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Random sets from the perspective of metric statistics
\address[D. Kurisu]{Center for Spatial Information Science, The University of Tokyo\\ 5-1-5, Kashiwanoha, Kashiwa-shi, Chiba 277-8568, Japan.} \email{[email removed]}
\address[Y. Okamoto]{Graduate School of Economics, Kyoto University, Yoshida Honmachi, Sakyo, Kyoto 606-8501, Japan. } \email{[email removed]}
\address[T. Otsu]{Department of Economics, London School of Economics, Houghton Street, London, WC2A 2AE, UK.} \email{[email removed]}
Since the seminal work by BeMo08, the random set theory and related inference methods have been widely applied in partially identified econometric models; see MoMo18 and Mo20 for surveys in econometrics, and Mo17 for a comprehensive overview on the random set theory. The random set approach characterizes an identification region of interest by utilizing the Au65 mean for set valued random variables (SVRVs) and constructs its sample analog estimator by using Minkowski averages of SVRVs. Obviously the key ingredient for partial identification analysis using random sets is the Aumann mean, which is a natural and convenient extension of the conventional mean for Euclidean random variables to SVRVs.
On the other hand, in the recent statistics literature for analyzing complex data, there is an emerging and rapidly growing field, called {\it metric statistics} (or statistics for random objects); see DuChMu24 for an overview and references therein. Metric statistics is concerned with complex data situated in a metric space, and popular examples include distributional data, network data, symmetric positive definite matrices, trees, data on Riemannian manifolds, among others. In this literature, a fundamental notion to characterize the population mean for metric valued random objects is the Fr48 mean.
Given these literatures, it is natural to ask whether there is a relationship between these notions of population means, and also whether the methodologies of metric statistics can shed new light on econometric analysis of random sets. This is of course not the first paper addressing these issues. For example, Section 3.2 of Mo17 introduced the Fr\'echet mean and mentioned that it can be applied to SVRVs that situated in the Hausdorff metric space. LiMoMoPe21 mentioned that their local regression smoother based on the Minkowski average can be interpreted as the sample Fr\'echet mean. Although these discussions are highly insightful, their main focuses are on the conventional random set analysis using the Aumann mean, and the analysis using the Fr\'echet mean is not pursued. On the other hand, to the best of our knowledge, there is no formal study in the literature of metric statistics on SVRVs. This paper is written to fill this gap.
In particular, this paper makes two contributions. First, we formally study the relationship between the Aumann and Fr\'echet means, and establish the equivalence of these notions of population means in the space of nonempty compact and convex sets equipped with the $L^2$-metric based on their support functions. A key ingredient to establish such equivalence is an isometric embedding of a general metric space into a Hilbert space. Second, we apply or extend some methodologies of metric statistics to econometric problems that involve random sets. After introducing the global Fr\'echet regression (GFR) by PeMu19, we apply the GFR to the projection model of Euclidean covariates, and clarify the relation with the set valued best linear predictor by BeMo08. Furthermore, we present some extensions of the GFR for an errors-in-variables model with set valued outcomes and a missing set valued data problem.
This paper is organized as follows. Section (ref) discusses our main result, equivalence of the Aumann and Fr\'echet means. Then Section (ref) presents applications of metric statistics for random sets.
This section is devoted to discuss the relationship between the Aumann mean and the Fr\'echet mean of SVRVs. After introducing our basic setup and the Aumann mean (Section (ref)), we introduce a metric space based on the support function (Section (ref)). Then we introduce the Fr\'echet mean and present a key property, isometric embedding (Section (ref)). Based on these preparations, Section (ref) establishes the equivalence of the Aumann and Fr\'echet means of SVRVs.
We follow the notation in BeMo08 and introduce our basic setup. Let $(\Omega, \mathcal{A}, \mu)$ be a measurable space, and $K(\mathbb{R}^d)$ be the collection of all nonempty closed subsets of $\mathbb{R}^d$. A random element $F:\Omega \to K(\mathbb{R}^d)$ is called an SVRV. Let $K_{k}(\mathbb{R}^d)$ be the set of nonempty compact subsets of $\mathbb{R}^d$, and $K_{kc}(\mathbb{R}^d)$ denote the set of nonempty compact and convex subsets of $\mathbb{R}^d$. For an SVRV $F \in K_k(\mathbb{R}^d)$, the Aumann mean is defined as follows.
Let $L^1=L^1(\Omega, \mathbb{R}^d)$ denote the space of measurable random variables with values in $\mathbb{R}^d$ s.t. $\|\xi\|_1 = \mathbb{E}[\|\xi\|]<\infty$, $\mathcal{S}(F)$ denote the set of all measurable selections (or points) from a set $F$, and $\mathcal{S}^1(F) = \mathcal{S}(F) \cap L^1$. The Aumann mean of an SVRV $F \in K_k(\mathbb{R}^d)$ is defined as
Similarly, the conditional Aumann mean of $F$ given $X \in \mathcal{X}$ is defined as $\mathbb{E}[F|X] = \left\{\mathbb{E}[f|X]: f \in \mathcal{S}^1(F)\right\}$, where $\mathcal{X}$ is a subset of $\mathbb{R}^p$. The Aumann mean is a natural generalization of the conventional mean for Euclidean random variables and plays a fundamental role in random set theory and its econometric and statistical applications; see Mo17, MoMo18, and Mo20.
Hereafter we focus on SVRVs on the set of nonempty compact and convex subsets, $K_{kc}(\mathbb{R}^d)$, and characterize the Aumann mean $\mathbb{E}[F]$ for $F\in K_{kc}(\mathbb{R}^d)$ by using the notion of the Fr\'echet mean for random objects in a metric space, which has been increasingly popular in recent literature on metric statistics; see DuChMu24 for example.
A key ingredient to clarify the relationship between the Aumann and Fr\'echet means is to choose a proper metric on the set of nonempty compact and convex subsets $K_{kc}(\mathbb{R}^d)$. To this end, we introduce the support function $s(p,F)=\sup_{f \in F}\langle p,f \rangle$ for $F\in K_{kc}(\mathbb{R}^d)$ over $p\in \mathbb{S}^{d-1}$, where $\mathbb{S}^{d-1} = \{x \in \mathbb{R}^d: \|x\|=1\}$ is the unit sphere in $\mathbb{R}^d$. For an SVRV $F\in K_{kc}(\mathbb{R}^d)$, the Aumann mean $\mathbb{E}[F]$ is equivalently characterized by its support function.
For $F, G \in K_{kc}(\mathbb{R}^d)$, define
This paper employs $d_{kc}$ as a metric on $K_{kc}(\mathbb{R}^d)$ and establishes the equivalence of the Aumann and Fr\'echet means. We close this subsection by presenting some properties related to $d_{kc}$. Let $d_H(\cdot, \cdot)$ be the Hausdorff metric on $K_k(\mathbb{R}^d)$ defined by $d_H(F,G)=\max\{\sup_{f\in F}\inf_{g \in G}\|f-g\|,\sup_{g\in G}\inf_{f \in F}\|f-g\|\}$ for $F,G \in K_k(\mathbb{R}^d)$, $K_{kc}^B(\mathbb{R}^d) = \{F \in K_{kc}(\mathbb{R}^d): \sup_{f \in F}\|f\|\leq B\}$ for a positive constant $B$, $L^2(\mathbb{S}^{d-1})$ denote the space of functions such that $\|f\|_{2,\mathbb{S}^{d-1}}<\infty$, and $\Psi:K_k(\mathbb{R}^d) \to L^2(\mathbb{S}^{d-1})$ be a map such that $\Psi(F) = s(\cdot, F)$.
Lemma (ref) establishes several topological properties of compact convex sets under the metric $d_{kc}$. In particular, Lemma (ref) (ii) shows that convergence under the Hausdorff metric, which can be defined as the sup-norm distance between support functions on $\mathbb{S}^{d-1}$ (see Lemma (ref)), is equivalent to convergence under the $L^2$-based distance $d_{kc}$. This equivalence motivates the use of $d_{kc}$, which is analytically more tractable, for statistical analysis of random compact convex sets. In Section (ref), we show that the Aumann mean (and its sample counterpart, the normalized Minkowski sum) can be characterized as the Fr\'echet mean, that is, the minimizer of $\mathbb{E}[d_{kc}^2(\nu,F)]$, the expectation of the squared distance between a data descriptor $\nu \in K_{kc}^B(\mathbb{R}^d)$ and an SVRV $F\in K_{kc}^B(\mathbb{R}^d)$ (and its sample version). Finally, Lemma (ref) (iii) and (iv) guarantee the existence and uniqueness of the Fr\'echet mean.
We now introduce the Fr\'echet mean for random objects. Let $\mathcal{X}$ be a subset of $\mathbb{R}^p$, $(\mathcal{M},d)$ be a separable metric space, and $(Y,X) \in \mathcal{M} \times \mathcal{X}$ be a pair of random elements. The Fr\'echet mean $\mathbb{E}_\oplus[Y]$ of $Y$ is defined as a minimizer of the population Fr\'echet function $Q(\nu)=\mathbb{E}[d^2(\nu,Y)]$, that is,
Similarly, the conditional Fr\'echet mean $\mathbb{E}_\oplus[Y|X]$ of $Y$ given $X$ is defined as
The Fr\'echet mean is a direct generalization of the conventional mean for the Euclidean space toward a general metric space, and its statistical analysis has been increasingly popular in recent literature. The key step to establish the relationship between the Aumann and Fr\'echet means is to consider an isometric embedding of a general metric space into a Hilbert space. To this end, we impose the following assumptions.
Note that $(K_{kc}^B(\mathbb{R}^d),d_{kc})$ satisfies Assumption (ref). Indeed, setting $\Psi(F) = s(\cdot, F)$ implies that $K_{kc}^B(\mathbb{R}^d)$ admits an isometric embedding into $L^2(\mathbb{S}^{d-1})$. Furthermore, by Lemma (ref) (iv), the image $\Psi(K_{kc}^B(\mathbb{R}^d))$ satisfies Assumption (ref) (ii). Let $(\mathcal{M},d)$ be a metric space. The metric $d^2$ is of negative type if, for all $n \geq 2$, $\nu_1,\dots,\nu_n \in \mathcal{M}$ and $\alpha_1,\dots, \alpha_n \in \mathbb{R}$ with $\sum_{i=1}^n\alpha_i = 0$, we have $\sum_{i=1}^n\sum_{j=1}^n \alpha_i\alpha_j d^2(\nu_i,\nu_j) \leq 0$. A sufficient condition for Assumption (ref) (i) is provided in Proposition 3 in sejd:13 for example, which implies that if $d^2$ is of negative type, then this condition is satisfied; see also scho:38. Many of the common metric spaces studied in metric statistics are known to satisfy Assumption (ref). Examples include the 2-Wasserstein space for univariate probability distributions, the space of symmetric positive (semi)definite matrices endowed with the Frobenius, power, or log-Euclidean metric, the space of graph Laplacians equipped with the Frobenius metric, and the space of compositional data endowed with the Aitchison metric. See also Appendix C.1 in KuZhOtMu25b.
Under these assumptions, we obtain the following characterizations of the Fr\'echet mean. For a random element $Z$ taking values in a Hilbert space $\mathcal{H}$, we define its expectation as the Riesz representation of the linear functional that maps $h \in \mathcal{H}$ to $\mathbb{E}[\langle h,Z \rangle_\mathcal{H}] \in \mathbb{R}$.
According to Proposition (ref) (i), the Fr\'echet mean of $Y$ is obtained as the pullback, through $\Psi$, of the expectation of $\Psi(Y)$ in the image space $\Psi(K_{kc}^B(\mathbb{R}^d))$. Proposition (ref) (ii) can be interpreted as a law of iterated expectation for random objects, which naturally generalizes the corresponding result for Euclidean random variables.
We now apply the Fr\'echet mean to SVRVs in the metric space $(K_{kc}^B(\mathbb{R}^d),d_{kc})$. Let $(F,X) \in K_{kc}^B(\mathbb{R}^d) \times \mathcal{X}$ be a pair of random elements. For $F,G \subset \mathbb{R}^d$, let $F \oplus G=\{f+g: f \in F, g \in G\}$ denote the Minkowski sum of $F$ and $G$. The population Fr\'echet mean $\mathbb{E}_\oplus[F]$ of $F$ with respect to the metric $d_{kc}$ is defined as a minimizer of the population Fr\'echet function $\mathcal{Q}(\nu)=\mathbb{E}[d_{kc}^2(\nu,F)]$, that is,
Similarly, the population conditional Fr\'echet mean of $F$ given $X$ is defined as \[ \mathbb{E}_\oplus[F|X] \in \mathrm{argmin}_{\nu \in K_{kc}^B(\mathbb{R}^d)}\mathcal{Q}(\nu|X), \quad \mathcal{Q}(\nu|X) = \mathbb{E}[d_{kc}^2(\nu,F)|X]. \] The main results of this paper are presented as follows.
Proposition (ref) (i) guarantees uniqueness of the Fr\'echet means $\mathbb{E}_\oplus[F]$ and $\mathbb{E}_\oplus[F|X]$. Proposition (ref) (ii) and (iii) are our main results, equivalence of the Fr\'echet and conditional Fr\'echet means to the Aumann and conditional Aumann means, respectively. Furthermore, Proposition (ref) (iii) also provides the law of iterated expectations for the Fr\'echet mean.
Note that an SVRV $F \in K_{kc}(\mathbb{R}^d)$ is a metric space-valued random element, which is called random object. See e.g., MaAl14 and MaDr21 for a review. By Proposition (ref), the (conditional) Aumann mean of $F$ can be defined as the (conditional) Fr\'echet mean of $F$ under the metric $d_{kc}$. From this viewpoint, statistical analysis of SVRVs falls within the scope of metric statistics, a field that has seen remarkable development in recent years; see DuChMu24 for example.
The equivalence of the sample Minkowski and Fr\'echet means is presented as follows.
This section presents several methods developed within metric statistics that can be applied to the analysis of SVRVs. For $t \in \mathbb{R}$, define $t F = \{tf: f \in F\}$. Let $\mathcal{X} \subset \mathbb{R}^p$ be a compact set.
To motivate, consider the linear projection model for an Euclidean outcome $Y\in\mathbb{R}$ and Euclidean covariates $X \in \mathbb{R}^p$, \[ Y=\theta_0^*+(\theta_1^*)'(X-\mu)+\varepsilon,\quad\mathbb{E}[X\varepsilon]=0, \] where $\mu=\mathbb{E}[X]$ and the vector $(\theta_0^*,\theta_1^*)$ solves \[ (\theta_0^*,\theta_1^*)=\mathrm{argmin}_{(\theta_0,\theta_1) \in\mathbb{R}^{p+1}}\mathbb{E}[(\mathbb{E}[Y|X]-\{\theta_{0}+\theta_{1}'(X-\mu)\})^{2}]. \] Let $\sigma_{YX}=\mathbb{E}[Y(X-\mu)]$ and $\Sigma=\mathbb{E}[(X-\mu)(X-\mu)']$. Assume that $\Sigma$ is invertible. Then we have $\theta_0^*=\mathbb{E}[Y]$ and $\theta_1^*=\Sigma^{-1}\sigma_{YX}$, and obtain the regression function
where $w(x,z)=1+(x-\mu)'\Sigma^{-1}(z-\mu)'$. Since $\mathbb{E}[w(x,X)]=1$, the regression function $m(x)$ is characterized as the solution \[ m(x)=\mathrm{argmin}_{y\in\mathbb{R}}\mathbb{E}[w(x,X)d_{E}^2(y,Y)], \] where $d_E$ is the standard Euclidean metric.
By extending the above representation of $m(x)$, the global Fr\'echet regression (GFR) function for an outcome $Y$ in a general metric space $(\mathcal{M},d)$ is defined as follows PeMu19:
Let $\{Y_i,X_i\}_{i=1}^n$ be an i.i.d. sample from the joint distribution of $(Y,X)\in \mathcal{M}\times \mathbb{R}^p$. Then the GFR estimator is given as the sample counterpart of $m_{\oplus}$, that is
where $\hat{w}(x,z)=1+(x-\bar{X})'\hat{\Sigma}^{-1}(z-\bar{X})$, $\bar{X}=n^{-1}\sum_{i=1}^n X_i$, and $\hat{\Sigma}=n^{-1}\sum_{i=1}^n (X_i-\bar{X})(X_i-\bar{X})'$.
We now apply the GFR for a pair of random elements $(F,X) \in K_{kc}^B(\mathbb{R}^d) \times \mathcal{X}$, which is defined as
In order to derive the explicit form of $m_{\oplus}(x)$, recall $\Psi: K_{kc}^B(\mathbb{R}^d) \ni F \mapsto s(\cdot, F) \in L^2(\mathbb{S}^{d-1})$, and define \[ m_{\oplus,\Psi}(x) = \mathrm{argmin}_{h \in L^2(\mathbb{S}^{d-1})}\mathbb{E}[w(x,X)\|h - \Psi(F)\|_{2,\mathbb{S}^{d-1}}]. \] Then we have
where the second equality follows from Proposition (ref).
Assume that $\inf_{z \in \mathcal{X}}w(x,z) \geq 1$, which guarantees $w(x,X) F \in K_{kc}^B(\mathbb{R}^d)$. Then from Proposition (ref) (ii) and Lemma (ref) (i), we have
which yields the expression of $m_\oplus(x)$ by the Aumann means:
Note that $\Theta$ corresponds to the population set valued best linear predictor in BeMo08. On the other hand, if $\inf_{z \in \mathcal{X}}w(x,z) < 1$, we have
Specifically, it is possible to provide an example such that $m_{\oplus,\Psi}(x)< \Psi(\mathbb{E}[w(x,X)F])$ (see Appendix (ref)). Therefore, for the GFR with a general weight function $w(x,z)$, we compute $m_\oplus(x)$ by projecting $m_{\oplus,\Psi}(x)$ on $\Psi(K_{kc}^B(\mathbb{R}^d))$, that is,
The following result shows that $m_\oplus(x)$ is in fact the GFR function.
Figure (ref) illustrates the relation between $m_\oplus(x)$ and $\mathbb{E}[w(x,X)F]$ when $d=1$. The colored region corresponds to the set of support functions. Note that, in the one-dimensional case, $F$ can be written as $F=[-a,b]$ with $b+a\geq0$. Note also that the support function $s(p,F)$ of $F \in K_{kc}(\mathbb{R})$ is characterized by $s(1,F)$ and $s(-1,F)$ and satisfies $b=s(1,F)\geq -s(-1,F)=-a$. Hence the set of support functions $\Psi(K_{kc}(\mathbb{R}))=\{s(\cdot, F): F \in K_{kc}(\mathbb{R})\}$ can be identified with $\{(a,b) \in \mathbb{R}^2: b+a \geq 0\}$. The blue region $R_{+,+}$, the purple region $R_{-,+}$, and the red region $R_{-,-}$ correspond to the sets $K_1,K_2$, and $K_3$ such that $K_1 \sqcup K_2 \sqcup K_3=K_{kc}(\mathbb{R})$ where $K_1=\{[x,y]:0< x\leq y <\infty\}$, $K_2=\{[x,y]:-\infty < x\leq 0 \leq y <\infty\}$, and $K_3=\{[x,y]:-\infty < x \leq y < 0\}$. In each figure, black points correspond to $m_{\oplus, \Psi}$, $\widetilde{m}_{\oplus,\Psi}$, and $\Psi(\mathbb{E}[w(x,X)F])$. The left figure represents the case $m_{\oplus, \Psi}(x) \in \Psi(K_{kc}(\mathbb{R}))$, in which case $m_{\oplus}(x) \subset \mathbb{E}[w(x,X)F]$. The middle figure corresponds to the case of $m_{\oplus, \Psi}(x) \notin \Psi(K_{kc}(\mathbb{R}))$ and $m_{\oplus}(x) \subset \mathbb{E}[w(x,X)F]$. The right figure corresponds to the case of $m_{\oplus, \Psi}(x) \notin \Psi(K_{kc}(\mathbb{R}))$ and $m_{\oplus}(x) \not\subset \mathbb{E}[w(x,X)F]$.
Let $(F,X) \in K_{kc}^B(\mathbb{R}^d) \times \mathcal{X}$ be a pair of random elements and $\{F_i,X_i\}_{i=1}^n$ be an i.i.d. sample from the joint distribution of $(F,X)$. The uniform convergence rate of the GFR estimator in ((ref)) can be established as follows.
In Proposition (ref), since $\alpha>2$ can be chosen arbitrarily, the convergence rate of the GFR estimator can be arbitrarily close to the parametric rate when $d=1$. The restriction $d\leq 4$ is due to the complexity of the metric space $(K_{kc}(\mathbb{R}^d),d_{kc})$. Specifically, let $N(\varepsilon, \mathcal{F},d)$ be the covering number, that is, the minimal number of balls of radius $\varepsilon$ (with respect to the metric $d$) needed to cover the set $\mathcal{F}$. We can see $N(\varepsilon, K_{kc}^B(\mathbb{R}^d),d_{kc}) \leq D\varepsilon^{-2}$ for $d=1$ but $N(\varepsilon, K_{kc}^B(\mathbb{R}^d),d_{kc}) \leq e^{D\varepsilon^{-(d-1)/2}}$ for $d\geq 2$ where $D$ is a positive constant which is independent of $\varepsilon$. To show Proposition (ref) for $d\geq 2$, we need to verify $\int_0^1 \sqrt{\varepsilon^{-(d-1)/2}}d\varepsilon<\infty$, which requires $d\leq 4$. See the proof of Proposition (ref) for details. The same comment applies to Proposition (ref) below.
As an illustration, consider the case of one-dimensional interval-valued random variable, i.e., $F\in K_{kc}^B(\mathbb{R})$. In this case, letting $F=[L,U]$, we have
Also since $w(x,X)F=[\min\{w(x,X)L, w(x,X)U\}, \max\{w(x,X)L, w(x,X)U\}]$, we obtain
Following BeMo08, we consider the best linear prediction of the returns to education of log-wages for white men between the ages of 20 and 50. We artificially divide the support of the outcome variable into five equal intervals to construct interval data. We employ the March 2009 Current Population Survey data from Hansen:2022_econ, and treat them as the population. Then we compare the GFR $m_{\oplus}(x)= \Psi^{-1}(\widetilde{m}_{\oplus,\Psi}(x))$ and the Aumann mean $\mathbb{E}[{w(x,X)F}]$ at $x=\mu\approx 13.727$ and $x=16$, corresponding to the Bachelor's degree. Note that $w(\mu,z)=1+(\mu-\mu)'\Sigma^{-1}(z-\mu)=1$ (see Figure (ref)) so that $m_{\oplus}(x)=\mathbb{E}[{w(x,X)F}]$. In this case, the Aumann mean and the GFR are computed as
On the other hand, Figure (ref) shows that $\inf_{z \in \mathcal{X}}w(16,z) < 1$. We can also confirm that $m_{\oplus,\Psi}(16)$ is a proper support function, i.e., $m_{\oplus,\Psi}(16) \in \Psi(K_{kc}(\mathbb{R}))$. These suggest that $m_{\oplus}(x)\subset\mathbb{E}[{w(x,X)F}]$ (Proposition (ref)). Indeed, we obtain that
The GFR approach can be naturally extended to errors-in-variables models. Suppose the object of interest is
However, Euclidean covariates $X$ are mismeasured, and we instead observe $W = X + \varepsilon$, where $\varepsilon$ is a measurement error such that $\mathbb{E}[\varepsilon|X,F]=0$ and $\mathbb{E}[\varepsilon \varepsilon'] = \Sigma_\varepsilon$. In this setup, note that $\mathbb{E}[W]=\mu$, $\Sigma_W = \mathbb{E}[(W - \mathbb{E}[W])(W - \mathbb{E}[W])']=\Sigma + \Sigma_\varepsilon$, and
Then the GFR estimator ((ref)) with $\mathcal{M}=K_{kc}^B(\mathbb{R}^d)$, $d=d_{kc}$, and $Y_i=F_i$ is not consistent for $m_\oplus(x)$.
Suppose we observe instrumental variables $Z = X + v$ (or repeated measurements on $X$) such that $\mathbb{E}[v|X,F]$ and $\mathbb{E}[v\varepsilon'|X,F]=0$. Then we can recover $\Sigma$ as \[ \Sigma_{ZW}=\mathbb{E}[(Z-\mathbb{E}[Z])(W-\mathbb{E}[W])'] = \mathbb{E}[(X-\mu + v)(X - \mu + \varepsilon)'] = \Sigma. \] Therefore, the GFR function $m_\oplus(x)$ can be estimated based on the alternative representation
where $\tilde{w}(x,z) = 1 + (x-\mu)'\Sigma_{ZW}^{-1}(z -\mu)$.
Other possible statistical methods for analyzing set-valued response variables with Euclidean covariates using the GFR or related regression models include TuWuMu23 and KuOt25 for variable selection and model averaging, BhMu23 for an extension of the GFR to single index models, PeMu19 and ChMu22 for extensions of the GFR to local linear regression.
Consider the missing at random setup. Let $T \in \{0,1\}$ be the indicator for missing data. We observe $TF$ and $X$, where $(F,X) \in K_{kc}^B(\mathbb{R}^d) \times \mathcal{X}$ and assume $F$ and $T$ are independent given $X$. Let $\{F_i,X_i,T_i\}_{i=1}^n$ be an i.i.d. sample from the joint distribution of $(F,X,T)$. We are interested in the Fr\'echet mean of $F$, that is, $\mathbb{E}_\oplus[F] = \mathrm{argmin}_{\nu \in K_{kc}^B(\mathbb{R}^d)}\mathbb{E}[d_{kc}^2(\nu,F)]$.
Letting $e(x) = \mathbb{P}(T=1|X=x)$, we can see that
Therefore, the Fr\'echet mean of $F$ can be alternatively written as \[ \mathbb{E}_\oplus[F] = \mathrm{argmin}_{\nu \in K_{kc}^B(\mathbb{R}^d)}\mathbb{E}\left[{T \over e(X)}d_{kc}^2(\nu,F)\right], \] which can be estimated by
and $\hat{e}(x)$ is a nonnegative nonparametric estimator of $e(x)$.
One can see that $\hat{\mathbb{E}}_\oplus[F]$ is a modified version of inverse probability weighting estimator proposed in KuZhOtMu24:
See Appendix (ref) for the derivation of ((ref)). The following result provides the convergence rate of $\hat{\mathbb{E}}_\oplus[F]$.
If the propensity score $e(x)$ belongs to a class of parametric models $\mathcal{M}_e=\{e(x;\varphi): \varphi \in \Phi\}$ with a compact parameter space $\Phi \subset \mathbb{R}^q$, Condition (iii) is satisfied with $\int_0^1 \sqrt{\log N(\delta \varepsilon, B_{\delta'}(e), \|\cdot\|_\infty)}d\varepsilon = O(-\log\delta)$ as $\delta \to 0$. In this case, it can be shown that convergence rate of $\hat{\mathbb{E}}_\oplus[F]$ is $O_p(n^{-{1 \over 2(\alpha-1)}})$ for any $\alpha >2$ when $d=1$ and $O_p(n^{-{1 \over 2(1 + (d-1)/4)}})$ when $d \in \{2,3,4\}$. For details on this point, see, for example, Section 4.1 of KuZhOtMu24.
In the recent literature, metric statistics has been extended to conduct causal inference for outcomes situated in a general metric space; see KuZhOtMu24, ZhKuOtMu25, KuZhOtMu25a, KuZhOtMu25b, and BhLiWuXu25. In sum, metric statistics can be a useful toolkit for econometric analysis on non-Euclidean economic data including SVRVs.