Optimal Bandwidth of the “Minkowski Content”-Based Estimator of the Mean Density of Random Closed Sets: Theoretical Results and Numerical Experiments

Abstract

The estimation of the mean density of random closed sets in $\mathbb {R}^d$ with integer Hausdorff dimension $n<d$ is a problem of interest from both a theoretical and an applicative point of view. In literature different kinds of estimators are available, mostly for the homogeneous case. Recently the non-homogeneous case has been faced by the authors; more precisely, two different kinds of estimators, asymptotically unbiased and weakly consistent, have been proposed: in Camerlenghi et al. (J Multivar Anal 125:65–88, 2014) a kernel-type estimator generalizing the well-known kernel density estimator for random variables, and in Villa (Stoch Anal Appl 28:480–504, 2010) an estimator based on the notion of Minkowski content of a set. The study of the optimal bandwidth of the “Minkowski content”-based estimator has been left as an open problem in Villa (Stoch Anal Appl 28:480–504, 2010, Sect. 6) and in Villa (Bernoulli 20:1–27, 2014, Remark 14), and only partially solved in Camerlenghi et al. (J Multivar Anal 125:65–88, 2014, Sect. 4), where a formula is available in the particular case of homogeneous Boolean models. We give here a solution of such an open problem, by providing explicit formulas for the optimal bandwidth for quite general random closed sets (i.e., not necessarily Boolean models or homogeneous germ-grain models). We also discuss a series of relevant examples and corresponding numerical experiments to validate our theoretical results.

Appendix

1.1 Positive reach, curvature measures, and related results

We summarize here some basic definitions and results on sets with positive reach and associated curvature measures, which might be useful for the non-expert reader for a more readability of the paper. We refer to the existent literature for a more exhaustive treatment of this subject.

Let $A \subset \mathbb {R}^d$ be a non-empty closed set and $z \in \mathbb {R}^d$, set $\mathrm{dist}(A,z):= \inf \{ ||a-z|| \, : \, a \in A \}$.

Denote by $\mathrm{Unp}(A):=\{x\in \mathbb {R}^d\,:\,\exists !\,a\in A\ \mathrm{such\ that }\ \mathrm{dist}$ $(x,A)=|| a-x||\}$ the set of points having a unique projection on $A$. The definition of $\mathrm{Unp}(A)$ implies the existence of a projection mapping $\xi _A :\mathrm{Unp}(A) \rightarrow A$ which assigns to $x \in \mathrm{Unp(A)}$ the unique point $\xi _A(x) \in A$ such that $\mathrm{dist}(z,A) = ||\xi _A(x)-x||$. Then for all $x \in \mathrm{Unp}(A)$ with $\mathrm{dist}(x,A)>0$ we may define $u_A(x):= (x- \xi _A(x))/\mathrm{dist }(x,A)$. The set of all $x \in \mathbb {R}^d \setminus A$ for which $\xi _A(x)$ is not defined is called the exoskeleton of $A$ and it is denoted by $\mathrm{exo}(A)$. The exoskeleton is a measurable subset of $\mathbb {R}^d$ and $\mathcal {H} ^d(\mathrm{exo}(A))=0$. Denoted by $\mathbf{S}^{d-1}:= \partial B_1(0)$ the unit sphere in $\mathbb {R}^d$, the normal bundle of $A$ is the subset of $\partial A \times \mathbf{S}^{d-1}$:

$$\begin{aligned} N(A) := \{ (\xi _A(x), u_A(x) ) \, : \, x \not \in A \cup \mathrm{exo}(A) \} \end{aligned}$$

The reach function $\delta (A, \cdot ) : \mathbb {R}^d \times \mathbf{S}^{d-1} \rightarrow [0, \infty ]$ is defined by $\delta (A, x,u):= \inf \{ t \ge 0 \, : \, x +tu \in \mathrm{exo}(A)\}$ if $(x,u) \in N(A)$, $\delta (A, x,u) := 0$ otherwise; then the reach of $A$ is defined by

$$\begin{aligned} \mathrm{reach} (A):= \inf \{ \delta (A,x,u) \, : \, (x,u) \in N(A) \}. \end{aligned}$$

If $\mathrm{reach}(A) > 0$ the set is said to be a set with positive reach.

For any non-empty closed set $A\subset \mathbb {R}^d$ there exist uniquely determined signed measures $\mu _0(A;\,\cdot \,),\ldots ,\mu _{d-1}(A;\,\cdot \,)$ on $N(A)$, said support measures of A, which arise as coefficient measures of a local Steiner formula; namely in [19, Theorem 2.1] it is proved that

$$\begin{aligned}&\int _{\mathbb {R}^d \setminus A} f(x) \mathcal {H}^{d}(\mathrm{d}x) = \sum _{i=0}^{d-1} b_{d-i}(d-i)\int _{0}^{\infty } \int _{N(A)} t^{d-1-i}\nonumber \\&\quad \mathbf 1 _{[0, \delta (A,x,u))}(t) f(x+tu) \mu _i(A; \mathrm{d}(x,u)) \mathrm{d}t, \end{aligned}$$

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for any measurable bounded function $f : \mathbb {R}^d \rightarrow \mathbb {R}$ with compact support.

Furthermore, if $\mathcal {H}^k(\partial A)=0$ for some $k\in \{1,..., d-1\}$, $\mu _k(A; \cdot ) \equiv 0$ (see [21, Proposition 2.4]).

We also remind that if the closed set $A$ has positive reach, the following relationship between the support measure $\mu _{i}(A;\,\cdot \,)$ and the curvature measure $\varPhi _{i}(A;\,\cdot \,)$ associated with A introduced in [17] holds:

$$\begin{aligned} \mu _i (A; \cdot \times \mathbf{S}^{d-1}) = \varPhi _i(A; \cdot ) \quad \forall i = 0, ..., d-1 ; \end{aligned}$$

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using Federer’s notation, $\varPhi _i(A):=\varPhi _i(A;A)$ is the total curvature measure of $A$. Morover, if $\mathrm{reach}(A)>0$, the following global Steiner formula holds

$$\begin{aligned} \mathcal {H}^d(A_{\oplus r}) =\sum _{i=0}^d r^{d-i}b_{d-i}\varPhi _i(A),\quad \forall r<\mathrm{reach}(A); \end{aligned}$$

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note that $\varPhi _i(A)=0,$ for any $i>n$, if $\mathrm{dim}(A)=n$.

Let $A$ be a closed subset of $\mathbb {R}^d$, define

$$\begin{aligned} \partial ^{+} A = \{ x \in \partial A \, : \, (x,u) \in N(A) \text{ for } \text{ some } u \in \mathbf{S}^{d-1}\} , \end{aligned}$$

it is also well known that $\partial A = \partial ^+ A$ if $\mathrm{reach} (A) > 0$. For each $x \in \partial ^+ A$, we can define

$$\begin{aligned} N(A,x) := \{ u \in \mathbf{S}^{d-1} \, : \, (x,u) \in N(A) \}; \end{aligned}$$

the normal cone of $A$ at $x$ is defined by $n(A,x) := \{ \lambda u \, : \, \lambda \ge 0 \,, u \in N(A,x) \} .$

Let

$$\begin{aligned} \partial ^{++} A := \{ x \in \partial ^+ A \, : \, \mathrm{dim} n(A,x) =1 \}, \end{aligned}$$

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where $\mathrm{dim} B$ denotes the dimension of the affine hull of $B \subset \mathbb {R}^d$; then it follows that $\partial ^{++} A$ is the disjoint union of the sets $\partial ^1 A$ and $\partial ^2 A$, defined by

$$\begin{aligned} \partial ^i A := \{ x \in \partial ^{++}A \, : \, \mathrm{card } N(A,x) = i \} \quad i = 1,2 . \end{aligned}$$

For $x \in \partial ^1 A$, the unique element of $N(A,x)$, say $\nu (A,x)$, is the outer normal to $A$ at $x$; for $x \in \partial ^2 A$ there exist two outer normal ($\nu (A,x)$ and $-\nu (A,x)$ by (48)) to $A$ at $x$. In particular we recall the following representation for $\mu _{d-1} (A; \cdot )$ (see [19, Proposition 4.1]):

$$\begin{aligned}&\mu _{d-1} (A; \cdot ) = \frac{1}{2} \int _{\partial ^{1} A} \mathbf 1 _{\{(x , \nu (A,x))\in \cdot \}} \mathcal {H}^{d-1} (\mathrm{d}x) +\nonumber \\&\quad \frac{1}{2} \int _{\partial ^{2} A} \mathbf 1 _{\{ (x , \nu (A,x))\in \cdot \}} +\mathbf 1 _{\{(x , -\nu (A,x))\in \cdot \}}\mathcal {H}^{d-1} (\mathrm{d}x) \end{aligned}$$

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Finally, we also recall that explicit description of the curvature measures of order zero is given in [19, Proposition 4.10]; in particular for any convex set $A$ it holds (see [25, Secton 4.4]) $\forall B\in \mathcal {B}_{\mathbb {R}^d}$

$$\begin{aligned} \varPhi _0(A,B)= \frac{1}{d b_d}\int _{\mathbf{S}^{d-1}} \sum _{x \in B} \mathbf 1 _{N(A)}(x,u) \mathcal {H}^{d-1} (\mathrm{d}u). \end{aligned}$$

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