Probability distribution estimation using control theoretic smoothing splines
In this paper, we examine the relationship between optimal control and statistics. We explore the use of control theoretic smoothing splines in the estimation of continuous probability distribution functions defined on a finite interval [0,T], where the data is summarized by empirical probability distributions. In particular, we consider the estimation of distributions of the form exp(f(t)), where there is no restriction on the sign of f(t). The construction of the optimal smoothed curve, y(t), is based on the minimization of an integral cost function done through the application of the Hilbert Projection Theorem, which guarantees that a unique minimum exists. Further spline construction is implemented in approximating cumulative distribution functions where the Hilbert space methodology is no longer applicable. This estimation is based on a process of iterative optimization through dynamic programming.