Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  consider the problem of estimating an unknown deterministic parameter vector in a linear model with a random model matrix, with known second-order statistics.  first seek the linear estimator that minimizes the worst-case mean-squared error (MSE) across all parameter vectors whose (possibly ighted) norm is bounded above.  show that the minimax MSE estimator can be found by solving a semidefinite programming problem and develop necessary and sufficient optimality conditions on the minimax MSE estimator. Using these conditions,  derive closed-form expressions for the minimax MSE estimator in some special cases.  then demonstrate, through examples, that the minimax MSE estimator can improve the performance over both a Baysian approach and a least-squares method.  then consider the case in which the norm of the parameter vector is also bounded below. Since the minimax MSE approach cannot account for a nonzero lor bound,  consider, in this case, a minimax regret method in which  seek the estimator that minimizes the worst-case difference beten the MSE attainable using a linear estimator that does not know the parameter vector, and the optimal MSE attained using a linear estimator that knows the parameter vector. For analytical tractability,  restrict our attention to the scalar case and develop a closed-form expression for the minimax regret estimator.
['Yonina C. Eldar']
I am writing a research paper on a idea Minimax estimation of deterministic parameters in linear models with a random model matrix
[{"from": "human", "value": "I am writing a research paper on a idea Minimax estimation of deterministic parameters in linear models with a random model matrix"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  consider the problem of estimating an unknown deterministic parameter vector in a linear model with a random model matrix, with known second-order statistics.  first seek the linear estimator that minimizes the worst-case mean-squared error (MSE) across all parameter vectors whose (possibly ighted) norm is bounded above.  show that the minimax MSE estimator can be found by solving a semidefinite programming problem and develop necessary and sufficient optimality conditions on the minimax MSE estimator. Using these conditions,  derive closed-form expressions for the minimax MSE estimator in some special cases.  then demonstrate, through examples, that the minimax MSE estimator can improve the performance over both a Baysian approach and a least-squares method.  then consider the case in which the norm of the parameter vector is also bounded below. Since the minimax MSE approach cannot account for a nonzero lor bound,  consider, in this case, a minimax regret method in which  seek the estimator that minimizes the worst-case difference beten the MSE attainable using a linear estimator that does not know the parameter vector, and the optimal MSE attained using a linear estimator that knows the parameter vector. For analytical tractability,  restrict our attention to the scalar case and develop a closed-form expression for the minimax regret estimator."}]