The standard solution of a set of linear algebraic simultaneous equations with empirical coefficients in the case of rather large variance of the coefficients may appear to be unstable or even nonexisting, which is due to the ill-posed solution and inconsistency of the simultaneous equations. The regularization methods due to Tikhonov impose the somewhat artificial constraint of minimal complexity on the solutions, and, while guaranteeing the existence of a solution, minimize neither the quadratic risk of the estimator of the solution nor the residual. There are methods of construction of maximum likelihood estimators for random sets of simultaneous linear algebraic equations. But these methods require averaging over a large number of realizations of the coefficients of the simultaneous equations, because good properties of the maximum likelihood estimators manifest themselves for large samples only. Moreover, it is obvious that a priori information on the distribution of errors in the coefficients of the linear algebraic simultaneous equations and on the right-hand sides can be used to refine the solutions.
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