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A Grid-Rate Condition for Valid Uniform Inference
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The practice of uniform inference on continuous functionals \(F(\bm{x})\), \(\bm{x} \in \mathcal{X} \subset \mathbb{R}^d\), \(d<\infty\), such as distribution functions chernozhukov-fernandez-melly-2013inference or quantile functions callaway-li-2019, via confidence bands, typically relies on estimating the underlying continuous object only at a finite collection of grid points \( \mathcal{X}^{L} \subset \mathcal{X} \). At first sight, this creates a tension between the inherently continuous nature of the target functional and its discrete implementation. In practice, however, this tension is reconciled by interpolating across the grid points: the estimates and associated “uniform” confidence bands are displayed as continuous curves obtained by linearly connecting the nodes. As emphasised by epperson-2013-introduction, “almost all graphs produced by computers are actually the result of something called piecewise linear interpolation...” Exploiting linear interpolation as an approximation device ensures the estimators of the linear interpolant $F_L(\cdot)$ (the estimand of $F(\cdot)$ de facto) and its confidence bands retain the continuity property even in sample (on $\bm{x} \in \mathcal{X} $). This paper derives a simple grid-growth rate, $L_n = \omega(r_n^{1/4})$, under which the resulting confidence bands are asymptotically valid.\footnote{$r_n=n$ for functionals whose estimators are pointwise consistent at the usual $\sqrt{n}$ parametric rate.} The analysis thus provides a theoretically grounded rule for choosing \(L_n\). While imai-2025-accuracy works in a non-Donsker non-parametric setting and delivers a case-specific coverage bound to be inverted by the practitioner, the Donsker framework in the current paper, which covers semiparametric estimators such as quantile and distribution regression, admits a clean rate-transparent condition: $L_n = \omega(n^{1/4})$, with $r_n = n$ pinned down by the $\sqrt{n}$-rate.
Obtaining a grid-growth rate for asymptotically valid uniform inference matters because a critical trade-off emerges. On the one hand, a sparse grid with slow-growing or fixed $L_n$ is computationally cheap but may fail to capture the function's variability between points, leading to confidence bands that are asymptotically invalid (i.e., their true coverage probability falls below the nominal level). On the other hand, a dense grid (fast-growing $L_n^d$ especially when $d>1$) is more likely to yield valid uniform coverage but can be computationally prohibitive and may introduce finite-sample issues, e.g., an increased incidence of quantile crossing in the quantile regression (QR) framework. The existing literature often acknowledges that the number of nodes $L_n$ ought to grow at a rate sufficient for the discrete maximum to converge to its continuous supremum, but the corresponding grid-growth rule is often left implicit. For example, chernozhukov-fernandez-melly-2013inference recommends $L_n = \omega(n^{1/2}) $ and kim-wooldridge-2024-difference suggests a default of $L_n=99$. This paper provides a clear guideline for practitioners, ensuring their computational methods are not only efficient but also statistically sound. It bridges the gap between theoretical continuity and practical discretisation by leveraging the (piecewise) continuity of linear interpolation.
The rest of the paper proceeds as follows. (ref) formalises the linear and multilinear interpolation framework and derives the deterministic interpolation-error bound. (ref) shows that, under the grid-growth condition, the interpolated empirical process inherits the weak limit of the underlying process and the resulting interpolated confidence bands have valid uniform coverage. The section also records the computational algorithm, discusses Hadamard differentiable transformations, and summarises Monte Carlo evidence, with full simulation details reported in (ref).
\paragraph{Notation:} For any two sequences with $\min\{L_n, K_n\} \to \infty$, $L_n = \omega(K_n)$ means $K_n / L_n \to 0$ as $n \to \infty$. Whenever necessary, quantities such as $L$ are indexed by $n$ (e.g., $L_n$) to emphasise their dependence on the sample size $n$. Let $C$ denote a positive constant whose value may vary across occurrences. For real numbers $a$ and $b$, define $a \wedge b := \min\{a,b\}$ and $a \vee b := \max\{a,b\}$. For a function $F(\mathbf{x})$, where $\mathbf{x} \in \mathcal{X}$ and $F \in \mathcal{F}$, the domain $\mathcal{X}$ is a compact metric space in $\mathbb{R}^d$, and $\mathcal{F} \subset \ell^\infty(\mathcal{X})$ is a class of functional parameters of interest, such as distribution functions (DF), quantile functions (QF), and expectile functions. For a vector $\bm{x} \in \mathbb{R}^d $, $ \displaystyle \lVert \bm{x} \rVert_\infty = \max_{1\leq k \leq d} |x_k|$ and for a function $f \in \mathcal{F} $, $ \displaystyle \lVert f \rVert_\infty = \sup_{\bm{x} \in \mathcal{X}} |F(\bm{x})|$. Finally, let $\mathbb{V}[W]$ denote the variance of a random variable $W$.
The framework is stated for an $\mathbb{R}^d$-indexed empirical process. The univariate case is recovered by setting $d=1$.
It is useful first to discuss the construction in the special case $d=1$. For tractability, consider an equally spaced grid of $L$ nodes over the domain $\mathcal{X} := [\underline{\mathrm{x}}, \, \bar{\mathrm{x}}]$: \[ \mathcal{X}^L := \{ \mathrm{x}_1, \mathrm{x}_2, \ldots, \mathrm{x}_L\}, \quad \text{where} \quad \mathrm{x}_{\ell+1} - \mathrm{x}_\ell = \varepsilon \text{ for } \ell = 1, \ldots, L-1, \] with mesh \[ \varepsilon = \frac{\bar{\mathrm{x}} - \underline{\mathrm{x}}}{L-1}. \] The endpoints are fixed as $\mathrm{x}_1 = \underline{\mathrm{x}}$ and $\mathrm{x}_L = \bar{\mathrm{x}}$, regardless of the choice of $L$. Denote the estimator of $F(\cdot)$ at $x \in \mathcal{X} $ by $ \widehat{F}(x) $. The inferential goal is uniform inference on $F(\cdot)$ over $\mathcal{X}$, while estimation occurs only on the discrete set of nodes $\mathcal{X}^L \subset \mathcal{X} $. Linear interpolation extends the grid-point estimates to a continuous estimator on $\mathcal{X}$.
For $x \in \mathcal{X} $, the linear interpolant of $F(x)$ is
with the convention $\ell_L(\mathrm{x}_L)=L-1$, and let $\widehat{F}_L(\cdot)$ be the same linear interpolant built from the estimated values $ \big\{ \widehat F(\mathrm{x}_1), \ldots, \widehat F(\mathrm{x}_L) \big\} $. $\widehat{F}_L(x), \, x\in\mathcal{X} $ is the estimator de facto of $ F(x), \, x\in \mathcal{X} $ from commonly displayed results. Linearly interpolating $F$ with $F_L$ on $\mathcal{X}$ incurs an interpolation error $R_n(\cdot)$:
where $(r_n)_{n\ge 1}$ is a non-stochastic sequence such that, for each $x \in \mathcal{X}$, the centred and scaled estimator $Z_n(x):=r_n^{1/2}\big(\widehat F - F\big)(x)$ converges in distribution as $n \to \infty$, \(\displaystyle Z_n^L(x):= \big(1-\delta_L(x)\big)Z_n(\mathrm{x}_{\ell_L(x)}) + \delta_L(x)Z_n(\mathrm{x}_{\ell_L(x)+1}) \) is the interpolated empirical process, and $R_n(x)$ is the interpolation error. The formal multivariate process is stated below, with metric \(\mathrm{d}(\bm{x},\bm{y})=\lVert \bm{x}-\bm{y}\rVert_{\infty}\). $r_n = n $ if pointwise convergence in distribution occurs at the usual $\sqrt{n}$ parametric rate.\footnote{Here \(r_n\) records the effective pointwise convergence rate: \(r_n=n\) in the usual \(\sqrt n\)-regular case, but may differ when the effective sample size is not \(n\), as under some clustered or dependent designs.}
A tractable interpolation device in the multivariate setting is multilinear interpolation, with linear interpolation in the preceding subsection as the case $d=1$. Multilinear interpolation extends linear, bilinear, and trilinear interpolation to $d$-dimensional axis-aligned hyper-rectangles: \[ \mathcal{X} := \prod_{k=1}^d [\underline{\mathrm{x}}_k,\, \bar{\mathrm{x}}_k], \] i.e., the Cartesian product of coordinate-wise intervals in $\mathbb{R}^d$. For multilinear interpolation on hyper-rectangles, $F$ is evaluated at the $2^d$ vertices of a cell \[ \mathfrak{C}_{\bm{\ell}} = \prod_{k=1}^d [\mathrm{x}_{\ell_k, k},\, \mathrm{x}_{\ell_k+1, k}], \] indexed by $\bm{\ell} = (\ell_1,\ldots,\ell_d) \in \{1,\ldots,L-1\}^d $. On each axis $k\in \{1,\ldots,d\}$, there are $L$ equally spaced nodes with mesh \( \varepsilon_k = \frac{\bar{\mathrm{x}}_{k} - \underline{\mathrm{x}}_{k}}{L-1}, \qquad k = 1,\dots,d, \) where $x_{1,k}$ and $x_{L,k}$ are set to the endpoints $\underline{\mathrm{x}}_k$ and $\bar{\mathrm{x}}_k$, respectively, for each $ k \in \{1,\ldots,d\} $. Then observe that the index set $ \mathcal{X} $ is the union of all cells: \[ \mathcal{X} = \bigcup_{\bm{\ell} \in \{1,\ldots,L-1\}^d} \mathfrak{C}_{\bm{\ell}} = \prod_{k=1}^d [\underline{\mathrm{x}}_k,\, \bar{\mathrm{x}}_k]. \]
For \(\bm{x}=(x_1,\ldots,x_d)\in\mathcal X\), let \( \bm\ell_L(\bm{x}) = \big(\ell_{L,1}(x_1),\ldots,\ell_{L,d}(x_d)\big) \) denote the cell index of \(\bm{x}\), where \[ \ell_{L,k}(x_k) := \sum_{\ell=1}^{L-1} \ell\,\mathbf 1\{x_k\in[\mathrm{x}_{\ell,k},\mathrm{x}_{\ell+1,k})\}, \] with the convention \(\ell_{L,k}(\mathrm{x}_{L,k})=L-1\). Define the local interpolation coordinates \[ \delta_{L,k}(\bm{x}) := \frac{x_k-\mathrm{x}_{\ell_{L,k}(x_k),k}} {\varepsilon_k}, \qquad \varepsilon_k = \mathrm{x}_{\ell_{L,k}(x_k)+1,k} - \mathrm{x}_{\ell_{L,k}(x_k),k}. \] The \(d\)-linear interpolant is then given by
where \[ \mathbf{x}_L(\bm{x};\bm{\iota}) = \Big( \mathrm{x}_{\ell_{L,1}(x_1)+\iota_1,1}, \dots, \mathrm{x}_{\ell_{L,d}(x_d)+\iota_d,d} \Big), \qquad \bm{\iota}=(\iota_1,\ldots,\iota_d)\in\{0,1\}^d \] This enumerates the $2^d$ vertices of the cell containing $\bm{x}$; see, e.g., weiser-1988-note. The interpolant \(F_L(\bm{x})\) is a convex combination of the vertex values \( F\big(\mathbf{x}_L(\bm{x};\bm{\iota})\big) \), since the weights \( w_L(\bm{x};\bm{\iota}) \), \(\bm{\iota}\in\{0,1\}^d\), are non-negative and satisfy \[ \sum_{\bm{\iota}\in\{0,1\}^d} w_L(\bm{x};\bm{\iota}) = 1. \]
Moreover, for any $\bm{x}\in\mathcal{X}^L$, \( F_L\big(\mathbf{x}_L(\bm{x};\bm{\iota})\big) = F\big(\mathbf{x}_L(\bm{x};\bm{\iota})\big) \quad \text{for all } \bm{\iota}\in\{0,1\}^d \).
For the multivariate interpolant defined above, set the resulting interpolation error as \[ R_n(\bm{x}):=r_n^{1/2}\big(F_L-F\big)(\bm{x}), \qquad \bm{x}\in\mathcal{X}. \] The following conditions on $\mathcal{X}$ and $F$ are useful in characterising this deterministic interpolation error.
The following result characterises the interpolation error and the growth rate of $L$ in $n$ needed to make it asymptotically negligible.
(ref) illustrates the unscaled interpolation error from linearly interpolating the function \(F(x)=\cos(5x)+\sqrt{x}\). In this one-dimensional illustration, the shaded area corresponds to the integrated absolute interpolation error \( \int_\mathcal{X} |(F_L-F)(x)|\,dx \), while \(m.x\) denotes the maximum absolute interpolation error \( \sup_{x\in\mathcal{X}}|(F_L-F)(x)| \). Both measures shrink as \(L\) increases.
Having isolated the deterministic interpolation error \(R_n(\bm{x})=r_n^{1/2}\{F_L(\bm{x})-F(\bm{x})\}\), this section studies its effect on inference for the interpolated estimator \(\widehat F_L\). The underlying centred empirical process is \(Z_n:=r_n^{1/2}(\widehat F-F)\). The section first shows that interpolation preserves the weak limit of this process under the grid-growth condition, then records the computational algorithm, and finally proves that the resulting interpolated confidence bands have the same asymptotic coverage as the corresponding continuous-index benchmark.
The stochastic component of the analysis is formulated in a Donsker framework.\footnote{The interested reader is referred to imai-2025-accuracy for a non-parametric treatment in a non-Donsker framework.}
(ref) gives \(Z_n\rightsquigarrow Z\) in \(\ell^\infty(\mathcal{X})\). In particular, it implies the finite-dimensional convergence and stochastic equicontinuity conditions, with respect to the Euclidean index metric on \(\mathcal{X}\), that are usually used to establish weak convergence of an \(\mathbb{R}^d\)-indexed empirical process, e.g., Kosorok-2008-introduction. The next result shows that the interpolated process inherits this weak limit once the deterministic interpolation error is asymptotically negligible.
(ref) shows that under the stated rate condition on the grid density over $\mathcal{X}$, the interpolated process converges weakly to the same limit as the infeasible empirical process. The univariate result follows immediately by taking $d=1$.
The algorithm follows standard sup-$t$ procedures for uniform confidence bands, e.g., chernozhukov-fernandez-melly-2013inference and meier-2025, but makes the interpolation step (as a final step) explicit. This records the operation that is otherwise implicit when estimates computed on a finite grid are plotted as continuous curves.
Steps 2 and 3 can be implemented using weighted-bootstrap draws. Let \(\xi_1,\ldots,\xi_n\) be non-negative bootstrap weights with mean one and variance one, independent of the data. For example, a grid-point scale estimate \`a la chernozhukov-fernandez-melly-2013inference is
where \(Q_F^*(u;\mathbf{x})\) denotes the \(u\)-th quantile of \(\{r_n^{1/2}(\widehat F^{*(b)}-\widehat F)(\mathbf{x}):1\leq b\leq B\}\) and $\Phi^{-1}(\cdot)$ denotes the quantile function of the standard normal. The output of (ref) is therefore a function-valued estimate and a function-valued band, both continuous on $\mathcal{X}$, obtained by interpolating the grid-point estimate and the grid-point bands.
Since $ \widehat{F}_L $ is the estimator of $F$ de facto, the inference procedure is completed with $\sup$-t confidence bands (CB) on $F$ using $ \widehat{F}_L $, which is continuous in $\bm{x}$ in-sample. This subsection focuses on the multivariate case, which generalises the univariate case. Define the infeasible confidence band by \[ \widetilde{\mathcal{C}}_{n,1-\alpha}(\bm{x}) = \left[ \widehat{F}(\bm{x}) \ - \ \widehat{\sigma}(\bm{x})\,\frac{t_{n,1-\alpha}}{r_n^{1/2}}, \ \widehat{F}(\bm{x}) \ + \ \widehat{\sigma}(\bm{x})\,\frac{t_{n,1-\alpha}}{r_n^{1/2}} \right], \quad \bm{x} \in \mathcal{X} \] where $ \widehat{\sigma}(\bm{x}) $ is the robust bootstrap standard-error function and $t_{n,1-\alpha}$ is the conditional bootstrap \((1-\alpha)\)-quantile of \( \displaystyle \sup_{\bm{x}\in\mathcal{X}} \frac{|Z_n^*(\bm{x})|}{\widehat{\sigma}(\bm{x})} \). The confidence bands $\widetilde{\mathcal{C}}_{n,1-\alpha}(\bm{x}), \, \bm{x}\in\mathcal{X} $ are infeasible because they require computation over a continuum of indices $ \bm{x}\in\mathcal{X} $. Interpolation of the feasible simultaneous confidence bands on the tensor grid $\mathcal{X}^L$ instead yields feasible continuous bands $\mathcal{C}_{n,1-\alpha}^L(\bm{x}), \bm{x}\in\mathcal{X}$ in-sample.
Following (ref), first construct the confidence bands simultaneously on the grid: \[ \mathcal C_{n,1-\alpha}(\mathbf{x}) = \left[ \widehat F(\mathbf{x})-\widehat\sigma(\mathbf{x})\frac{t_{n,1-\alpha}^L}{r_n^{1/2}}, \ \widehat F(\mathbf{x})+\widehat\sigma(\mathbf{x})\frac{t_{n,1-\alpha}^L}{r_n^{1/2}} \right] =: \big[\widehat{\mathcal L}(\mathbf{x}),\widehat{\mathcal U}(\mathbf{x})\big], \qquad \mathbf{x}\in\mathcal{X}^L. \] Next, for \(\bm{x}\in\mathcal{X}\), define the interpolated lower and upper endpoints by \[ \widehat{\mathcal L}_L(\bm{x}) = \sum_{\bm{\iota}\in\{0,1\}^d} w_L(\bm{x};\bm{\iota})\,\widehat{\mathcal L}\big(\mathbf{x}_L(\bm{x};\bm{\iota})\big), \qquad \widehat{\mathcal U}_L(\bm{x}) = \sum_{\bm{\iota}\in\{0,1\}^d} w_L(\bm{x};\bm{\iota})\,\widehat{\mathcal U}\big(\mathbf{x}_L(\bm{x};\bm{\iota})\big). \] The feasible interpolated confidence band is then \[ \mathcal{C}_{n,1-\alpha}^L(\bm{x}) = \big[\widehat{\mathcal L}_L(\bm{x}),\widehat{\mathcal U}_L(\bm{x})\big], \quad \bm{x} \in \mathcal{X}. \] Equivalently, since the same critical value is used at every grid point, define the interpolant of the grid-point standard error by
Then \[ \mathcal{C}_{n,1-\alpha}^L(\bm{x}) = \left[ \widehat{F}_L(\bm{x}) \ - \ \widehat{\sigma}_L(\bm{x})\,\frac{t_{n,1-\alpha}^L}{r_n^{1/2}}, \ \widehat{F}_L(\bm{x}) \ + \ \widehat{\sigma}_L(\bm{x})\,\frac{t_{n,1-\alpha}^L}{r_n^{1/2}} \right], \quad \bm{x} \in \mathcal{X}. \]
In order to examine the asymptotic validity of the interpolated confidence bands $\mathcal{C}_{n,1-\alpha}^L$, the following condition imposes a weighted-bootstrap approximation and robust studentisation used to compute the sup-\(t\) critical value.
Since \(\Psi\) is \(P\)-Donsker by (ref), the conditional multiplier central limit theorem yields \[ \mathbb G_n^*\psi_{\cdot} \rightsquigarrow_{\widehat{\mathbb{P}}} Z \qquad\text{in }\ell^\infty(\mathcal{X}), \] see, e.g., Kosorok-2008-introduction. Together with the bootstrap expansion in (ref), this gives \[ Z_n^* \rightsquigarrow_{\widehat{\mathbb{P}}} Z \qquad\text{in }\ell^\infty(\mathcal{X}). \] By chernozhukov-fernandez-melly-2013inference, this conditional weak convergence implies \[ \sup_{\bm{x}\in\mathcal{X}}\big|\widehat\sigma(\bm{x})-\sigma(\bm{x})\big|=o_p(1), \qquad W_n^*:=Z_n^*/\widehat\sigma \rightsquigarrow_{\widehat{\mathbb{P}}} W:=Z/\sigma \quad\text{in }\ell^\infty(\mathcal{X}). \] Let \(t_{1-\alpha}\) be the \((1-\alpha)\)-quantile of \(T:=\sup_{\bm{x}\in\mathcal{X}}|W(\bm{x})|\). Then \(T\) has a continuous distribution at \(t_{1-\alpha}\), \(t_{n,1-\alpha}\to_p t_{1-\alpha}\), and \[ \lim_{n\rightarrow \infty}\mathbb{P}\Big[F(\bm{x}) \in \widetilde{\mathcal{C}}_{n,1-\alpha}(\bm{x}) \ \forall \ \bm{x}\in\mathcal{X} \Big] = (1-\alpha). \] Thus, after (ref), the remaining task in (ref) is to show that interpolation does not alter the band event asymptotically under the rate condition $ L_n=\omega(r_n^{1/4}) $.
The following result shows that the in-sample continuous uniform bands based on the interpolated process, which emanate from the usual practice of displaying bands computed on a finite tensor grid $\mathcal{X}$ as continuous curves, have asymptotically valid coverage under the rate condition.
A main takeaway of the above result is that the usually displayed interpolated continuous curves as confidence bands are asymptotically valid under the rate condition $ L_n=\omega(r_n^{1/4}) $. This fills an important gap in the expanding literature on uniform inference with estimators whose associated function classes are Donsker.
Although the preceding results apply directly to \(F\), they extend to transformations of the form \(\phi\circ F\). Specifically, suppose \(\phi\) is Hadamard differentiable at \(F\), tangentially to $\ell^{\infty}(\mathcal{X})$, with derivative $\dot{\phi}_F$. Then, by standard functional delta-method arguments, e.g., van-der-vaart-2000, \( \phi(F_L) - \phi(F) = \dot{\phi}_F(F_L - F) + o\!\big(\lVert F_L - F \rVert_{\infty}\big). \) Consequently, the present analysis extends directly to such transformations of $F$.
As a leading example, let $\phi$ denote the quantile map, and let $F_X$ be a strictly increasing and continuously differentiable distribution function with derivative $f_X$ such that $\displaystyle \inf_{\bm{x}\in\mathcal{X}} f_X(x)>0$. Fix $\tau\in(0,1)$ and write $x:=F_X^{-1}(\tau)$. Then under the rate condition $L_n=\omega(r_n^{1/4})$,
where $\widehat{Z}_n^L(x):=r_n^{1/2}\big(\widehat{F}_{X,L} - F_X\big)(x)$. It follows under the conditions of (ref) and the continuous mapping theorem that \[ r_n^{1/2}\big(\widehat{F}_{X,L}^{-1} - F_X^{-1}\big)(\tau) \ \rightsquigarrow\ \frac{1}{f_X(x)}\,Z(x) \qquad\text{in } \ell^{\infty}(\mathcal{X}). \]
(ref) in the appendix reports Monte Carlo evidence for the interpolated confidence band construction in (ref). The simulations use two-period difference-in-differences designs with univariate and bivariate continuously distributed outcomes. In each design, the procedure is applied to two target parameters: the Distribution Treatment Effect on the Treated, \(F^{\mathrm{DTT}}\), and the (counterfactual) distribution function of the untreated potential outcomes for the treated group, \(F^{\mathrm{CF}}\). The DGP sets the treatment effect to zero, so \(F^{\mathrm{DTT}}\equiv0\), while \(F^{\mathrm{CF}}\) is non-zero and known from the simulation design.
For each target \(m\in\{\mathrm{DTT},\mathrm{CF}\}\), the estimator is computed on a finite grid \(\mathcal{X}_m^L\), the sup-\(t\) critical value is computed on that grid, and the resulting estimator and confidence band are interpolated over the evaluation region \(\mathcal{X}_m\). The object-specific grid size follows \(L_{m,n}=\lceil a\,(\bar y_{m,1}-\underline y_{m,1}) n^b\rceil\vee2\), with \((a,b)\in\{1,2,4\}\times\{0.30,0.35,0.40\}\), allowing the simulations to examine both sample-size behaviour and sensitivity to grid density.
The results are reassuring for the proposed implementation. Across the univariate and bivariate designs, the continuous \(\mathcal{L}_2\) error of the interpolated estimator declines steadily with \(n\) for both \(F^{\mathrm{DTT}}\) and \(F^{\mathrm{CF}}\). Uniform coverage is close to nominal overall, with mild conservativeness in parts of the bivariate DTT design and only limited sensitivity to the precise grid rule once the grid is moderately dense. In particular, choices such as \(L_{m,n}\approx n^{0.3}\), which lie above the \(n^{1/4}\) threshold, perform well in the simulations while remaining computationally less taxing than substantially finer tensor grids. The dominant improvement comes from increasing \(n\), rather than from fine-tuning \((a,b)\), supporting the practical use of the interpolated bands described above.
This paper provides a theoretical basis for choosing the grid density over the index set $\mathcal{X}$ when conducting uniform inference on a continuous target functional $F$. The starting point is the practical observation that feasible simultaneous confidence bands are computed on a finite tensor grid but are typically reported as continuous curves by linearly interpolating between grid points. Treating this interpolation step as part of the statistical procedure, rather than as a purely graphical convention, makes the approximation error explicit.
Under twice continuous differentiability and weak convergence of the underlying empirical process, the paper shows that the interpolation error is asymptotically negligible whenever $L_n = \omega(r_n^{1/4})$. This rate condition is sufficient for the interpolated estimator to inherit the weak limit of the grid-point process and for the resulting interpolated confidence bands to attain their nominal uniform coverage. For estimators with the usual $r_n=n$ rate, the rule becomes $L_n=\omega(n^{1/4})$, giving a simple and theoretically grounded guide for balancing computational cost against asymptotic validity.
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