Approximation of Distributions by Bounded Sets
- PDF / 329,524 Bytes
- 9 Pages / 439.37 x 666.142 pts Page_size
- 21 Downloads / 220 Views
Approximation of Distributions by Bounded Sets Meelis Käärik · Kalev Pärna
Published online: 27 March 2007 © Springer Science + Business Media B.V. 2007
Abstract Let P be a probability distribution on a locally compact separable metric space (S, d). We study the following problem of approximation of a distribution P by a set A from a given class A ⊂ 2S : W (A, P ) ≡ ϕ(d(x, A))P (dx) → min, S
A∈A
where ϕ is anondecreasing function. A special case where A consists of unions of bounded sets, A = { ki=1 Ai : (Ai ) ≤ K, i = 1, . . . , k}, is considered in detail. We give sufficient conditions for the existence of an optimal approximative set and for the convergence of the sequence of optimal sets An found for measures Pn which satisfy Pn ⇒ P . Current article is a follow-up to Käärik and Pärna (Acta Appl. Math. 78, 175–183, 2003; Acta Comment. Univ. Tartu. 8, 101–112, 2004) where the case of parametric sets was studied. Keywords Approximation of distributions · Fitting sets to distributions · Loss-function · Discrepancy function · Consistency · M-estimation · k-centres
1 Introduction Approximation of distributions by sets is an old issue which offers various theoretical and practical problems. Classical examples are the mean and the median which are best onepoint approximative sets for a distribution P (w.r.t. L2 - and L1 -norm, respectively). Also the class of k-point approximative sets (which leads to k-centres or k-means) has been studied thoroughly during last decades, including for probabilities in Rd [5, 7, 20, 24], general Supported by the Estonian Science Foundation Grant No. 5277. M. Käärik () · K. Pärna Institute of Mathematical Statistics, University of Tartu, J. Liivi 2, Tartu 50409, Estonia e-mail: [email protected] K. Pärna e-mail: [email protected]
16
M. Käärik, K. Pärna
Banach spaces [3, 4, 14, 18], and separable metric spaces [15]. In practice, k-centres are particularly important in information transmission where they are used as a tool for discetization (or quantization) of a continuous signal [7, 8]. Besides k-centres, more complex approximative have been investigated as well. Mathematically, given a probability P on a metric space (S, d), we try to find a subset A from a given class A ⊂ 2S which minimizes the mean distance from a random point X ∼ P to the subset A: W (A, P ) ≡ ϕ(d(x, A))P (dx) → min, A∈A
S +
+
where d(x, A) = inf{d(x, a) : a ∈ A} and ϕ : R → R is a nondecreasing discrepancy function. For instance, approximation of distributions by spheres with fixed radius in finitedimensional Euclidean spaces is considered in [9]. A closely related problem (called median ball problem) is studied in [1] and the approximation by multiple balls is considered in [19]. The problem of fitting various conical sections is considered in [6, 22, 23]. Fitting circles is also widely used in practice: e.g. in particle physics [2], microwave calibrations [13], astronomy [17], archeology [21] and more. The examples above give a motivation to study reasonably general classes of approximative set
Data Loading...
