal{E}_{1,\mathcal{C}}\cap\mathcal{E}_{2,\mathcal{C}})\geqslant
1-M(b_{\mathcal{C}},b_{\mathcal{C}}^{\prime})$ whenever
$b_{\mathcal{C}},b_{\mathcal{C}}^{\prime}$ are sufficiently small. Then on
$\mathcal{E}_{1,\mathcal{C}}$, a sufficient condition for
$\psi(\tilde{S}_{\mathcal{C}})=\psi(\tilde{S}^{*}_{\mathcal{C}})$ is that
$|t(\tilde{S}_{\mathcal{C}})-t(\tilde{S}^{*}_{\mathcal{C}})|<b_{\mathcal{C}}.$
By
$\sup_{\mathcal{C}\in\mathscr{C}_{n}}\|\tilde{S}^{*}_{\mathcal{C}}-\tilde{S}_{\mathcal{C}}\|_{2}\rightarrow
0$, and by continuity properties of $t(\cdot)$ outside of
$\mathcal{E}_{2,\mathcal{C}}$, there are $b_{\mathcal{C}}<b_{n}\rightarrow
0,b_{\mathcal{C}}^{\prime}<b_{n}^{\prime}\rightarrow 0$ such that
$|t(\tilde{S}_{\mathcal{C}})-t(\tilde{S}^{*}_{\mathcal{C}})|<b_{\mathcal{C}}$
is satisfied on $\mathcal{E}_{1,\mathcal{C}}\cap\mathcal{E}_{2,\mathcal{C}}$.
Therefore,
$|\tilde{\mathrm{E}}[\psi(\tilde{S}_{\mathcal{C}})]-\tilde{\mathrm{E}}[\psi(\tilde{S}_{\mathcal{C}}^{*})]|\leqslant
M(b_{n},b_{n}^{\prime})$, for all $\mathcal{C}\in\mathscr{C}_{n}$ from which
the above desired uniform limit result follows.

Finally, ${\mathrm{P}}_{0}(\widehat{T}_{\text{IM}(\alpha),n}=\text{ Reject
})={\mathrm{P}}_{0}(T_{\text{IM}(\widehat{\alpha}),\widehat{\mathcal{C}}}=\text{
Reject
})\leqslant{\mathrm{P}}_{0}(T_{\text{IM}(\alpha),\widehat{\mathcal{C}}}=\text{
Reject })$
$=\sum_{\mathcal{C}\in\mathscr{C}_{n}}{\mathrm{P}}_{0}(T_{\text{IM}(\alpha),{\mathcal{C}}}=\text{
Reject
}|\widehat{\mathcal{C}}=\mathcal{C}){\mathrm{P}}_{0}(\widehat{\mathcal{C}}={\mathcal{C}}).$
Bound the above expression by
$o(1)+\sum_{\mathcal{C}\in\mathscr{C}_{0,n}}{\mathrm{P}}_{0}(T_{\text{IM}(\alpha),{\mathcal{C}}}=\text{
Reject }){\mathrm{P}}_{0}(\widehat{\mathcal{C}}={\mathcal{C}})\leqslant
o(1)+\alpha$ by using both Condition 6(i) and Condition 6(ii) and the previous
bounds. This concludes the proof of Theorem 1. ∎

###  S.3 Proof of Theorem 7

Before giving the proof, note that the following more general condition is
used in place of Condition 7(iii) in the main text.

######  Condition  7.

(iii)’ Either

  * a.

$|\mathscr{C}_{n}|$ and $\max_{\mathcal{C}\in\mathscr{C}_{n}}|\mathcal{C}|$
are each bounded by a constant independent of $n$; or

  * b.

For any sequence $\check{k}_{n}\rightarrow\infty$ sufficiently slowly, the
following hold. There is a sequence $\check{\delta}_{n}\rightarrow 0$ and sets
$\mathcal{A}_{\mathcal{C}}\subseteq\mathbb{R}^{\mathcal{C}}$ for each
$\mathcal{C}\in\mathscr{C}_{n}$ with $|\mathcal{C}|>\check{k}_{n}$ which are
closed under the action of $\mathcal{H}_{\mathcal{C}}$ such that,
$\lim_{n\rightarrow\infty}\inf_{\mathcal{C}\in\mathscr{C}_{n},|\mathcal{C}|>\check{k}}\tilde{\mathrm{P}}(\mathcal{A}_{\mathcal{C}})=1$,
$w(\cdot)$ can be renormalized to have Lipschitz constant 1 respect to the
Euclidean norm on all $\mathcal{A}_{\mathcal{C}}$, and it holds that
$\sup_{\mathcal{C}\in\mathscr{C}_{n}}\sup_{r\in\mathbb{R}}{\mathrm{P}}_{\mathcal{C}}(|r-w(h\tilde{S}^{*}_{\mathcal{C}})|<\check{\delta}_{n})\rightarrow
0$ where ${\mathrm{P}}_{\mathcal{C}}$ be the uniform probability measure over
$h\in\mathcal{H}_{\mathcal{C}}$. Finally,
$|\\{\mathcal{C}\in\mathscr{C}_{n}:|\mathcal{C}|\leqslant\check{k}_{n}\\}|$
depends only $\check{k}_{n}$.

The next proposition verifies that the test statistic, $w$, defined in the
text above and used for the empirical example and simulation study satisfy the
anticoncentration requirement of Condition 7(iii)’.

######  Proposition 2.

For $S\in\mathbb{R}^{\mathcal{C}}$, let $w(S)=|t(S)|$. Let
$S^{*}_{\mathcal{C}}$ be a Gaussian random variables with independent
components with variances bounded from below and above (as in Condition 7(i)).
Suppose that
$\mathscr{C}_{n}=\\{\mathcal{C}^{(2)},...,\mathcal{C}^{(k_{\mathrm{max},n})}\\}$
where $k_{\mathrm{max},n}$ may be an arbitrary function of $n$. Then Condition
7(iii)’a is satisfied if $k_{\mathrm{max},n}$ is bounded and Condition
7(iii)’b is satisfied if $k_{\mathrm{max},n}\rightarrow\infty$.

The proof below is given assuming Condition 7(iii)’.

###### Proof.

Define functions $\phi(S)$ and $\tilde{\phi}(S,U)$ for
$S\in\mathbb{R}^{\mathcal{C}}$ and $U\in\mathbb{R}$ according to

$\phi(S)=\begin{cases}1,\text{\qquad\ if }w(S)>w^{({j_{\alpha}})}(S)\\\
\tilde{a}(S),\text{\quad if }w(S)=w^{({j_{\alpha}})}(S)\\\ 0,\text{\qquad\ if
}w(S)<w^{({j_{\alpha}})}(S)\\\ \end{cases}$

and define $\tilde{\phi}(S,U)=\phi(S)$ if $W(S)\neq W^{({j_{\alpha}})}(S)$ and
$\textbf{1}_{U<a(S)}$ otherwise.

Without loss of generality, assume that $\tilde{\Omega}$ has a uniformly
distributed random variable $\tilde{U}\in[0,1]$ independent of all previously
defined random variables defined on it. By Theorem 2.1 in Canay et al.,
(2017), using the fact that
$h(\tilde{S}^{*}_{\mathcal{C}})=_{d}\tilde{S}^{*}_{\mathcal{C}}$ for all $h$
by the fact that $h$ acts as component-wise multiplication by signs, it
follows that ${\mathrm{E}}_{0}[\phi(\tilde{S}_{\mathcal{C}}^{*})]=\alpha$.
Note,

${\mathrm{E}}_{0}[\phi({S}_{\mathcal{C}})]-\alpha=\tilde{\mathrm{E}}[\tilde{\phi}(\tilde{S}_{\mathcal{C}},{\tilde{U}})-\tilde{\phi}(\tilde{S}_{\mathcal{C}}^{*},{\tilde{U}})].$

Therefore, it is sufficient to show
$\limsup_{n\rightarrow\infty}\sup_{\mathcal{C}\in\mathscr{C}_{n}}\tilde{\mathrm{E}}[\tilde{\phi}(\tilde{S}_{\mathcal{C}},{\tilde{U}})-\tilde{\phi}(\tilde{S}_{\mathcal{C}}^{*},{\tilde{U}})]=0$.
Consider an integer $\check{k}$, and decompose
$\mathscr{C}_{n}=\mathscr{C}_{n,\check{k}}\cup\mathscr{C}_{n,\check{k}}^{c}$
where
$\mathscr{C}_{n,\check{k}}=\\{\mathcal{C}\in\mathscr{C}_{n}:|\mathcal{C}|\leqslant\check{k}\\}$
and
$\mathscr{C}_{n,\check{k}}^{c}=\\{\mathcal{C}\in\mathscr{C}_{n}:|\mathcal{C}|>\check{k}\\}$,
which are handled separately. For each
$\mathcal{C}\in\mathscr{C}_{n,\check{k}}$, let ${\mathcal{E}}_{\mathcal{C}}$
be the event in which the order statistics of
$\\{w(h\tilde{S}_{\mathcal{C}}):h\in\mathcal{H}_{\mathcal{C}}\\}$ and
$\\{w(h\tilde{S}_{\mathcal{C}}^{*}):h\in\mathcal{H}_{\mathcal{C}}\\}$
correspond to the same transformations
$h^{(1)},\dots,h^{(|\mathcal{H}_{\mathcal{C}}|)}$. Then the previous
expression can be controlled by

$\displaystyle|\tilde{\mathrm{E}}[\tilde{\phi}(\tilde{S}_{\mathcal{C}},{\tilde{U}})-\tilde{\phi}(\tilde{S}_{\mathcal{C}}^{*},{\tilde{U}})]|=$
$\displaystyle|\tilde{\mathrm{E}}[\tilde{\phi}(\tilde{S}_{\mathcal{C}},{\tilde{U}}){\mathbf{1}}_{{\mathcal{E}}_{\mathcal{C}}}+\tilde{\phi}(\tilde{S}_{\mathcal{C}},{\tilde{U}}){\mathbf{1}}_{{\mathcal{E}}_{\mathcal{C}}^{c}}-\tilde{\phi}(\tilde{S}_{\mathcal{C}}^{*},{\tilde{U}}){\mathbf{1}}_{{\mathcal{E}}_{\mathcal{C}}}-\tilde{\phi}(\tilde{S}_{\mathcal{C}}^{*},{\tilde{U}}){\mathbf{1}}_{{\mathcal{E}}_{\mathcal{C}}^{c}}]|$
$\displaystyle=$
$\displaystyle|\tilde{\mathrm{E}}[\tilde{\phi}(\tilde{S}_{\mathcal{C}},{\tilde{U}}){\mathbf{1}}_{{\mathcal{E}}_{\mathcal{C}}^{c}}-\tilde{\phi}(\tilde{S}_{\mathcal{C}}^{*},{\tilde{U}}){\mathbf{1}}_{{\mathcal{E}}_{\mathcal{C}}^{c}}]|$
$\displaystyle=$
$\displaystyle|\tilde{\mathrm{E}}[(\tilde{\phi}(\tilde{S}_{\mathcal{C}},{\tilde{U}})-\tilde{\phi}(\tilde{S}_{\mathcal{C}}^{*},{\tilde{U}})){\mathbf{1}}_{{\mathcal{E}}_{\mathcal{C}}^{c}}]|\leq
2\tilde{\mathrm{E}}[{\mathbf{1}}_{{\mathcal{E}}_{\mathcal{C}}^{c}}].$

Furthermore,
$\sup_{\mathcal{C}\in\mathscr{C}_{n,\check{k}}}\tilde{\mathrm{E}}[{\mathbf{1}}_{{\mathcal{E}}_{\mathcal{C}}^{c}}]$
can be controlled using Fatou’s Lemma by

$\limsup_{n\rightarrow\infty}\sup_{\mathcal{C}\in\mathscr{C}_{n,\check{k}}}\tilde{\mathrm{E}}[{\mathbf{1}}_{{\mathcal{E}}_{\mathcal{C}}^{c}}]\leq\limsup_{n\rightarrow\infty}\tilde{\mathrm{E}}[\sup_{\mathcal{C}\in\mathscr{C}_{n,\check{k}}}{\mathbf{1}}_{{\mathcal{E}}_{\mathcal{C}}^{c}}]\leqslant\tilde{\mathrm{E}}[\limsup_{n\rightarrow\infty}\sup_{\mathcal{C}\in\mathscr{C}_{n,\check{k}}}{\mathbf{1}}_{{\mathcal{E}}_{\mathcal{C}}^{c}}].$

To further bound the right-hand side of the above expression, consider any
fixed $\omega\in\tilde{\Omega}$ which satisfies the conditions
$\sup_{\mathcal{C}\in\mathscr{C}_{n}}\|\tilde{S}_{\mathcal{C}}(\omega)-\tilde{S}_{\mathcal{C}}^{*}(\omega)\|_{2}\rightarrow
0$ and $\inf_{\mathcal{C}\in\mathscr{C}_{n,\check{k}}}\inf_{h\nsim
h^{\prime}\in\mathcal{H}_{\mathcal{C}}}|w_{\mathcal{C}}(h(\tilde{S}_{\mathcal{C}}^{*})(\omega))-w_{\mathcal{C}}(h^{\prime}(\tilde{S}_{\mathcal{C}}^{*})(\omega))|>\delta_{\omega}>0$
for some $\delta_{\omega}>0$. Here, the equivalence $\sim$ is defined by
$h\sim h^{\prime}$ whenever $w\circ h=w\circ h^{\prime}$. By Condition 7, the
set of such $\omega\in\tilde{\Omega}$ has $\tilde{\mathrm{P}}$-measure 1.
Note, more explicitly, choosing such $\delta_{\omega}>0$ is possible due to
(1) the continuity of the function $w$ and the continuity of the action of $h$
outside a set of $\tilde{P}$-measure 0, and (2) the fact that
$|\mathscr{C}_{n,\check{k}}|$ depends only on $\check{k}$.

Let
$h^{(1)}_{\mathcal{C}}(\omega),\dots,h^{(|\mathcal{H}_{\mathcal{C}}|)}_{\mathcal{C}}(\omega)$
be such that
$w(h^{(1)}_{\mathcal{C}}(\omega)\tilde{S}_{\mathcal{C}}^{*}(\omega))\leqslant\dots\leqslant
w(h^{(|\mathcal{H}_{\mathcal{C}}|)}_{\mathcal{C}}(\omega)\tilde{S}_{\mathcal{C}}^{*}(\omega)).$
Then the expression

$\displaystyle
w(h^{(j+1)}_{n,\mathcal{C}}(\omega)\tilde{S}_{\mathcal{C}}(\omega))-w(h^{({j)}}_{n,\mathcal{C}}(\omega)\tilde{S}_{\mathcal{C}}(\omega))=$
$\displaystyle\
w(h^{(j+1)}_{n,\mathcal{C}}(\omega)\tilde{S}_{\mathcal{C}}(\omega))-w(h^{(j+1)}_{n,\mathcal{C}}(\omega)\tilde{S}_{\mathcal{C}}^{*}(\omega))$
$\displaystyle+w(h^{(j+1)}_{n,\mathcal{C}}(\omega)\tilde{S}^{*}_{n,\mathcal{C}}(\omega))-w(h^{({j)}}_{n,\mathcal{C}}(\omega)\tilde{S}^{*}_{n,\mathcal{C}}(\omega))$
$\displaystyle+w(h^{({j)}}_{n,\mathcal{C}}(\omega)\tilde{S}^{*}_{n,\mathcal{C}}(\omega))-w(h^{({j)}}_{n,\mathcal{C}}(\omega)\tilde{S}_{\mathcal{C}}(\omega))$

is nonnegative and is furthermore strictly positive for at least one $j$
unless $h\sim h^{\prime}$ for all $h,h^{\prime}\in\mathcal{H}_{\mathcal{C}}$
in which case the theorem holds trivially. Positivity of the above expression
in the case of non-equivalent-under-$\sim$ action follows from noting that the
first and the third terms are smaller than $\delta_{\omega}/2$ in absolute
value for $n$ sufficiently large, while the second term is greater than
$\delta_{\omega}$.

The claim
$\limsup_{n\rightarrow\infty}\sup_{\mathcal{C}\in\mathscr{C}_{n}}{\mathbf{1}}_{{\mathcal{E}}_{\mathcal{C}}^{c}}(\omega)=0$
$\tilde{\mathrm{P}}$-a.s. follows. Thus,
$\limsup_{n\rightarrow\infty}\sup_{\mathcal{C}\in\mathscr{C}_{n,\check{k}}}\tilde{\mathrm{E}}[{\mathbf{1}}_{{\mathcal{E}}_{\mathcal{C}}^{c}}]=0$.
Therefore, there is a sequence $\check{k}_{n}\rightarrow\infty$ sufficiently
slowly such that
$\limsup_{n\rightarrow\infty}\sup_{\mathcal{C}\in\mathscr{C}_{n,\check{k}_{n}}}\tilde{\mathrm{E}}[{\mathbf{1}}_{{\mathcal{E}}_{\mathcal{C}}^{c}}]=0$.

Next consider the partitions in the complement
$\mathcal{C}\in\mathscr{C}_{\check{k}_{n}}^{c}$. Let
$\mathcal{A}_{s}\subseteq\mathbb{R}^{\mathcal{C}}$ be of the form
$\mathcal{A}_{S}=\\{hS:h\in\mathcal{H}_{\mathcal{C}}\\}$ for some
$S\in\mathbb{R}^{\mathcal{C}}$. Then $w^{j_{\alpha}}$ is constant over
$\mathcal{A}_{S}$ and thus $w^{j_{\alpha}}(\mathcal{A}_{S})$ is well defined.
Then for any $\delta>0$,

$\tilde{\mathrm{P}}(|w^{j_{\alpha}}(\mathcal{A}_{S})-w(\tilde{S}^{*}_{\mathcal{C}})|<\delta|\tilde{S}^{*}_{\mathcal{C}}\in\mathcal{A}_{S})\leqslant\sup_{r\in\mathbb{R}}\tilde{\mathrm{P}}(|r-w(\tilde{S}^{*}_{\mathcal{C}})|<\delta|\tilde{S}^{*}_{\mathcal{C}}\in\mathcal{A}_{S}).$

Let ${\mathrm{P}}_{\mathcal{C}}$ be the uniform probability over
$h\in\mathcal{H}_{\mathcal{C}}$. Then the above expression is bounded by

$p_{\mathcal{C},S}(\delta):=\sup_{r\in\mathbb{R}}{\mathrm{P}}_{\mathcal{C}}(|r-w(h{S})|<\delta).$

Next, for any $\mathcal{A}_{\mathcal{C}}$ which is a disjoint union of sets of
the form $\mathcal{A}_{S}$. Then

$\tilde{\mathrm{P}}(|w^{j_{\alpha}}(\tilde{S}^{*}_{\mathcal{C}})-w(\tilde{S}^{*}_{\mathcal{C}})|\geqslant\delta)\geqslant\tilde{\mathrm{P}}(|w^{j_{\alpha}}(\tilde{S}^{*}_{\mathcal{C}})-w(\tilde{S}^{*}_{\mathcal{C}})|\geqslant\delta|\tilde{S}^{*}_{\mathcal{C}}\in\mathcal{A}_{\mathcal{C}})\tilde{\mathrm{P}}(\mathcal{A}_{\mathcal{C}})$
$\geqslant(1-\sup
p_{\mathcal{C},S}(\delta))\tilde{\mathrm{P}}(\mathcal{A}_{\mathcal{C}})$

where the supremum runs over $S$ indexing
$\mathcal{A}_{S}\subseteq\mathcal{A}_{\mathcal{C}}$.

Note that a sufficient condition for
$\tilde{\phi}(\tilde{S}^{*}_{\mathcal{C}},\tilde{U})=\tilde{\phi}(\tilde{S}_{\mathcal{C}},\tilde{U})$
is that
$\mathrm{sign}(w^{j_{\alpha}}(\tilde{S}^{*}_{\mathcal{C}})-w(\tilde{S}^{*}_{\mathcal{C}}))=\mathrm{sign}(w^{j_{\alpha}}(\tilde{S}_{\mathcal{C}})-w(\tilde{S}_{\mathcal{C}}))$.
A further sufficient condition for this is that for some $\delta>0$,
$|w^{j_{\alpha}}(\tilde{S}^{*}_{\mathcal{C}})-w(\tilde{S}^{*}_{\mathcal{C}})|\geqslant\delta$,
$|w^{j_{\alpha}}(\tilde{S}^{*}_{\mathcal{C}})-w^{j_{\alpha}}(\tilde{S}_{\mathcal{C}})|<\delta/2$,
and $|w(\tilde{S}^{*}_{\mathcal{C}})-w(\tilde{S}_{\mathcal{C}})|<\delta/2$. By
Condition 7(iii)’, $w$ has Lipschitz constant 1 with respect to Euclidean norm
on $\mathbb{R}^{\mathcal{C}}$ on a suitably chosen sequence of events of
probability approaching 1. Then
$|w^{j_{\alpha}}(\tilde{S}^{*}_{\mathcal{C}})-w^{j_{\alpha}}(\tilde{S}_{\mathcal{C}})|<\delta/2$
whenever
$\|\tilde{S}^{*}_{\mathcal{C}}-\tilde{S}_{\mathcal{C}}\|_{2}<\delta/2$ in
which case it also holds that
$|w(\tilde{S}^{*}_{\mathcal{C}})-w(\tilde{S}_{\mathcal{C}})|<\delta/2$. A
sequence $\check{\delta}_{n}$ may be chosen such that
$\check{\delta}_{n}\rightarrow 0$ sufficiently slowly such that for $n$
sufficiently large, each of the above three inequalities may be achieved with
common bounds on an event with probability $1-o_{\tilde{\mathrm{P}}}(1)$.

Finally, using the same reasoning as at the conclusion of Theorem 6, and
invoking Condition 6,
${\mathrm{P}}_{0}(\widehat{T}_{\text{CRS}(\alpha),n}=\text{ Reject
})=o(1)+\sum_{\mathcal{C}\in\mathscr{C}_{0,n}}{\mathrm{P}}_{0}(T_{\text{CRS}(\alpha),{\mathcal{C}}}=\text{
Reject }){\mathrm{P}}_{0}(\widehat{\mathcal{C}}={\mathcal{C}})\leqslant
o(1)+\alpha.$ concluding the proof of Theorem 2. ∎

###  S.4 Proof of Proposition 2

###### Proof.

Fix $S\in\mathbb{R}^{\mathcal{C}}$. Let $D$ be a constant chosen later, and
consider a rescaled version of $w$ defined by
$w(S)=|\mathcal{C}|^{-1/2}|\bar{S}|/(D\mathrm{sd}(S))$. Next bound the
following anticoncentration quantity:

$\sup_{r\in\mathbb{R}}{\mathrm{P}}_{\mathcal{C}}(w(hS)\in[r,r+\delta]).$

Because $\mathcal{H}_{\mathcal{C}}$ is finite, the range of the supremum above
can be restricted and the above quantity is further reduced by

$=\sup_{r\in[\min_{h\in\mathcal{H}_{\mathcal{C}}}w(hS),\max_{h\in\mathcal{H}_{\mathcal{C}}}w(hS)]}{\mathrm{P}}_{\mathcal{C}}(w(hS)\in[r,r+\delta])$
$\leqslant\sup_{r\in[\min_{h\in\mathcal{H}_{\mathcal{C}}}w(hS),\max_{h\in\mathcal{H}_{\mathcal{C}}}w(hS)]}{\mathrm{P}}_{\mathcal{C}}(|\bar{hS}|\in[\min_{h\in\mathcal{H}_{\mathcal{C}}}aD\mathrm{sd}(hS),\max_{h\in\mathcal{H}_{\mathcal{C}}}(a+\delta)D\mathrm{sd}(hS)])$

Note that the length of the interval inside the ${\mathrm{P}}_{\mathcal{C}}$
never exceeds

$\ell=\ell(\delta,D,S):=\sup_{r\in[\min_{h\in\mathcal{H}_{\mathcal{C}}}w(hS),\max_{h\in\mathcal{H}_{\mathcal{C}}}w(hS)]}\max_{h\in\mathcal{H}_{\mathcal{C}}}(r+\delta)D\mathrm{sd}(hS)-\min_{h\in\mathcal{H}_{\mathcal{C}}}rD\mathrm{sd}(hS).$

Therefore,
$\sup_{r\in\mathbb{R}}{\mathrm{P}}_{\mathcal{C}}(w(hS)\in[r,r+\delta])\leqslant\sup_{r\in\mathbb{R}}{\mathrm{P}}_{\mathcal{C}}(|\mathcal{C}|^{-1/2}|\bar{hS}|\in[r,r+\ell]).$
By considering both branches of the absolute value function, (which gives an
additional factor of 2), and expressing the above quantity more explicitly in
terms of components $h_{\mathsf{c}}$, it follows that

$\sup_{r\in\mathbb{R}}{\mathrm{P}}_{\mathcal{C}}(|r-|\mathcal{C}|^{-1/2}|\sum_{\mathsf{C}\in\mathcal{C}}h_{\mathsf{C}}S_{\mathcal{C}})||<\ell)\leqslant
2\sup_{r\in\mathbb{R}}{\mathrm{P}}_{\mathcal{C}}(|r-|\mathcal{C}|^{-1/2}\sum_{\mathsf{C}\in\mathcal{C}}h_{\mathsf{C}}S_{\mathcal{C}})|<\ell).$

By Corollary 2.9 in Rudelson and Vershynin, (2008), which states an
anticoncentration bound for bernoulli sums, this is bounded by

$2\sup_{r\in\mathbb{R}}{\mathrm{P}}_{\mathcal{C}}(|r-{|\mathcal{C}|}^{-1/2}\sum_{\mathsf{C}\in\mathcal{C}}h_{\mathsf{C}}S_{\mathsf{C}}|<\ell)\leqslant
2\left[\sqrt{\frac{2}{\pi}}\frac{\ell}{\||\mathcal{C}|^{-1/2}S\|_{2}}+C_{1}B\left(\frac{\||\mathcal{C}|^{-1/2}S\|_{3}}{\||\mathcal{C}|^{-1/2}S\|_{2}}\right)^{3}\right]$

where $C_{1}$ is an absolute constant, and $B=1$ is the third moment of a
Bernoulli random variable.

For a sequence $\check{k}\rightarrow\infty$, Gaussian concentration properties
coupled with the assumed bounds on the variances of the components of
$\tilde{S}^{*}_{\mathcal{C}}$ imply that there is are sets
$\mathcal{A}_{\mathcal{C}}$ closed under the action of
$\mathcal{H}_{\mathcal{C}}$ with
$\tilde{\mathrm{P}}(\mathcal{A}_{\mathcal{C}})\rightarrow_{\check{k}\rightarrow\infty}1$
and a fixed choice $D$ which makes $w$ Lipschitz with constant 1 on all of
$\mathcal{A}_{\mathcal{C}}$, such that $\ell$ is bounded by a fixed constant
times $\delta$ for all $S$ in all $\mathcal{A}_{\mathcal{C}}$. Gaussian
concentration properties for
$\||\mathcal{C}|^{-1/2}\tilde{S}^{*}_{\mathcal{C}}\|_{2}$ and
$(\||\mathcal{C}|^{-1/2}\tilde{S}^{*}_{\mathcal{C}}\|_{3}/\||\mathcal{C}|^{-1/2}\tilde{S}^{*}_{\mathcal{C}}\|_{2})^{3}$
then also allow the construction of $\check{\delta}_{n}\rightarrow 0$ such
that the right hand of the above expression side also $\rightarrow 0$ provided
that $\check{k}$ grows sufficiently slowly.

∎

###  S.5 Additional Simulation Results

This section provides additional simulation results to compliment those in the
main text.

The same settings as in Section 4 are considered; though here we provide
results with number of locations given by $N_{\mathrm{pan}}=820$ in addition
to $N_{\mathrm{pan}}=205$. In the $N_{\mathrm{pan}}=820$ settings, four copies
of the locations from the empirical example are created by reflecting the
original locations over the $29^{\circ}$ latitude and $75^{\circ}$ longitude
lines. The data generating process follows Section 4. The maximal number of
groups to be considered in CCE, IM, and CRS is chosen to be
${G_{\mathrm{max}}}=12.$ In all cases, we consider 1000 simulation
replications.

The reported results in this section also display more detailed information
about the simulation studies than given in the main text in both the
$N_{\mathrm{pan}}=820$ and $N_{\mathrm{pan}}=205$ cases.

Table 6: Summary: OLS - BASELINE ($N_{\mathrm{pan}}=205$)

Method | Estim. Mean | Estim. RMSE | Size | Power
---|---|---|---|---
|  |  |  | -1 | -0.5 | 0.5 | 1
SK | 0.015 | 0.337 | 0.381 | 0.974 | 0.730 | 0.727 | 0.981
UNIT-U | 0.015 | 0.337 | 0.577 | 0.992 | 0.844 | 0.832 | 0.987
UNIT | 0.015 | 0.337 | 0.047 | 0.831 | 0.335 | 0.319 | 0.818
CCE | 0.015 | 0.337 | 0.046 | 0.717 | 0.292 | 0.248 | 0.704
IM | 0.014 | 0.213 | 0.044 | 0.979 | 0.553 | 0.523 | 0.962
CRS | 0.014 | 0.213 | 0.042 | 0.957 | 0.514 | 0.519 | 0.970

  * •

Notes: Simulation results for estimation in the design described in Section 2.
The nominal size is 0.05. Estimates are presented for the estimators, SK,
UNIT-U, UNIT, CCE, IM, CRS described in the text. Columns display method,
estimated mean, estimated RMSE, size, and power against four alternatives (-1,
-0.5, 0.5, 1).

Table 7: Summary: OLS- SAR ($N_{\mathrm{pan}}=205$)

Method | Estim. Mean | Estim. RMSE | Size | Power
---|---|---|---|---
|  |  |  | -1 | -0.5 | 0.5 | 1
SK | -0.013 | 0.866 | 0.447 | 0.688 | 0.514 | 0.517 | 0.691
UNIT-U | -0.013 | 0.866 | 0.639 | 0.761 | 0.683 | 0.681 | 0.766
UNIT | -0.013 | 0.866 | 0.359 | 0.657 | 0.461 | 0.452 | 0.656
CCE | -0.013 | 0.866 | 0.047 | 0.362 | 0.160 | 0.167 | 0.372
IM | -0.006 | 0.385 | 0.038 | 0.586 | 0.216 | 0.189 | 0.588
CRS | -0.003 | 0.354 | 0.049 | 0.674 | 0.231 | 0.256 | 0.670

  * •

Notes: Simulation results for estimation in the design described in Section 2.
The nominal size is 0.05. Estimates are presented for the estimators, SK,
UNIT-U, UNIT, CCE, IM, CRS described in the text. Columns display method,
estimated mean, estimated RMSE, size, and power against four alternatives (-1,
-0.5, 0.5, 1).

Table 8: Summary: IV - BASELINE ($N_{\mathrm{pan}}=205$)

Method | Estim. Median | Estim. MAD | Size | Power
---|---|---|---|---
|  |  |  | -1 | -0.5 | 0.5 | 1
SK | 0.002 | 0.114 | 0.362 | 1.000 | 0.999 | 0.913 | 0.993
UNIT-U | 0.002 | 0.114 | 0.568 | 0.996 | 0.950 | 1.000 | 1.000
UNIT | 0.002 | 0.114 | 0.074 | 0.954 | 0.711 | 0.910 | 1.000
CCE | 0.002 | 0.114 | 0.062 | 0.927 | 0.652 | 0.729 | 0.987
IM | -0.059 | 0.091 | 0.046 | 0.930 | 0.736 | 0.951 | 0.985
CRS | -0.061 | 0.095 | 0.040 | 0.992 | 0.992 | 0.680 | 0.921

  * •

Notes: Simulation results for estimation in the design described in Section 2.
The nominal size is 0.05. Estimates are presented for the estimators, SK,
UNIT-U, UNIT, CCE, IM, CRS described in the text. Columns display method,
estimated median, estimated MAD, size, and power against four alternatives
(-1, -0.5, 0.5, 1).

Table 9: Summary: IV - SAR ($N_{\mathrm{pan}}=205$)

Method | Estim. Median | Estim. MAD | Size | Power
---|---|---|---|---
|  |  |  | -1 | -0.5 | 0.5 | 1
SK | 0.002 | 0.280 | 0.324 | 0.898 | 0.655 | 0.627 | 0.768
UNIT-U | 0.002 | 0.280 | 0.502 | 0.797 | 0.675 | 0.798 | 0.942
UNIT | 0.002 | 0.280 | 0.256 | 0.737 | 0.604 | 0.567 | 0.839
CCE | 0.002 | 0.280 | 0.083 | 0.634 | 0.464 | 0.282 | 0.626
IM | -0.046 | 0.158 | 0.025 | 0.658 | 0.420 | 0.470 | 0.773
CRS | -0.095 | 0.213 | 0.041 | 0.778 | 0.703 | 0.331 | 0.563

  * •

Notes: Simulation results for estimation in the design described in Section 2.
The nominal size is 0.05. Estimates are presented for the estimators, SK,
UNIT-U, UNIT, CCE, IM, CRS described in the text. Columns display method,
estimated median, estimated MAD, size, and power against four alternatives
(-1, -0.5, 0.5, 1).

Table 10: Clustering: OLS - BASELINE ($N_{\mathrm{pan}}=205$)

|  | $k$ | $\widehat{k}$
---|---|---|---
|  | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
CCE | size (usual cv) | 0.085 | 0.126 | 0.154 | 0.176 | 0.192 | 0.201 | 0.222 |
| size (simulated cv) | 0.056 | 0.049 | 0.048 | 0.046 | 0.050 | 0.042 | 0.044 | 0.046
|  $\widehat{k}$ frequency | 0.000 | 0.000 | 0.001 | 0.098 | 0.311 | 0.260 | 0.330 |
| p(sim_size>.05) | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000
|  $\widehat{\alpha}$ quantile |  |  |  |  |  |  |  |
| q10 | 0.025 | 0.016 | 0.009 | 0.008 | 0.007 | 0.005 | 0.003 | 0.004
| q25 | 0.027 | 0.017 | 0.010 | 0.009 | 0.007 | 0.006 | 0.004 | 0.006
| q50 | 0.030 | 0.019 | 0.011 | 0.011 | 0.009 | 0.007 | 0.005 | 0.008
| q75 | 0.033 | 0.021 | 0.012 | 0.012 | 0.010 | 0.008 | 0.006 | 0.010
| q90 | 0.035 | 0.023 | 0.014 | 0.014 | 0.011 | 0.009 | 0.007 | 0.012
IM | size (usual cv) | 0.041 | 0.058 | 0.063 | 0.062 | 0.054 | 0.050 | 0.054 |
| size (simulated cv) | 0.040 | 0.053 | 0.059 | 0.054 | 0.047 | 0.045 | 0.042 | 0.044
|  $\widehat{k}$ frequency | 0.000 | 0.000 | 0.000 | 0.007 | 0.018 | 0.113 | 0.862 |
| p(sim_size>.05) | 0.383 | 0.752 | 0.832 | 0.909 | 0.888 | 0.627 | 0.791 | 0.762
|  $\widehat{\alpha}$ quantile |  |  |  |  |  |  |  |
| q10 | 0.044 | 0.038 | 0.037 | 0.035 | 0.035 | 0.040 | 0.037 | 0.039
| q25 | 0.048 | 0.041 | 0.040 | 0.038 | 0.039 | 0.044 | 0.040 | 0.042
| q50 | 0.050 | 0.046 | 0.044 | 0.042 | 0.043 | 0.048 | 0.044 | 0.046
| q75 | 0.050 | 0.050 | 0.048 | 0.046 | 0.047 | 0.050 | 0.049 | 0.050
| q90 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050
CRS | size (usual cv) | 0.000 | 0.000 | 0.000 | 0.000 | 0.031 | 0.040 | 0.048 |
| size (simulated cv) | 0.000 | 0.000 | 0.000 | 0.000 | 0.030 | 0.033 | 0.041 | 0.042
|  $\widehat{k}$ frequency | 0.000 | 0.000 | 0.000 | 0.000 | 0.001 | 0.099 | 0.900 |
| p(sim_size>.05) | 0.000 | 0.000 | 0.000 | 0.000 | 0.054 | 0.530 | 0.726 | 0.641
|  $\widehat{\alpha}$ quantile |  |  |  |  |  |  |  |
| q10 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.047 | 0.039 | 0.039
| q25 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.047 | 0.039 | 0.043
| q50 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.047 | 0.047 | 0.047
| q75 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050
| q90 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050

  * •

Notes: Simulation results for the design described in Section 2. Inferential
properties are presented for the estimators, CCE, IM, CRS, described in the
text. Columns under $k$ report results with the number of groups fixed at a
certain $k$. $\widehat{k}$ is the number of clusters chosen by the criterion
on size-power tradeoff described in the text. The rows “$\widehat{k}$
frequency” is the frequency of a particular $k$ achieving the highest
simulated power among candidate $k$’s in the setting.

Table 11: Clustering: OLS - SAR ($N_{\mathrm{pan}}=205$)

|  | $k$ | $\widehat{k}$
---|---|---|---
|  | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
CCE | size (usual cv) | 0.041 | 0.053 | 0.090 | 0.067 | 0.080 | 0.082 | 0.095 |
| size (simulated cv) | 0.034 | 0.035 | 0.037 | 0.037 | 0.049 | 0.040 | 0.043 | 0.047
|  $\widehat{k}$ frequency | 0.000 | 0.001 | 0.022 | 0.051 | 0.113 | 0.268 | 0.545 |
| p(sim_size>.05) | 0.940 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000
|  $\widehat{\alpha}$ quantile |  |  |  |  |  |  |  |
| q10 | 0.034 | 0.026 | 0.020 | 0.024 | 0.021 | 0.021 | 0.018 | 0.021
| q25 | 0.037 | 0.029 | 0.022 | 0.027 | 0.023 | 0.024 | 0.021 | 0.024
| q50 | 0.041 | 0.032 | 0.025 | 0.030 | 0.026 | 0.027 | 0.024 | 0.027
| q75 | 0.045 | 0.035 | 0.028 | 0.033 | 0.030 | 0.031 | 0.027 | 0.031
| q90 | 0.048 | 0.038 | 0.031 | 0.037 | 0.033 | 0.034 | 0.031 | 0.034
IM | size (usual cv) | 0.022 | 0.024 | 0.021 | 0.027 | 0.038 | 0.049 | 0.051 |
| size (simulated cv) | 0.021 | 0.024 | 0.020 | 0.026 | 0.038 | 0.044 | 0.047 | 0.038
|  $\widehat{k}$ frequency | 0.000 | 0.003 | 0.069 | 0.554 | 0.040 | 0.176 | 0.158 |
| p(sim_size>.05) | 0.389 | 0.419 | 0.385 | 0.426 | 0.385 | 0.536 | 0.521 | 0.336
|  $\widehat{\alpha}$ quantile |  |  |  |  |  |  |  |
| q10 | 0.043 | 0.043 | 0.044 | 0.043 | 0.044 | 0.041 | 0.042 | 0.047
| q25 | 0.047 | 0.047 | 0.047 | 0.047 | 0.048 | 0.045 | 0.045 | 0.049
| q50 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.049 | 0.050 | 0.050
| q75 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050
| q90 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050
CRS | size (usual cv) | 0.000 | 0.000 | 0.000 | 0.000 | 0.027 | 0.050 | 0.055 |
| size (simulated cv) | 0.000 | 0.000 | 0.000 | 0.000 | 0.027 | 0.041 | 0.046 | 0.049
|  $\widehat{k}$ frequency | 0.000 | 0.000 | 0.000 | 0.000 | 0.001 | 0.401 | 0.598 |
| p(sim_size>.05) | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.434 | 0.408 | 0.268
|  $\widehat{\alpha}$ quantile |  |  |  |  |  |  |  |
| q10 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.047 | 0.047 | 0.047
| q25 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.047 | 0.047 | 0.047
| q50 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050
| q75 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050
| q90 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050

  * •

Notes: Simulation results for the design described in Section 2. Inferential
properties are presented for the estimators, CCE, IM, CRS, described in the
text. Columns under $k$ report results with the number of groups fixed at a
certain $k$. $\widehat{k}$ is the number of clusters chosen by the criterion
on size-power tradeoff described in the text. The rows “$\widehat{k}$
frequency” is the frequency of a particular $k$ achieving the highest
simulated power among candidate $k$’s in the setting.

Table 12: Clustering: IV - BASELINE ($N_{\mathrm{pan}}=205$)

|  | $k$ | $\widehat{k}$
---|---|---|---
|  | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
CCE | size (usual cv) | 0.079 | 0.123 | 0.157 | 0.162 | 0.176 | 0.186 | 0.213 |
| size (simulated cv) | 0.041 | 0.053 | 0.053 | 0.052 | 0.058 | 0.059 | 0.055 | 0.062
|  $\widehat{k}$ frequency | 0.000 | 0.001 | 0.001 | 0.039 | 0.242 | 0.261 | 0.456 |
| p(sim_size>.05) | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000
|  $\widehat{\alpha}$ quantile |  |  |  |  |  |  |  |
| q10 | 0.025 | 0.016 | 0.009 | 0.008 | 0.006 | 0.004 | 0.003 | 0.004
| q25 | 0.028 | 0.017 | 0.010 | 0.009 | 0.007 | 0.005 | 0.004 | 0.005
| q50 | 0.030 | 0.019 | 0.011 | 0.010 | 0.008 | 0.006 | 0.004 | 0.007
| q75 | 0.033 | 0.021 | 0.012 | 0.012 | 0.010 | 0.008 | 0.005 | 0.008
| q90 | 0.036 | 0.023 | 0.014 | 0.014 | 0.011 | 0.010 | 0.007 | 0.010
IM | size (usual cv) | 0.045 | 0.060 | 0.062 | 0.062 | 0.045 | 0.035 | 0.044 |
| size (simulated cv) | 0.045 | 0.055 | 0.060 | 0.054 | 0.042 | 0.033 | 0.042 | 0.046
|  $\widehat{k}$ frequency | 0.000 | 0.000 | 0.005 | 0.051 | 0.062 | 0.066 | 0.816 |
| p(sim_size>.05) | 0.300 | 0.683 | 0.702 | 0.786 | 0.640 | 0.280 | 0.414 | 0.375
|  $\widehat{\alpha}$ quantile |  |  |  |  |  |  |  |
| q10 | 0.045 | 0.039 | 0.038 | 0.037 | 0.040 | 0.045 | 0.043 | 0.045
| q25 | 0.049 | 0.042 | 0.042 | 0.040 | 0.043 | 0.049 | 0.047 | 0.048
| q50 | 0.050 | 0.047 | 0.046 | 0.045 | 0.048 | 0.050 | 0.050 | 0.050
| q75 | 0.050 | 0.050 | 0.050 | 0.049 | 0.050 | 0.050 | 0.050 | 0.050
| q90 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050
CRS | size (usual cv) | 0.000 | 0.000 | 0.000 | 0.000 | 0.032 | 0.041 | 0.047 |
| size (simulated cv) | 0.000 | 0.000 | 0.000 | 0.000 | 0.029 | 0.032 | 0.036 | 0.040
|  $\widehat{k}$ frequency | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.157 | 0.843 |
| p(sim_size>.05) | 0.000 | 0.000 | 0.000 | 0.000 | 0.038 | 0.618 | 0.820 | 0.682
|  $\widehat{\alpha}$ quantile |  |  |  |  |  |  |  |
| q10 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.047 | 0.039 | 0.039
| q25 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.047 | 0.039 | 0.039
| q50 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.047 | 0.047 | 0.047
| q75 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.047 | 0.050
| q90 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050

  * •

Notes: Simulation results for the design described in Section 2. Inferential
properties are presented for the estimators, CCE, IM, CRS, described in the
text. Columns under $k$ report results with the number of groups fixed at a
certain $k$. $\widehat{k}$ is the number of clusters chosen by the criterion
on size-power tradeoff described in the text. The rows “$\widehat{k}$
frequency” is the frequency of a particular $k$ achieving the highest
simulated power among candidate $k$’s in the setting.

Table 13: Clustering: IV - SAR ($N_{\mathrm{pan}}=205$)

|  | $k$ | $\widehat{k}$
---|---|---|---
|  | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
CCE | size (usual cv) | 0.039 | 0.091 | 0.148 | 0.095 | 0.113 | 0.127 | 0.138 |
| size (simulated cv) | 0.030 | 0.044 | 0.062 | 0.058 | 0.069 | 0.079 | 0.082 | 0.083
|  $\widehat{k}$ frequency | 0.006 | 0.027 | 0.048 | 0.047 | 0.071 | 0.257 | 0.544 |
| p(sim_size>.05) | 0.929 | 0.998 | 1.000 | 0.998 | 0.998 | 0.998 | 1.000 | 0.999
|  $\widehat{\alpha}$ quantile |  |  |  |  |  |  |  |
| q10 | 0.033 | 0.023 | 0.014 | 0.014 | 0.009 | 0.008 | 0.006 | 0.014
| q25 | 0.037 | 0.027 | 0.020 | 0.024 | 0.019 | 0.020 | 0.016 | 0.020
| q50 | 0.041 | 0.031 | 0.024 | 0.028 | 0.025 | 0.026 | 0.022 | 0.025
| q75 | 0.045 | 0.034 | 0.027 | 0.033 | 0.029 | 0.030 | 0.027 | 0.029
| q90 | 0.049 | 0.038 | 0.031 | 0.037 | 0.033 | 0.034 | 0.031 | 0.033
IM | size (usual cv) | 0.025 | 0.022 | 0.027 | 0.022 | 0.031 | 0.031 | 0.034 |
| size (simulated cv) | 0.025 | 0.022 | 0.027 | 0.022 | 0.031 | 0.027 | 0.031 | 0.025
|  $\widehat{k}$ frequency | 0.005 | 0.078 | 0.240 | 0.568 | 0.030 | 0.038 | 0.041 |
| p(sim_size>.05) | 0.364 | 0.391 | 0.257 | 0.194 | 0.159 | 0.175 | 0.179 | 0.249
|  $\widehat{\alpha}$ quantile |  |  |  |  |  |  |  |
| q10 | 0.043 | 0.041 | 0.042 | 0.043 | 0.031 | 0.020 | 0.013 | 0.032
| q25 | 0.047 | 0.046 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050
| q50 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050
| q75 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050
| q90 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050
CRS | size (usual cv) | 0.000 | 0.000 | 0.000 | 0.000 | 0.027 | 0.045 | 0.049 |
| size (simulated cv) | 0.000 | 0.000 | 0.000 | 0.000 | 0.017 | 0.034 | 0.037 | 0.041
|  $\widehat{k}$ frequency | 0.000 | 0.000 | 0.000 | 0.000 | 0.032 | 0.480 | 0.488 |
| p(sim_size>.05) | 0.000 | 0.000 | 0.000 | 0.000 | 0.209 | 0.620 | 0.641 | 0.497
|  $\widehat{\alpha}$ quantile |  |  |  |  |  |  |  |
| q10 | 0.050 | 0.050 | 0.050 | 0.050 | 0.031 | 0.016 | 0.008 | 0.008
| q25 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.047 | 0.031 | 0.043
| q50 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.047 | 0.047 | 0.050
| q75 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050
| q90 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050

  * •

Notes: Simulation results for the design described in Section 2. Inferential
properties are presented for the estimators, CCE, IM, CRS, described in the
text. Columns under $k$ report results with the number of groups fixed at a
certain $k$. $\widehat{k}$ is the number of clusters chosen by the criterion
on size-power tradeoff described in the text. The rows “$\widehat{k}$
frequency” is the frequency of a particular $k$ achieving the highest
simulated power among candidate $k$’s in the setting.

Table 14: Summary: OLS - BASELINE ($N_{\mathrm{pan}}=820$)

Method | Estim. Mean | Estim. RMSE | Size | Power
---|---|---|---|---
|  |  |  | -1 | -0.5 | 0.5 | 1
SK | 0.002 | 0.259 | 0.395 | 0.998 | 0.856 | 0.847 | 0.998
UNIT-U | 0.002 | 0.259 | 0.700 | 1.000 | 0.938 | 0.944 | 1.000
UNIT | 0.002 | 0.259 | 0.039 | 0.962 | 0.517 | 0.496 | 0.960
CCE | 0.002 | 0.259 | 0.052 | 0.917 | 0.449 | 0.449 | 0.899
IM | -0.001 | 0.146 | 0.058 | 1.000 | 0.877 | 0.881 | 1.000
CRS | -0.001 | 0.146 | 0.058 | 1.000 | 0.871 | 0.868 | 1.000

  * •

Notes: Simulation results for estimation in the design described in Section 2.
The nominal size is 0.05. Estimates are presented for the estimators, SK,
UNIT-U, UNIT, CCE, IM, CRS described in the text. Columns display method,
estimated mean, estimated RMSE, size, and power against four alternatives (-1,
-0.5, 0.5, 1).

Table 15: Summary: OLS- SAR ($N_{\mathrm{pan}}=820$)

Method | Estim. Mean | Estim. RMSE | Size | Power
---|---|---|---|---
|  |  |  | -1 | -0.5 | 0.5 | 1
SK | 0.003 | 0.229 | 0.154 | 0.996 | 0.780 | 0.787 | 0.990
UNIT-U | 0.003 | 0.229 | 0.405 | 0.999 | 0.909 | 0.903 | 1.000
UNIT | 0.003 | 0.229 | 0.123 | 0.992 | 0.756 | 0.743 | 0.995
CCE | 0.003 | 0.229 | 0.020 | 0.860 | 0.363 | 0.364 | 0.848
IM | -0.013 | 0.249 | 0.035 | 0.908 | 0.465 | 0.524 | 0.911
CRS | -0.013 | 0.229 | 0.062 | 0.959 | 0.612 | 0.553 | 0.938

  * •

Notes: Simulation results for estimation in the design described in Section 2.
The nominal size is 0.05. Estimates are presented for the estimators, SK,
UNIT-U, UNIT, CCE, IM, CRS described in the text. Columns display method,
estimated mean, estimated RMSE, size, and power against four alternatives (-1,
-0.5, 0.5, 1).

Table 16: Summary: IV - BASELINE ($N_{\mathrm{pan}}=820$)

Method | Estim. Median | Estim. MAD | Size | Power
---|---|---|---|---
|  |  |  | -1 | -0.5 | 0.5 | 1
SK | 0.004 | 0.089 | 0.390 | 1.000 | 1.000 | 0.973 | 1.000
UNIT-U | 0.004 | 0.089 | 0.700 | 1.000 | 0.995 | 1.000 | 1.000
UNIT | 0.004 | 0.089 | 0.048 | 0.998 | 0.870 | 1.000 | 1.000
CCE | 0.004 | 0.089 | 0.051 | 0.989 | 0.837 | 0.963 | 1.000
IM | -0.041 | 0.061 | 0.055 | 0.994 | 0.940 | 0.998 | 0.999
CRS | -0.042 | 0.061 | 0.056 | 0.999 | 0.999 | 0.930 | 0.993

  * •

Notes: Simulation results for estimation in the design described in Section 2.
The nominal size is 0.05. Estimates are presented for the estimators, SK,
UNIT-U, UNIT, CCE, IM, CRS described in the text. Columns display method,
estimated median, estimated MAD, size, and power against four alternatives
(-1, -0.5, 0.5, 1).

Table 17: Summary: IV - SAR ($N_{\mathrm{pan}}=820$)

Method | Estim. Median | Estim. MAD | Size | Power
---|---|---|---|---
|  |  |  | -1 | -0.5 | 0.5 | 1
SK | 0.003 | 0.076 | 0.144 | 1.000 | 1.000 | 0.953 | 0.997
UNIT-U | 0.003 | 0.076 | 0.389 | 1.000 | 0.983 | 1.000 | 1.000
UNIT | 0.003 | 0.076 | 0.118 | 0.998 | 0.948 | 1.000 | 1.000
CCE | 0.003 | 0.076 | 0.030 | 0.979 | 0.804 | 0.898 | 0.999
IM | -0.052 | 0.130 | 0.022 | 0.774 | 0.561 | 0.756 | 0.864
CRS | -0.052 | 0.123 | 0.037 | 0.921 | 0.891 | 0.652 | 0.847

  * •

Notes: Simulation results for estimation in the design described in Section 2.
The nominal size is 0.05. Estimates are presented for the estimators, SK,
UNIT-U, UNIT, CCE, IM, CRS described in the text. Columns display method,
estimated median, estimated MAD, size, and power against four alternatives
(-1, -0.5, 0.5, 1).

(a) BASELINE (b) SAR

Figure 4: OLS power curves ($N_{\text{pan}}=820$).

(a) BASELINE (b) SAR

Figure 5: IV power curves ($N_{\text{pan}}=820$). Table 18: Clustering: OLS -
BASELINE ($N_{\mathrm{pan}}=820$)

|  | $k$ | $\widehat{k}$
---|---|---|---
|  | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
CCE | size (usual cv) | 0.056 | 0.093 | 0.117 | 0.146 | 0.180 | 0.178 | 0.177 | 0.188 | 0.209 | 0.209 | 0.218 |
| size (simulated cv) | 0.041 | 0.045 | 0.046 | 0.040 | 0.044 | 0.050 | 0.051 | 0.050 | 0.047 | 0.047 | 0.049 | 0.052
|  $\widehat{k}$ frequency | 0.000 | 0.003 | 0.006 | 0.045 | 0.086 | 0.068 | 0.119 | 0.098 | 0.189 | 0.150 | 0.236 |
| p(sim_size>.05) | 0.994 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000
|  $\widehat{\alpha}$ quantile |  |  |  |  |  |  |  |  |  |  |  |
| q10 | 0.030 | 0.021 | 0.015 | 0.011 | 0.008 | 0.008 | 0.007 | 0.006 | 0.004 | 0.004 | 0.004 | 0.005
| q25 | 0.033 | 0.023 | 0.017 | 0.011 | 0.009 | 0.009 | 0.008 | 0.007 | 0.005 | 0.005 | 0.004 | 0.006
| q50 | 0.036 | 0.025 | 0.019 | 0.013 | 0.010 | 0.011 | 0.010 | 0.008 | 0.006 | 0.006 | 0.005 | 0.008
| q75 | 0.039 | 0.028 | 0.021 | 0.014 | 0.011 | 0.012 | 0.011 | 0.009 | 0.007 | 0.007 | 0.006 | 0.010
| q90 | 0.042 | 0.030 | 0.023 | 0.016 | 0.012 | 0.013 | 0.012 | 0.010 | 0.009 | 0.008 | 0.007 | 0.013
IM | size (usual cv) | 0.046 | 0.060 | 0.059 | 0.064 | 0.064 | 0.075 | 0.085 | 0.091 | 0.096 | 0.090 | 0.076 |
| size (simulated cv) | 0.041 | 0.053 | 0.054 | 0.057 | 0.052 | 0.062 | 0.069 | 0.057 | 0.067 | 0.059 | 0.053 | 0.058
|  $\widehat{k}$ frequency | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.002 | 0.013 | 0.013 | 0.005 | 0.195 | 0.772 |
| p(sim_size>.05) | 0.492 | 0.565 | 0.727 | 0.675 | 0.828 | 0.966 | 0.966 | 0.998 | 1.000 | 1.000 | 0.999 | 0.997
|  $\widehat{\alpha}$ quantile |  |  |  |  |  |  |  |  |  |  |  |
| q10 | 0.042 | 0.041 | 0.039 | 0.039 | 0.037 | 0.032 | 0.031 | 0.025 | 0.021 | 0.024 | 0.024 | 0.026
| q25 | 0.046 | 0.045 | 0.042 | 0.043 | 0.039 | 0.036 | 0.035 | 0.028 | 0.024 | 0.027 | 0.027 | 0.029
| q50 | 0.050 | 0.049 | 0.046 | 0.047 | 0.044 | 0.039 | 0.038 | 0.032 | 0.027 | 0.030 | 0.031 | 0.032
| q75 | 0.050 | 0.050 | 0.050 | 0.050 | 0.048 | 0.043 | 0.042 | 0.035 | 0.031 | 0.034 | 0.035 | 0.036
| q90 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.047 | 0.046 | 0.039 | 0.035 | 0.038 | 0.038 | 0.040
CRS | size (usual cv) | 0.000 | 0.000 | 0.000 | 0.000 | 0.034 | 0.074 | 0.082 | 0.088 | 0.100 | 0.093 | 0.077 |
| size (simulated cv) | 0.000 | 0.000 | 0.000 | 0.000 | 0.033 | 0.047 | 0.062 | 0.053 | 0.058 | 0.059 | 0.053 | 0.058
|  $\widehat{k}$ frequency | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.001 | 0.004 | 0.009 | 0.004 | 0.170 | 0.812 |
| p(sim_size>.05) | 0.000 | 0.000 | 0.000 | 0.000 | 0.027 | 0.882 | 0.923 | 0.993 | 1.000 | 0.998 | 0.999 | 0.992
|  $\widehat{\alpha}$ quantile |  |  |  |  |  |  |  |  |  |  |  |
| q10 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.047 | 0.031 | 0.027 | 0.021 | 0.023 | 0.023 | 0.024
| q25 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.047 | 0.039 | 0.029 | 0.023 | 0.026 | 0.026 | 0.027
| q50 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.047 | 0.039 | 0.033 | 0.027 | 0.030 | 0.030 | 0.031
| q75 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.047 | 0.047 | 0.035 | 0.031 | 0.034 | 0.034 | 0.035
| q90 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.047 | 0.039 | 0.035 | 0.038 | 0.038 | 0.039

  * •

Notes: Simulation results for the design described in Section 2. Inferential
properties are presented for the estimators, CCE, IM, CRS, described in the
text. Columns under $k$ report results with the number of groups fixed at a
certain $k$. $\widehat{k}$ is the number of clusters chosen by the criterion
on size-power tradeoff described in the text. The rows “$\widehat{k}$
frequency” is the frequency of a particular $k$ achieving the highest
simulated power among candidate $k$’s in the setting.

Table 19: Clustering: OLS - SAR ($N_{\mathrm{pan}}=820$)

|  | $k$ | $\widehat{k}$
---|---|---|---
|  | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
CCE | size (usual cv) | 0.040 | 0.043 | 0.037 | 0.033 | 0.033 | 0.035 | 0.035 | 0.045 | 0.047 | 0.048 | 0.052 |
| size (simulated cv) | 0.036 | 0.037 | 0.026 | 0.020 | 0.021 | 0.017 | 0.012 | 0.017 | 0.019 | 0.020 | 0.020 | 0.020
|  $\widehat{k}$ frequency | 0.000 | 0.000 | 0.004 | 0.008 | 0.028 | 0.052 | 0.105 | 0.113 | 0.167 | 0.220 | 0.303 |
| p(sim_size>.05) | 0.767 | 0.987 | 0.992 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000
|  $\widehat{\alpha}$ quantile |  |  |  |  |  |  |  |  |  |  |  |
| q10 | 0.038 | 0.030 | 0.028 | 0.024 | 0.021 | 0.023 | 0.022 | 0.020 | 0.019 | 0.018 | 0.016 | 0.020
| q25 | 0.041 | 0.033 | 0.031 | 0.027 | 0.024 | 0.025 | 0.024 | 0.022 | 0.021 | 0.020 | 0.019 | 0.022
| q50 | 0.045 | 0.036 | 0.035 | 0.030 | 0.027 | 0.028 | 0.027 | 0.025 | 0.024 | 0.023 | 0.021 | 0.026
| q75 | 0.050 | 0.040 | 0.038 | 0.033 | 0.030 | 0.031 | 0.030 | 0.028 | 0.027 | 0.026 | 0.024 | 0.029
| q90 | 0.050 | 0.044 | 0.041 | 0.036 | 0.033 | 0.034 | 0.033 | 0.031 | 0.029 | 0.028 | 0.026 | 0.033
IM | size (usual cv) | 0.051 | 0.058 | 0.027 | 0.024 | 0.027 | 0.037 | 0.040 | 0.039 | 0.034 | 0.034 | 0.042 |
| size (simulated cv) | 0.047 | 0.055 | 0.026 | 0.020 | 0.024 | 0.034 | 0.037 | 0.036 | 0.033 | 0.032 | 0.040 | 0.035
|  $\widehat{k}$ frequency | 0.000 | 0.000 | 0.002 | 0.006 | 0.073 | 0.044 | 0.409 | 0.223 | 0.061 | 0.067 | 0.115 |
| p(sim_size>.05) | 0.495 | 0.366 | 0.515 | 0.383 | 0.471 | 0.522 | 0.535 | 0.491 | 0.453 | 0.482 | 0.483 | 0.354
|  $\widehat{\alpha}$ quantile |  |  |  |  |  |  |  |  |  |  |  |
| q10 | 0.042 | 0.043 | 0.041 | 0.044 | 0.042 | 0.041 | 0.041 | 0.042 | 0.043 | 0.042 | 0.042 | 0.047
| q25 | 0.045 | 0.047 | 0.045 | 0.048 | 0.046 | 0.045 | 0.045 | 0.046 | 0.046 | 0.046 | 0.046 | 0.049
| q50 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.049 | 0.049 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050
| q75 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050
| q90 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050
CRS | size (usual cv) | 0.000 | 0.000 | 0.000 | 0.000 | 0.022 | 0.053 | 0.050 | 0.054 | 0.048 | 0.047 | 0.058 |
| size (simulated cv) | 0.000 | 0.000 | 0.000 | 0.000 | 0.022 | 0.049 | 0.048 | 0.054 | 0.046 | 0.040 | 0.053 | 0.062
|  $\widehat{k}$ frequency | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.019 | 0.199 | 0.143 | 0.127 | 0.188 | 0.324 |
| p(sim_size>.05) | 0.000 | 0.000 | 0.000 | 0.000 | 0.001 | 0.355 | 0.342 | 0.383 | 0.487 | 0.515 | 0.522 | 0.277
|  $\widehat{\alpha}$ quantile |  |  |  |  |  |  |  |  |  |  |  |
| q10 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.047 | 0.047 | 0.043 | 0.042 | 0.042 | 0.041 | 0.047
| q25 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.047 | 0.047 | 0.047 | 0.046 | 0.045 | 0.045 | 0.050
| q50 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050
| q75 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050
| q90 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050

  * •

Notes: Simulation results for the design described in Section 2. Inferential
properties are presented for the estimators, CCE, IM, CRS, described in the
text. Columns under $k$ report results with the number of groups fixed at a
certain $k$. $\widehat{k}$ is the number of clusters chosen by the criterion
on size-power tradeoff described in the text. The rows “$\widehat{k}$
frequency” is the frequency of a particular $k$ achieving the highest
simulated power among candidate $k$’s in the setting.

Table 20: Clustering: IV - BASELINE ($N_{\mathrm{pan}}=820$)

|  | $k$ | $\widehat{k}$
---|---|---|---
|  | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
CCE | size (usual cv) | 0.059 | 0.091 | 0.113 | 0.150 | 0.171 | 0.156 | 0.169 | 0.186 | 0.201 | 0.197 | 0.210 |
| size (simulated cv) | 0.044 | 0.047 | 0.048 | 0.043 | 0.046 | 0.047 | 0.054 | 0.052 | 0.047 | 0.040 | 0.041 | 0.051
|  $\widehat{k}$ frequency | 0.000 | 0.002 | 0.008 | 0.009 | 0.053 | 0.038 | 0.069 | 0.096 | 0.185 | 0.207 | 0.333 |
| p(sim_size>.05) | 0.994 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000
|  $\widehat{\alpha}$ quantile |  |  |  |  |  |  |  |  |  |  |  |
| q10 | 0.030 | 0.021 | 0.015 | 0.010 | 0.007 | 0.008 | 0.007 | 0.005 | 0.004 | 0.004 | 0.003 | 0.004
| q25 | 0.033 | 0.023 | 0.017 | 0.011 | 0.008 | 0.009 | 0.008 | 0.006 | 0.005 | 0.005 | 0.004 | 0.005
| q50 | 0.036 | 0.025 | 0.019 | 0.012 | 0.009 | 0.010 | 0.009 | 0.008 | 0.006 | 0.006 | 0.005 | 0.007
| q75 | 0.039 | 0.028 | 0.021 | 0.014 | 0.011 | 0.012 | 0.011 | 0.009 | 0.007 | 0.007 | 0.006 | 0.009
| q90 | 0.042 | 0.030 | 0.023 | 0.016 | 0.012 | 0.014 | 0.012 | 0.010 | 0.009 | 0.009 | 0.007 | 0.011
IM | size (usual cv) | 0.050 | 0.054 | 0.058 | 0.064 | 0.062 | 0.069 | 0.081 | 0.087 | 0.093 | 0.079 | 0.074 |
| size (simulated cv) | 0.048 | 0.051 | 0.057 | 0.059 | 0.054 | 0.056 | 0.073 | 0.063 | 0.065 | 0.057 | 0.054 | 0.055
|  $\widehat{k}$ frequency | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.001 | 0.009 | 0.000 | 0.128 | 0.862 |
| p(sim_size>.05) | 0.483 | 0.544 | 0.636 | 0.528 | 0.694 | 0.890 | 0.903 | 0.988 | 0.998 | 0.993 | 0.998 | 0.993
|  $\widehat{\alpha}$ quantile |  |  |  |  |  |  |  |  |  |  |  |
| q10 | 0.042 | 0.042 | 0.039 | 0.041 | 0.039 | 0.035 | 0.035 | 0.028 | 0.025 | 0.028 | 0.028 | 0.028
| q25 | 0.046 | 0.045 | 0.043 | 0.045 | 0.043 | 0.039 | 0.038 | 0.032 | 0.028 | 0.031 | 0.031 | 0.031
| q50 | 0.050 | 0.049 | 0.048 | 0.049 | 0.047 | 0.043 | 0.042 | 0.035 | 0.032 | 0.034 | 0.034 | 0.035
| q75 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.046 | 0.046 | 0.039 | 0.035 | 0.038 | 0.038 | 0.039
| q90 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.043 | 0.039 | 0.042 | 0.041 | 0.042
CRS | size (usual cv) | 0.000 | 0.000 | 0.000 | 0.000 | 0.034 | 0.072 | 0.084 | 0.087 | 0.107 | 0.098 | 0.089 |
| size (simulated cv) | 0.000 | 0.000 | 0.000 | 0.000 | 0.033 | 0.043 | 0.060 | 0.058 | 0.060 | 0.057 | 0.053 | 0.056
|  $\widehat{k}$ frequency | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.001 | 0.001 | 0.008 | 0.001 | 0.125 | 0.864 |
| p(sim_size>.05) | 0.000 | 0.000 | 0.000 | 0.000 | 0.030 | 0.889 | 0.941 | 0.996 | 1.000 | 1.000 | 1.000 | 0.996
|  $\widehat{\alpha}$ quantile |  |  |  |  |  |  |  |  |  |  |  |
| q10 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.047 | 0.031 | 0.027 | 0.021 | 0.022 | 0.022 | 0.022
| q25 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.047 | 0.039 | 0.029 | 0.023 | 0.025 | 0.024 | 0.025
| q50 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.047 | 0.039 | 0.031 | 0.026 | 0.028 | 0.028 | 0.028
| q75 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.047 | 0.047 | 0.035 | 0.029 | 0.031 | 0.031 | 0.032
| q90 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.047 | 0.039 | 0.033 | 0.035 | 0.034 | 0.036

  * •

Notes: Simulation results for the design described in Section 2. Inferential
properties are presented for the estimators, CCE, IM, CRS, described in the
text. Columns under $k$ report results with the number of groups fixed at a
certain $k$. $\widehat{k}$ is the number of clusters chosen by the criterion
on size-power tradeoff described in the text. The rows “$\widehat{k}$
frequency” is the frequency of a particular $k$ achieving the highest
simulated power among candidate $k$’s in the setting.

Table 21: Clustering: IV - SAR ($N_{\mathrm{pan}}=820$)

|  | $k$ | $\widehat{k}$
---|---|---|---
|  | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
CCE | size (usual cv) | 0.034 | 0.042 | 0.033 | 0.035 | 0.041 | 0.030 | 0.040 | 0.041 | 0.041 | 0.043 | 0.047 |
| size (simulated cv) | 0.029 | 0.035 | 0.021 | 0.021 | 0.019 | 0.020 | 0.022 | 0.025 | 0.027 | 0.029 | 0.030 | 0.030
|  $\widehat{k}$ frequency | 0.000 | 0.000 | 0.000 | 0.000 | 0.007 | 0.016 | 0.046 | 0.076 | 0.151 | 0.268 | 0.436 |
| p(sim_size>.05) | 0.800 | 0.992 | 0.994 | 1.000 | 1.000 | 0.999 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000
|  $\widehat{\alpha}$ quantile |  |  |  |  |  |  |  |  |  |  |  |
| q10 | 0.037 | 0.030 | 0.028 | 0.024 | 0.022 | 0.022 | 0.022 | 0.020 | 0.019 | 0.018 | 0.016 | 0.019
| q25 | 0.041 | 0.033 | 0.031 | 0.026 | 0.024 | 0.025 | 0.024 | 0.022 | 0.021 | 0.020 | 0.019 | 0.021
| q50 | 0.044 | 0.036 | 0.035 | 0.029 | 0.027 | 0.028 | 0.027 | 0.025 | 0.024 | 0.023 | 0.021 | 0.024
| q75 | 0.049 | 0.040 | 0.039 | 0.033 | 0.030 | 0.031 | 0.031 | 0.028 | 0.027 | 0.026 | 0.024 | 0.027
| q90 | 0.050 | 0.043 | 0.042 | 0.036 | 0.033 | 0.034 | 0.034 | 0.031 | 0.030 | 0.029 | 0.027 | 0.031
IM | size (usual cv) | 0.041 | 0.056 | 0.029 | 0.021 | 0.020 | 0.037 | 0.030 | 0.028 | 0.026 | 0.019 | 0.026 |
| size (simulated cv) | 0.039 | 0.055 | 0.025 | 0.019 | 0.019 | 0.035 | 0.029 | 0.028 | 0.026 | 0.019 | 0.026 | 0.022
|  $\widehat{k}$ frequency | 0.000 | 0.004 | 0.008 | 0.008 | 0.153 | 0.003 | 0.777 | 0.046 | 0.001 | 0.000 | 0.000 |
| p(sim_size>.05) | 0.481 | 0.321 | 0.399 | 0.234 | 0.256 | 0.224 | 0.240 | 0.176 | 0.115 | 0.104 | 0.139 | 0.160
|  $\widehat{\alpha}$ quantile |  |  |  |  |  |  |  |  |  |  |  |
| q10 | 0.042 | 0.045 | 0.043 | 0.046 | 0.046 | 0.047 | 0.046 | 0.048 | 0.049 | 0.050 | 0.049 | 0.049
| q25 | 0.046 | 0.048 | 0.047 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050
| q50 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050
| q75 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050
| q90 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050
CRS | size (usual cv) | 0.000 | 0.000 | 0.000 | 0.000 | 0.025 | 0.054 | 0.051 | 0.045 | 0.042 | 0.039 | 0.052 |
| size (simulated cv) | 0.000 | 0.000 | 0.000 | 0.000 | 0.025 | 0.044 | 0.041 | 0.042 | 0.040 | 0.032 | 0.038 | 0.037
|  $\widehat{k}$ frequency | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.031 | 0.469 | 0.320 | 0.090 | 0.048 | 0.042 |
| p(sim_size>.05) | 0.000 | 0.000 | 0.000 | 0.000 | 0.002 | 0.389 | 0.446 | 0.607 | 0.780 | 0.899 | 0.957 | 0.352
|  $\widehat{\alpha}$ quantile |  |  |  |  |  |  |  |  |  |  |  |
| q10 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.047 | 0.047 | 0.039 | 0.038 | 0.036 | 0.034 | 0.045
| q25 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.047 | 0.047 | 0.043 | 0.041 | 0.039 | 0.036 | 0.047
| q50 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.047 | 0.045 | 0.042 | 0.040 | 0.050
| q75 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.046 | 0.043 | 0.050
| q90 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.047 | 0.050

  * •

Notes: Simulation results for the design described in Section 2. Inferential
properties are presented for the estimators, CCE, IM, CRS, described in the
text. Columns under $k$ report results with the number of groups fixed at a
certain $k$. $\widehat{k}$ is the number of clusters chosen by the criterion
on size-power tradeoff described in the text. The rows “$\widehat{k}$
frequency” is the frequency of a particular $k$ achieving the highest
simulated power among candidate $k$’s in the setting.


§ PROOF OF LEMMA <REF>

Throughout this section, fix $\boldsymbol \nu\in \nurates$ with $\boldsymbol \nu(\diff \xi) = \boldsymbol \lambda \diff \xi$ for some $\boldsymbol \lambda \in \rates$. In that case, $N^F_{\boldsymbol \nu}$ for $F\in \mathbb{F}$ is a map $E^{\lf(F)}_\circ \to (0,\infty)$ defined by
\[
N^F_{\boldsymbol \nu}(\boldsymbol x) = \int\limits_{\dct(F)} \int\limits_{\dcs(F)} \ftm(\tau) \fsp(\xi \,\vert\,\tau,\boldsymbol x) \diff \xi \diff \tau.
\]
The goal of this section is to show that $N^F_{\boldsymbol \nu}$ is (finite and) continuous.

Let $F\in \mathbb{F}$. For fixed $\tau \in \dct(F)$, the map
\[
E^{\lf(F)}_\circ \to (0,\infty);\quad \boldsymbol x \mapsto \smashoperator[r]{\int_{\dcs(F)}} \fsp(\xi \,\vert\,\tau,\boldsymbol x) \diff \xi
\]
is continuous.

For fixed $\tau \in \dct(F)$, $\fsp(\xi \,\vert\,\tau,\boldsymbol x)$ is continuous in $\boldsymbol x$ for every $\xi \in \dcs(F)$, and bounded uniformly in $\boldsymbol x$ and $\xi$, so the statement follows from the dominated (or bounded) convergence theorem.

Let $\fpart\in \mathcal{P}$. Then $E^{\fpart}_\circ $ is an open subset of $E^\fpart$, so the distance between some $\boldsymbol x\in E^\fpart$ to $E^\fpart \setminus E^\fpart_\circ $ is well-defined and denoted $\varepsilon_d(\boldsymbol x)$. More explicitly,

\begin{equation}\label{eq:epsd}
\varepsilon_d(\boldsymbol x) =
\begin{cases}
\min_{u\neq v\in \fpart} \rho(\boldsymbol x_u , \boldsymbol x_v), & d \ge 2,\\
\max_{u_0,v_0\in \fpart} \min_{u\neq v\in \fpart, \{ u,v \} \neq \left\{ u_0,v_0 \right\} } \rho(\boldsymbol x_{u},\boldsymbol x_{v}), & d = 1,
\end{cases}
\end{equation}

where $\rho$ denotes the Euclidean metric on $E$, see <ref>. If $d\ge 2$ then $\varepsilon_d(\boldsymbol x)$ is the smallest, if $d = 1$ it is the second smallest distance among all pairs of two particles.

Let $F \in \mathbb{F}$. Then there exists $C = C(F,\boldsymbol \lambda,d) > 0$ such that, for any $A \subset E^{\lf(F)}_\circ $,

\begin{equation}\label{eq:NFboundtorus}
\smashoperator{\int_{\dct(F)}} \ftm(\tau) \sup_{\boldsymbol x\in A} \bigg( \smashoperator[r]{\int_{\dcs(F)}} \fsp(\xi \,\vert\,\tau,\boldsymbol x) \diff \xi \bigg) \diff \tau
\le C \bigg[ 1 + \left( \inf_{\boldsymbol x\in A} \varepsilon_d(\boldsymbol x) \right) ^{-C} \bigg] .
\end{equation}

In particular, $N_{\boldsymbol \nu}^F \colon E^{\lf(F)}_\circ \to (0,\infty)$ is continuous and, specialising $A$ in <ref> to a singleton,
\[
N^F_{\boldsymbol \nu}(\boldsymbol x) \le C \left( 1 + \varepsilon_d(\boldsymbol x)^{-C} \right) ,\qquad \boldsymbol x\in E^{\lf(F)}_\circ .
\]

The continuity of $N^F_{\boldsymbol \nu}$ follows from <ref> and <ref> by the dominated convergence theorem. We do not prove <ref> directly. Instead, we prove an analogous statement when the torus is replaced by $\R^d$, where a certain heat kernel trick can be used that isn't available on the torus. Write $\rfy{E} = \R^d$, and denote by $\rfy{p}_t \colon \R^d \to (0,\infty)$ for $t > 0$ the heat kernel, defined by
\[
\rfy{p}_t(x) = (2\pi t)^{-d / 2} \exp\left(-\frac{|x|^2}{2t}\right),\qquad x\in \R^d.
\]
Define $\rfy{E}^{\fpart}_\circ $ for $\fpart\in \mathcal{P}$ analogously to $E^{\fpart}_\circ $, and for a forest $F\in \mathbb{F}$, denote by $\rfy{\dcs}(F)$ the set of maps $\xi\colon \nd^\circ (F) \to \rfy{E}$ such that $\xi\big\vert_{\fpart} \in \rfy{E}^\fpart_\circ $ for all $\fpart \in F$. For $\xi\in \rfy{\dcs}(F)$, $\tau\in \dct(F)$ and $\boldsymbol x\in \rfy{E}^{\lf(F)}_\circ $, write

\begin{equation*}
\rfy{\fsp}(\xi\,\vert\, \tau,\boldsymbol x) \coloneqq  \smashop{\prod_{u\in \nd(F)\setminus \rt(F) }} \rfy{p}(\tau_{\pr_F(u)} - \tau_u, (\boldsymbol x \xi)_u - \xi_{\pr_F(u)}).
\end{equation*}

For every $x,y\in E$ there exists a $\vec{k}(x,y) \in \{-1,0\}$ (not necessarily unique) such that $\rho(x,y) = |x-y+\vec{k}(x,y)|$ (on the left is the Euclidean metric on the torus $E$, on the right is the Euclidean norm on $\R^d$). Here and in the following, if we treat an element $x\in E$ as an element of $\R^d$, then we identify it with its unique representative in $[0,1)^d$. Then, for some $C = C(d) > 0$,

\begin{align}\label{eq:app:ptbound}
&\le C(1 + \sqrt{t} )^d \rfy{p}_t(x-y+\vec{k}(x,y))\nonumber \\
&\le C(1+\sqrt{t} )^d \smashop{\sum_{\vec{k}\in \{-1,0,1\}^d}} \rfy{p}_t(x - y + \vec{k}),
\end{align}

where the first inequality follows e.g. from <cit.>.

For every $F = (\fpart^F_0, \ldots ,\fpart^F_m) \in \mathbb{F}$ there is $C = C(d,F) > 0$ such that for every $\tau\in \dct(F)$ and $\boldsymbol x\in E^{\lf(F)}_\circ $,

\begin{equation}\label{eq:appNF:fsp1}
\fsp(\xi \,\vert\,\tau,\boldsymbol x) \le C \left(1 + \sqrt{\max \tau} \right)^C \smashop{\sum_{\substack{\vec{\boldsymbol \ell}\colon \nd^\circ (F) \to  \{-m, \ldots ,m\}^{d} \\ \vec{\boldsymbol k} \colon \lf(F) \to \{-m, \ldots ,m\}^d}  }} \rfy{\fsp}(\xi +  \vec{\boldsymbol \ell}  \,\vert\,\tau,\boldsymbol x +  \vec{\boldsymbol k})
\end{equation}

for all $\xi \in \dcs(F)$, and

\begin{equation}\label{eq:appNF:fsp2}
\smashop{\int_{\dcs(F) }} \fsp(\xi \,\vert\,\tau,\boldsymbol x) \diff \xi \le C (1 + \sqrt{\max \tau})^C \smashoperator[l]{\sum_{\vec{\boldsymbol k} \colon \lf(F) \to \{-m, \ldots ,m\}^d}}\,\, \int\limits_{\rfy{\dcs}(F) } \rfy{\fsp}(\xi \,\vert\,\tau,\boldsymbol x +  \vec{\boldsymbol k}) \diff \xi.
\end{equation}

Denote by $C > 0$ a constant depending only on $d$ and $F$ whose value may increase from line to line. Let $v\in \rt(F)$ with $\ch_F(v) \neq \emptyset $, and suppose for simplicity that $\ch_F(u) \neq \emptyset $ for all $u\in \ch_F(v)$. Then, using <ref>,

\begin{align*}
\prod_{u\in \ch_F(v)} p_{\tau_v-\tau_u}(\xi_u - \xi_v)
&\le \prod_{u\in \ch_F(v)} \left( C (1 + \sqrt{\tau_v - \tau_u}  ) ^d \sum_{\vec{k}_u\in \{-1,0,1\}^d} \rfy{p}_{\tau_v-\tau_u}(\xi_u - \xi_v +  \vec{k}_u)\right)\\
&\le C (1 + \sqrt{\max \tau} )^{C} \prod_{u\in \ch_F(v)} \sum_{\vec{k}_u \in \{-1,0,1\}^d} \rfy{p}_{\tau_v-\tau_u}[(\xi_u +  \vec{k}_u) - \xi_v].
\end{align*}

We put $\vec{\boldsymbol \ell}_u \coloneqq \vec{k}_u$ for $u\in \ch_F(v)$. Now let $u\in \ch_F(v)$ and suppose again for simplicity that $\ch_F(w) \neq \emptyset $ for all $w\in \ch_F(u)$. Then

\begin{align*}
\prod_{w\in \ch_F(u)}& p_{\tau_u - \tau_w}(\xi_w - \xi_u)\\
&= \prod_{w\in \ch_F(u)} p_{\tau_u - \tau_w}[\xi_w +  \vec{\boldsymbol \ell}_u - (\xi_u +  \vec{\boldsymbol \ell}_u)]\\
&\le C (1 + \sqrt{\max \tau} )^C \smashoperator[l]{\prod_{w\in \ch_F(u)}} \smashoperator[r]{\sum_{\vec{k}_w \in \{-1,0,1\}^d}} \rfy{p}_{\tau_u - \tau_w} [(\xi_w +  \vec{k}_w +  \vec{\boldsymbol \ell}_u) - (\xi_u +  \vec{\boldsymbol \ell}_u)].
\end{align*}

We put $\vec{\boldsymbol \ell}_w \coloneqq \vec{k}_w + \vec{\boldsymbol \ell}_u$ for $w\in \ch_F(u)$. Proceeding inductively in this way down towards the leaves yields <ref>, from which the second claim easily follows (note that the integral on the left-hand side is w.r.t. Lebesgue measure on the torus, and on the right-hand side w.r.t. Lebesgue measure on all of $\R^d$).

This reduction lets us make use of the following statement. Define $\rfy{\varepsilon_d}(\boldsymbol x)$ for $\boldsymbol x\in \rfy{E}^\fpart$ for some $\fpart\in \mathcal{P}$ analogously to $\varepsilon_d$, that is by replacing $\rho(\cdot ,\cdot )$ in <ref> by the Euclidean distance $|\cdot-\cdot  |$ on $\R^d$.

Let $F \in \mathbb{F}$ and $K > 0$. Then there exists $C = C(F,\boldsymbol \lambda,d, K) > 0$ such that, for any $A \subset \rfy{E}^{\lf(F)}_\circ $,

\begin{equation*}%\label{eq:NFboundRd}
\smashoperator{\int_{\dct(F)}} \ftm(\tau) (1 + \sqrt{\max \tau} )^K \sup_{\boldsymbol x\in A} \bigg( \smashoperator[r]{\int_{\rfy{\dcs}(F)}} \rfy{\fsp}(\xi \,\vert\,\tau,\boldsymbol x) \diff \xi \bigg) \diff \tau
\le C \bigg[ 1 + \left( \inf_{\boldsymbol x\in A} \rfy{\varepsilon_d}(\boldsymbol x) \right) ^{-C} \bigg] .
\end{equation*}

The proof of <ref> hinges on an explicit representation of $\int \rfy{\fsp}(\xi \,\vert\,\tau,\boldsymbol x) \diff \xi$ based on the following heat kernel trick.

Let $x_1, \ldots ,x_n \in \R^d$ and $s_1, \ldots ,s_n > 0$. Then,
\[
\prod_{i=1}^n \rfy{p}_{s_i}(x_i-z) = \frac{\rfy{p}_s(z-\overline{x})}{\rfy{p}_s(0)} \prod_{i=1}^n \rfy{p}_{s_i}(x_i - \interp),
\]
where $\frac{1}{s} = \sum_{i=1}^n \frac{1}{s_i}$ and $\interp = \sum_{i=1}^n \frac{s}{s_i}x_i$.

By definition of the heat kernel,

\begin{align}\label{eqprf:NFaux:1}
\prod_{i=1}^n \rfy{p}_{s_i}(x_i-z)
&= \left(\prod_{i=1}^n (2\pi s_i)^{-d / 2}\right) \exp \left( -\frac{1}{2} \sum_{i=1}^n \frac{(x_i-z)^2}{s_i} \right) ,
\end{align}

\begin{align}\begin{split}\label{eqprf:NFaux:2}
\sum_{i=1}^n \frac{(x_i-z)^2}{s_i}
= \frac{z^2}{s} -2 \frac{\interp z}{s}+ \sum_{i=1}^n \frac{x_i^2}{s_i}
&= \frac{1}{s}(z-\interp)^2 -\frac{1}{s}\interp^2 + \sum_{i=1}^n \frac{x_i^2}{s_i}\\
&= \frac{1}{s}(z-\interp)^2 + \sum_{i=1}^n \frac{(x_i-\interp)^2}{s_i},
\end{split}\end{align}

where in the last step we used that

\begin{align*}
\sum_{i=1}^n \frac{(x_i-\interp)^2}{s_i} = \sum_{i=1}^n \frac{x_i^2}{s_i} - 2 \interp \sum_{i=1}^n \frac{x_i}{s_i} + \frac{1}{s} \interp^2 = \sum_{i=1}^n \frac{x_i^2}{s_i} - \frac{1}{s}\interp^2.
\end{align*}

Plugging <ref> back into <ref> finishes the proof.

For a non-trivial forest $F\in \mathbb{F}$ and $\boldsymbol x \in \rfy{E}^{\lf(F)}_\circ $, $\tau \in \dct(F)$,

\begin{align}\label{eq:NFaux}
\int_{\rfy{\dcs}(F)} \rfy{\fsp}(\xi \,\vert\,\tau,\boldsymbol x) \diff \xi = \prod_{v\in \nd^\circ (F)} \frac{1}{\rfy{p}_{r_v}(0)} \prod_{u\in \ch_F(v)} \rfy{p}_{r_u + \tau_v - \tau_u} (\interp_v -\interp_u),
\end{align}

where $\interp_u = \boldsymbol x_u$ and $r_u = 0$ for $u\in \lf(F)$ and, inductively for $v\in \nd^\circ (F)$,
\[
\frac{1}{r_v} \coloneqq \sum_{u\in \ch_F(v)} \frac{1}{r_u+\tau_v - \tau_u},\qquad \interp_v \coloneqq \sum_{u\in \ch_F(v)} \frac{r_v}{r_u+\tau_v - \tau_u} \interp_u.
\]
The following hold.

* If $v\in \nd^\circ (F)$ with $\ch(v) \subset \lf(F)$, then
\[
r_v = \frac{\tau_v}{|\ch_F(v)|},\qquad \interp_v = \frac{1}{|\ch_F(v)|} \sum_{u\in \ch_F(v)} \boldsymbol x_u.
\]

* There is $c = c(F) > 0$ such that $c \tau_v \le r_v < \tau_v$ for all $v\in \nd^\circ (F)$.

* If $v\in \nd^\circ (F)$ then
\[
\max_{u\in \ch(v)} |\interp_u-\interp_v| \ge \frac{1}{2}\max_{u,u'\in \ch(v)}|\interp_u-\interp_{u'}|.
\]

<Ref> follows by applying <ref> inductively from leaves to root, which is tedious but straightforward.

* is obvious from the definitions.

* If $v$ is not a leaf and $r_u \le \tau_u$ holds for all children, then

\begin{align*}
\frac{1}{r_v} \ge \sum_{u\in \ch_F(v)} \frac{1}{\tau_u + \tau_v - \tau_u} = \frac{|\ch_F(v)|}{\tau_v} \ge \frac{2}{\tau_v} > \frac{1}{\tau_v}.
\end{align*}

If $v$ is not a leaf and $r_u \ge c \tau_u$ holds for all children, then

\begin{align*}
\frac{1}{r_v} \le \sum_{u\in \ch_F(v)} \frac{1}{c \tau_u + \tau_v - \tau_u} \le \sum_{u\in \ch_F(v)} \frac{1}{c(\tau_u + \tau_v - \tau_u)} = \frac{|\ch_F(v)|}{c} \frac{1}{\tau_v}.
\end{align*}

* If $v\in \nd^\circ (F)$ and $u_1,u_2\in \ch(v)$, then
\[
|\interp_{u_1}-\interp_{u_2}| \le |\interp_{u_1} - \interp_v| + |\interp_v - \interp_{u_2}| \le 2 \max_{u\in \ch(v)}|\interp_u-\interp_v|.
\]

Fix a non-trivial forest $F=(\fpart^F_0, \ldots ,\fpart^F_m)\in \mathbb{F}$, and abbreviate $\ch = \ch_F$, $\pr = \pr_F$ and, for $\tau\in \dct(F)$ and $i\in [m]$, $\tau_i = \tau_{\fpart^F_i}$. Throughout the proof, $c,C > 0$ are constants that only depend on $F$, $d$, $\boldsymbol \lambda$, and $K$, and whose value may respectively decrease and increase from line to line. First let $d \ge 2$, then for $\boldsymbol x \in E^{\lf(F)}_\circ $ and $\tau\in \dct(F)$, using <ref>,

\begin{align}\label{eqprf:NFlem:4}
\int \rfy{\fsp}(\xi \,\vert\,\tau,\boldsymbol x) \diff \xi
%&\le C \prod_{v\in \nd^\circ (F)} r_v^{d / 2} \prod_{u\in \ch(v)} \rfy{p}_{r_u+\tau_v-\tau_u}(\interp_v-\interp_u)\\
&\le C \prod_{i=1}^m \underbrace{\smashoperator[r]{\prod_{v\in \fpart^F_i\setminus \fpart^F_{i-1}}}\, r_v^{d / 2} \prod_{u\in \ch(v)} \rfy{p}_{r_u+\tau_v-\tau_u}(\interp_v-\interp_u)}_{\eqqcolon G_i(\tau,\boldsymbol x)}.
\end{align}

Denote $G^{(i)}(\tau,\boldsymbol x) \coloneqq \prod_{j=i+1}^m G_j(\tau,\boldsymbol x)$, and ${\ftm}^{(i)}(\tau) \coloneqq \prod_{j=i+1}^{m} \lambda_{\fpart^F_{j-1},\fpart^F_j} \e^{-\lambda_{\fpart^F_j}(\tau_j-\tau_{j-1})}$ for $i\in \left\{ 0, \ldots ,m \right\} $, so ${\ftm}^{(0)}= \ftm$ and ${\ftm}^{(m)} \equiv 1$.

We show inductively that for $i\in \left\{ 1, \ldots ,m \right\} $ there exists $b_i \ge 0$ such that

\begin{align}\label{eqprf:NFlem:1}
\int {\ftm}^{(i)}(\tau) (1+\sqrt{\tau_m} )^K\sup_{\boldsymbol x\in A}G^{(i)}(\tau,\boldsymbol x) \diff \tau_{i+1} \ldots \diff \tau_m \le C \tau_1^{-b_i} (1 + \tau_i)^C.
\end{align}

For $i = m$ the LHS is equal to $(1+\sqrt{\tau_m} )^K \le 2^K(1 + \tau_m)^K$ and we can choose $b_m = 0$. Suppose we have proved the claim for some fixed $i\in \left\{ 2, \ldots ,m \right\} $, then by induction hypothesis

\begin{multline*}
\int {\ftm}^{(i-1)}(\tau) (1+\sqrt{\tau_m} )^K \sup_{\boldsymbol x\in A} G^{(i-1)}(\tau,\boldsymbol x) \diff \tau_i \ldots \diff \tau_m\\
\le C \tau_1^{-b_i} \int\limits_{\tau_{i-1}}^{\infty} \diff \tau_i\, (1 + \tau_i)^{C}\e^{-\lambda_i(\tau_i - \tau_{i-1})}\sup_{\boldsymbol x\in A}\,\smashoperator[r]{\prod_{v\in \fpart^F_i\setminus \fpart^F_{i-1}}}\, r_v^{d / 2}\prod_{u\in \ch(v)}\rfy{p}_{r_u+\tau_v-\tau_u}(\interp_v-\interp_u),
\end{multline*}

where we abbreviated $\lambda_i \coloneqq \lambda_{\fpart^F_i}$. We bound $r_v \le \tau_v = \tau_i$, and $\rfy{p}_{r_u+\tau_v-\tau_u}(\interp_v-\interp_u) \le C (r_u+\tau_v-\tau_u)^{-d / 2}$, and by <ref>,

\begin{equation}\label{eqprf:NFlem:3}
r_u + \tau_v - \tau_u \ge c\tau_u + \tau_v-\tau_u \ge c\tau_v \ge c\tau_1.
\end{equation}

Let $l$ be the total number of children of nodes $v\in \fpart^F_i \setminus \fpart^F_{i-1}$. Then our bound is no larger than $\tau_1^{-b_i}$ times

\begin{align*}
C \int\limits_{\tau_{i-1}}^{\infty}& \diff \tau_i \, (1 +\tau_i)^{C}\e^{-\lambda(\tau_i-\tau_{i-1})} \tau_i^{|\fpart^F_i\setminus \fpart^F_{i-1}|d / 2} (c\tau_1)^{-ld / 2}\\
&\le C \tau_1^{-ld / 2} \int_0^\infty \e^{-\lambda_i s}(1+\tau_{i-1} + s)^C \diff s\\
&\le C \tau_1^{-ld / 2} (1 + \tau_{i-1})^C,
\end{align*}

which completes the inductive step. Using <ref> with $i = 1$, recalling that $r_u = \tau_u = \tau_1$ for $u\in \lf(F)$, and putting $s \coloneqq \tau_1, b\coloneqq b_1, \lambda\coloneqq \lambda_1$, <ref> becomes

\begin{align*}
\int \ftm(\tau) & (1+\sqrt{\tau_m} )^K \sup_{\boldsymbol x\in A} \int \rfy{\fsp}(\xi \,\vert\,\tau,\boldsymbol x)\diff \xi\diff \tau\\
&\le C \int_0^\infty  s^{-b} (1+ s)^C \e^{-\lambda s} \sup_{\boldsymbol x\in A}\,\smashoperator[r]{\prod_{v\in \fpart^F_1\setminus \fpart^F_{0}}}\,  s^{d / 2}\prod_{u\in\ch(v)}\rfy{p}_{r_u+ s_v- s_u}(\interp_v-\interp_u)\diff s\\
&\le C \int_0^\infty  s^{-b} (1+ s)^C \e^{-\lambda s} \sup_{\boldsymbol x\in A}\prod_{v\in \fpart^F_1\setminus \fpart^F_{0}}\prod_{u\in\ch(v)}\rfy{p}_s(\interp_v-\interp_u)\diff s.
\end{align*}

Fix any $v_0\in \fpart^F_1\setminus \fpart^F_0$ and pick, for fixed $\boldsymbol x\in A$, $u_0\in \ch(v)$ for which $|\interp_{v_0}-\boldsymbol x_{u_0}|$ is largest, so that $|\boldsymbol x_u - \boldsymbol x_{u'}| \le 2 |\boldsymbol x_{u_0}-\interp_{v_0}|$ for all $u,u'\in \ch(v)$, in particular $2|\boldsymbol x_{u_0}-\interp_{v_0}| \ge \inf_{\boldsymbol x\in A} \min_{u\neq v}|\boldsymbol x_u-\boldsymbol x_v| \eqqcolon D$. Bound $\rfy{p}_s(\interp_v - \interp_u) \le C s^{-d / 2}$ if $(u,v) \neq (u_0,v_0)$, so we can further bound by

\begin{align}\label{eqprf:NFlem:2}
%N_F(\boldsymbol x)
C \int_0^\infty  s^{-b'} (1+ s)^C \e^{-\lambda  s} \e^{-c D^2 /  s } \diff  s \le C \left( 1 + D^{-2b'} \right) ,
\end{align}

for some $b' > b$, using <ref>.

Now let $d = 1$. The approach above fails when the first transition $\fpart^F_0\to\fpart^F_1$ is a binary merger of two identical particles, say $u$ and $u'$ with parent $v = u\cup u'$, in which case $\interp_v = \interp_u = \interp_{u'} = \boldsymbol x_u$, so $\rfy{p}_{\tau_1}(\interp_v-\interp_u) = \rfy{p}_{\tau_1}(\interp_v-\interp_{u'}) = (2\pi \tau_1)^{-d / 2}$, and the final integral (of the form <ref> but without the factor $\e^{-cD^2 / s}$) diverges. We split $A$ into a finite number of sets according to which pair of leaves are closest to each other (and potentially identical):
\[
A_{uv} \coloneqq \Big\{ \boldsymbol x\in A\colon |\boldsymbol x_u-\boldsymbol x_v| = \,\smashoperator[l]{\min_{\substack{u',v'\in \lf(F)\\ \text{distinct}}}}|\boldsymbol x_{u'}-\boldsymbol x_{v'}| \Big\},
\]
for distinct $u,v\in\lf(F)$. There might be overlap due to ties, but it will always be true that in $A_{uv}$ all particle locations are distinct except possibly $\boldsymbol x_u = \boldsymbol x_v$, and
\[
\forall \boldsymbol x\in A_{uv}\colon \rfy{\varepsilon_1}(\boldsymbol x) = \,\, \smashoperator[l]{\min_{\substack{u'\neq v'\in \lf(F)\\ \{u',v'\} \neq \{u,v\}}}} |\boldsymbol x_{u'}-\boldsymbol x_{v'}|.
\]
Since we can bound $\sup_{\boldsymbol x\in A} \le \sum_{u \neq v} \sup_{\boldsymbol x \in A_{uv}}$, it suffices to restrict our attention to $A_{u_0v_0}$ for fixed but arbitrary, distinct $u_0,v_0\in \lf(F)$. With a slight abuse of notation we denote $A = A_{u_0v_0}$ from now. If the first merge event is not a pure binary merge of $u_0$ and $v_0$, that is $\fpart^F_1\setminus \fpart^F_0 \neq \left\{ u_0\cup v_0 \right\} $, then we can copy the proof of the $d\ge 2$ case, and in the final step we choose $v\in \fpart^F_1\setminus \fpart^F_0$ with $v\neq u_0\cup v_0$, so there are $u,u'\in \ch(v)$ with $\left\{ u,u' \right\} \neq \left\{ u_0,v_0 \right\} $ and thus $|\boldsymbol x_u-\boldsymbol x_{u'}| \ge \rfy{\varepsilon_1}(\boldsymbol x)$ for all $\boldsymbol x\in A$. Now suppose the first event is a binary merge of $u_0$ and $v_0$, that is $\fpart^F_1\setminus \fpart^F_0$ is a singleton set containing their parent $w_0\coloneqq u_0\cup v_0$, so

\begin{align}\label{eqprf:NFlem:5}
G_1(\tau,\boldsymbol x)
%&= \prod_{v\in \fpart^F_1\setminus \fpart^F_0} r_v^{d / 2} \prod_{u\in \ch(v)} \rfy{p}_{\tau_1}(\interp_u - \interp_v) \\
&= (\tau_1 / 2)^{1 / 2} \rfy{p}_{\tau_1}(\boldsymbol x_{u_0} - \interp_{w_0}) \rfy{p}_{\tau_1}(\boldsymbol x_{v_0}-\interp_{w_0})
\le C \tau_1^{-1 / 2}
\end{align}

for all $\boldsymbol x\in A$. Then as before, we can prove inductively that for $i\in \left\{ 2, \ldots ,m \right\} $ there is $b_i \ge 0$ with

\begin{equation}\label{eqprf:NFlem:6}
\int {\ftm}^{(i)}(\tau) (1+\sqrt{\tau_m} )^K \sup_{\boldsymbol x \in A} G^{(i)}(\tau,\boldsymbol x) \diff \tau_{i+1}\ldots \diff \tau_m \le C \tau_{2}^{-b_i} (1 + \tau_i)^C,
\end{equation}

where we only need to replace the lower bound $c\tau_1$ by $c\tau_2$ in <ref>. Let $s\coloneqq \tau_1$ and $r\coloneqq \tau_2-\tau_1$, then plugging <ref> into <ref> gives

\begin{align}\begin{split}\label{eqprf:NFlem:7}
\int \ftm(\tau) & \sup_{\boldsymbol x\in A} \int \rfy{\fsp}(\xi \,\vert\,\tau,\boldsymbol x) \diff \xi \diff \tau\\
%&\le C \int_0^\infty \diff s \int_0^\infty \diff r (s+r)^{-b}(1+s+r)^C \e^{-\lambda_1 s}\e^{-\lambda_2 r} \sup_{\boldsymbol x\in A} G_1(\tau,\boldsymbol x) G_2(\tau,\boldsymbol x)\\
&\le C \int_0^\infty \diff s\, \e^{-\lambda_1 s}s^{-1 / 2} \int_0^\infty \diff r\, (s+r)^{-b_2}(1+s+r)^C \e^{-\lambda_2 r} \sup_{\boldsymbol x\in A} G_2(\tau,\boldsymbol x).
\end{split}\end{align}

If the second merge involves at least two leaves, say $w\in \fpart^F_2\setminus \fpart^F_1$ with $u,v\in \ch(w)\cap \lf(F)$, then necessarily $\left\{ u,v \right\} \cap \left\{ u_0,v_0 \right\} = \emptyset $, so $\max_{u' \in \ch(w)}|\boldsymbol x_{u'}-\interp_{w}| \ge \frac{1}{2}|\boldsymbol x_u-\boldsymbol x_v| \ge \rfy{\varepsilon_1}(\boldsymbol x)$ for all $\boldsymbol x\in A$ and $\tau$, so we can bound
\[
\sup_{\boldsymbol x\in A} G_2(\tau,\boldsymbol x) \le C (1+\tau_2)^C \tau_2^{-b_1} \e^{-cD^2 / \tau_2}
\]
for some $b_1 \ge 0$, where $D = \inf_{\boldsymbol x\in A}\rfy{\varepsilon_1}(\boldsymbol x)$. With $b\coloneqq b_1+b_2$ we can further bound <ref> by

\begin{align}\begin{split}\label{eqprf:NFlem:8}
C \int_0^\infty & \diff s\, \e^{-\lambda_1 s} s^{-1 / 2} \int_0^\infty \diff r\, \underbrace{(s+r)^{-b} \e^{-cD^2 / (s+r)}}_{\le C D^{-2b} \, \forall s,r > 0} (1+s+r)^C \e^{-\lambda_2 r}\\[3pt]
&\le C D^{-2b} \int_0^\infty \diff s\, \e^{-\lambda_1 s}s^{-1 / 2} \int_0^\infty \diff r\, (1+s+r)^C \e^{-\lambda_2 r}\\
&\le C D^{-2b} \int_0^\infty \e^{-\lambda_1 s}s^{-1 / 2} (1+s)^C \diff s\\
&\le C D^{-2b}.
\end{split}\end{align}

If the second merge involves at most one additional leaf, then it must be a binary merge of $w_0$ and a leaf $u \not\in \left\{ u_0,v_0 \right\} $ with parent $w_1 = u \cup w_0$. Then $|\boldsymbol x_u - \interp_{w_1}| \ge \frac{1}{2} |\boldsymbol x_u - \interp_{w_0}| \ge c |\boldsymbol x_u - \boldsymbol x_{u_0}| \ge c \varepsilon_1(\boldsymbol x)$ for all $\boldsymbol x\in A$ and $\tau$, where we used <ref> in the second step. Then we can bound

\begin{align*}
G_2(\tau,\boldsymbol x) = \underbrace{r_{w_1}^{1 / 2}}_{\le \tau_2^{1 / 2}} \underbrace{\rfy{p}_{\tau_2 - \tau_1 / 2} (\interp_{w_0}-\interp_{w_1})}_{\le C \tau_2^{-1 / 2}}\underbrace{\rfy{p}_{\tau_2}(\boldsymbol x_u - \interp_{w_1})}_{\le C\tau_2^{-1 / 2}\e^{-cD^2 / \tau_2}}
%&\le C \rfy{p}_{\tau_2}(\boldsymbol x_u-\interp_{w_1})
\le C \tau_2^{-1 / 2} \e^{-c D^2 / \tau_2},
\end{align*}

and finish the proof as in <ref>.

The following lemma finishes the proof of <ref> together with <ref>.

For every $\fpart\in \mathcal{P}$, $\boldsymbol x\in E^\fpart_\circ $, and $\vec{\boldsymbol k}\colon \fpart \to \Z^d$,
\[
\rfy{\varepsilon_d}(\boldsymbol x +  \vec{\boldsymbol k}) \ge \varepsilon_d(\boldsymbol x).
\]

If $u,v\in \fpart$, then

\begin{align*}
| \boldsymbol x_u +  \vec{\boldsymbol k}_u - (\boldsymbol x_v +  \vec{\boldsymbol k}_v)| \ge \inf \{|\boldsymbol x_u - \boldsymbol x_v +  \vec{\ell}| \colon :\vec{\ell}\in \Z^d\} = \rho(\boldsymbol x_u,\boldsymbol x_v).
\end{align*}

With that, the claim follows directly from the definitions of $\varepsilon_d$ and $\rfy{\varepsilon_d}$.

We finish by proving some technical lemmas used in the proof of <ref>.

For $a,b,\lambda,y> 0$,

\begin{align*}
\int_0^\infty \e^{-\lambda s - y^2 / s} (1+s)^a s^{-b}\diff s \le C \left( 1 + y^{-2b} \right) ,
\end{align*}

where $C = C(a,b,\lambda) > 0$.

First note that
\[
\sup_{s > 0}\left( s^{-b} \e^{-y^2 / s}\right) = y^{-2b} \sup_{s > 0}\left( s^{-b} \e^{-1 / s} \right) = C y^{-2b},
\]

\begin{align*}
\int_0^\infty \e^{-\lambda s - y^2 / s} (1+s)^a s^{-b} \diff s
&\le 2^a \int_0^1 s^{-b}\e^{-y^2 / s} \diff s + \int_1^\infty \e^{-\lambda s} (1+s)^a \diff s \le C y^{-2b} + C.
\end{align*}

If $x_1, \ldots ,x_n\in \R^d$ and $|x_1-x_2| \le |x_i-x_j|$ for all distinct $i,j\in [n]$, then
\[
|x_i - \frac{x_1+x_2}{2}| \ge \frac{3}{\sqrt{2} } |x_i - x_1|
\]
for all $i\in \left\{ 3, \ldots ,n \right\} $.

Put $a \coloneqq |x_1-x_2|$ and $\overline{x} \coloneqq (x_1+x_2) / 2$, and let $i\in \left\{ 3, \ldots ,n \right\} $. Then $x_i \not\in B(x_1,a) \cup B(x_2,a)$, which means that $|x_i - \overline{x}|$ is minimised if $x_i \in B(x_1,a) \cap B(x_2,a)$, in which case $|x_i - x_1| = |x_i - x_2| = |x_1-x_2| = a$, so by Pythagoras $|x_i - \overline{x}| = \frac{3}{\sqrt{2} } a$.

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I would like to thank my supervisor Alison Mary Etheridge for numerous helpful discussions, valuable feedback, guidance, and encouragement. I would further like to thank Michal Bassan and Félix Foutel-Rodier for many insightful conversations.

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