Cuevas González, AntonioPateiro López, Beatriz2019-04-152019-12-082018Antonio Cuevas, Beatriz Pateiro-López (2018) Polynomial volume estimation and its applications, Journal of Statistical Planning and Inference, Volume 196, pp 174-184, DOI: 10.1016/j.jspi.2017.11.0050378-3758http://hdl.handle.net/10347/18631Given a compact set S ⊂ R d we consider the problem of estimating, from a random sample of points, the Lebesgue measure of S, µ(S), and its boundary measure, L(S) (as defined by the Minkowski content of ∂S). This topic has received some attention, especially in the two-dimensional case d = 2, motivated by applications in image analysis. A new method to simultaneously estimate µ(S) and L(S) from a sample of points inside S is proposed. The basic idea is to assume that S has a polynomial volume, that is, that V (r) := µ{x : d(x, S) ≤ r} is a polynomial in r of degree d, for all r in some interval [0, R). We develop a minimum distance approach to estimate the coefficients of V (r) and, in particular µ(S) and L(S), which correspond, respectively, to the independent term and the first degree coefficient of V (r). The strong consistency of the proposed estimators is proved. Some numerical illustrations are giveneng© 2017 Elsevier B.V. All Rights reserved. This manuscript version is made available under the CC-BY-NC-ND 4.0 license (http://creativecommons.org/licenses/by-ncnd/4.0/)Attribution-NonCommercial-NoDerivatives 4.0 Internacionalhttp://creativecommons.org/licenses/by-nc-nd/4.0/Set estimationVolume estimationBoundary length estimationjournal article10.1016/j.jspi.2017.11.005open access