## Electronic Journal of Statistics

- Electron. J. Statist.
- Volume 10, Number 2 (2016), 2066-2096.

### Asymptotic results for multivariate estimators of the mean density of random closed sets

Federico Camerlenghi, Claudio Macci, and Elena Villa

#### Abstract

The problem of the evaluation and estimation of the mean density of random closed sets in $\mathbb{R} ^{d}$ with integer Hausdorff dimension $0<n<d$, is of great interest in many different scientific and technological fields. Among the estimators of the mean density available in literature, the so-called “Minkowski content”-based estimator reveals its benefits in applications in the non-stationary cases. We introduce here a multivariate version of such estimator, and we study its asymptotical properties by means of large and moderate deviation results. In particular we prove that the estimator is strongly consistent and asymptotically Normal. Furthermore we also provide confidence regions for the mean density of the involved random closed set in $m\geq1$ distinct points $x_{1},\ldots,x_{m}\in\mathbb{R} ^{d}$.

#### Article information

**Source**

Electron. J. Statist., Volume 10, Number 2 (2016), 2066-2096.

**Dates**

Received: January 2016

First available in Project Euclid: 18 July 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ejs/1468849971

**Digital Object Identifier**

doi:10.1214/16-EJS1159

**Mathematical Reviews number (MathSciNet)**

MR3522669

**Zentralblatt MATH identifier**

1345.62050

**Subjects**

Primary: 62F12: Asymptotic properties of estimators 60F10: Large deviations 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

**Keywords**

Minkowski content large deviations moderate deviations random closed sets confidence regions

#### Citation

Camerlenghi, Federico; Macci, Claudio; Villa, Elena. Asymptotic results for multivariate estimators of the mean density of random closed sets. Electron. J. Statist. 10 (2016), no. 2, 2066--2096. doi:10.1214/16-EJS1159. https://projecteuclid.org/euclid.ejs/1468849971