Linear and Nonlinear Functionsof Norm Minimizing Estimation in Set Indexed Stochastic Processes

Abstract

This article discusses the norm minimizing estimation of a set indexed stochastic process using another set indexed stochastic process as a reference. The collection A, which generates Borel sets of a topological space, plays a critical role in the selection of the indexing collection. We introduce the norm minimizing estimation of the set indexed stochastic process in terms of linear and nonlinear functions of another set indexed stochastic process. We also prove the orthogonality principle under certain assumptions. Additionally, we define the inner product and equivalence relation on square integrable set indexed stochastic processes and introduce an inner product and a norm on the quotient set. We present Theorem 2, which provides the necessary conditions for the set indexed norm minimizing estimation of the set indexed stochastic process using a constant set indexed process, a linear function of set indexed process, and a nonlinear function of set indexed process.