LAUSR

Abstract Given 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 given.