A PDE Approach to the Problems of Optimality of Expectations
Küçük Resim Yok
Tarih
2023
Yazarlar
Dergi Başlığı
Dergi ISSN
Cilt Başlığı
Yayıncı
Etamaths Publ
Erişim Hakkı
info:eu-repo/semantics/openAccess
Özet
Let (X, Z) be a bivariate random vector. A predictor of X based on Z is just a Borel function g(Z). The problem of least squares prediction of X given the observation Z is to find the global minimum point of the functional E[(X - g(Z))2] with respect to all random variables g(Z), where g is a Borel function. It is well known that the solution of this problem is the conditional expectation E(X|Z). We also know that, if for a nonnegative smooth function F : RxR & RARR; R, arg ming(Z)E[F (X, g(Z))] = E[X|Z], for all X and Z, then F (x, y) is a Bregmann loss function. It is also of interest, for a fixed & phi; to find F (x, y ), satisfying, arg ming(Z)E[F (X, g(Z))] = & phi;(E[X|Z]), for all X and Z. In more general setting, a stronger problem is to find F (x, y) satisfying arg miny & ISIN;RE[F (X, y)] = & phi;(E[X]), & FORALL;X. We study this problem and develop a partial differential equation (PDE) approach to solution of these problems.
Açıklama
Anahtar Kelimeler
expectation, conditional expectation, random variables, Bregman loss functions, partial differential equations
Kaynak
International Journal Of Analysis And Applications
WoS Q Değeri
N/A
Scopus Q Değeri
Q4
Cilt
21
