A PDE Approach to the Problems of Optimality of Expectations
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...
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| Format: | Article |
| Language: | English |
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Etamaths Publishing
2023-06-01
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| Series: | International Journal of Analysis and Applications |
| Online Access: | http://etamaths.com/index.php/ijaa/article/view/2824 |
| _version_ | 1797795995855093760 |
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| author | Mahir Hasanov |
| author_facet | Mahir Hasanov |
| author_sort | Mahir Hasanov |
| collection | DOAJ |
| description | 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: R×R → 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 ϕ to find F (x, y), satisfying, arg ming(Z)E[F(X, g(Z))] = ϕ(E[X|Z]), for all X and Z. In more general setting, a stronger problem is to find F (x, y) satisfying arg miny∈RE[F (X, y)] = ϕ(E[X]), ∀X. We study this problem and develop a partial differential equation (PDE) approach to solution of these problems. |
| first_indexed | 2024-03-13T03:26:28Z |
| format | Article |
| id | doaj.art-9251ca2cb7d64b56ad50bd58e3097ed9 |
| institution | Directory Open Access Journal |
| issn | 2291-8639 |
| language | English |
| last_indexed | 2024-03-13T03:26:28Z |
| publishDate | 2023-06-01 |
| publisher | Etamaths Publishing |
| record_format | Article |
| series | International Journal of Analysis and Applications |
| spelling | doaj.art-9251ca2cb7d64b56ad50bd58e3097ed92023-06-25T06:29:46ZengEtamaths PublishingInternational Journal of Analysis and Applications2291-86392023-06-0121575710.28924/2291-8639-21-2023-572209A PDE Approach to the Problems of Optimality of ExpectationsMahir Hasanov0a:1:{s:5:"en_US";s:18:"Beykent University";}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: R×R → 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 ϕ to find F (x, y), satisfying, arg ming(Z)E[F(X, g(Z))] = ϕ(E[X|Z]), for all X and Z. In more general setting, a stronger problem is to find F (x, y) satisfying arg miny∈RE[F (X, y)] = ϕ(E[X]), ∀X. We study this problem and develop a partial differential equation (PDE) approach to solution of these problems.http://etamaths.com/index.php/ijaa/article/view/2824 |
| spellingShingle | Mahir Hasanov A PDE Approach to the Problems of Optimality of Expectations International Journal of Analysis and Applications |
| title | A PDE Approach to the Problems of Optimality of Expectations |
| title_full | A PDE Approach to the Problems of Optimality of Expectations |
| title_fullStr | A PDE Approach to the Problems of Optimality of Expectations |
| title_full_unstemmed | A PDE Approach to the Problems of Optimality of Expectations |
| title_short | A PDE Approach to the Problems of Optimality of Expectations |
| title_sort | pde approach to the problems of optimality of expectations |
| url | http://etamaths.com/index.php/ijaa/article/view/2824 |
| work_keys_str_mv | AT mahirhasanov apdeapproachtotheproblemsofoptimalityofexpectations AT mahirhasanov pdeapproachtotheproblemsofoptimalityofexpectations |