Some Variational Methods And Numerical Solutions For Estimation Of Nonparametric Distributions

This paper ?rst proposes and studies the following maximum penalized likelihood method for solving a density function: where α > 0 is a regularization parameter, L > 0 is an upper bound of the density function f, Φ(f) is the penalized term of the density function, and H([a, b])is the function space in which f lies. If Φ(f) =∫b af′2dx or Φ(f) =∫b af′′2dx, it is proved the method has a unique solution in a certain Sobolev space. Then,we use the ?nite element method(resp. the B-spline method) to discretize the above method associated with Φ(f) =∫b af′2dx(resp. Φ(f) =∫b af′′2dx), and discuss the in?uence on the accuracy of the numerical solution in terms of the sample capacity m, the mesh size h, and the regularization parameter α, through a series of numerical experiments. Compared to the existing methods, our methods ensure the unique existence of the solution, are convenient for algorithm implementation, and have desired computational performance.Next, we apply the optimization theory in Banach spaces to derive the ?rst order necessary condition for the following minimization problem de?ned in the Sobolev space H1([a, b]): where J : H1([a, b]) â†' R is a functional with required smoothness. In particular,we obtain the result corresponding to J(f) =∫b aL(F(x), f(x), x)dx, and compare it with the existing E-L equation for the problem. Meanwhile, we point out the mathematical mistakes implied in Theorem 2 in [16] and give the correct result in this discrete case.