Pool Maximum Violation Region Algorithm to Obtain Closest Nonincreasing Function under Various Norms
Author : PY Patil
Abstract :
We came across estimation of an unknown probability density function. This unknown density function itself may have some known constraints. Therefore, it is trivial to expect the same constraints from its density estimator too. In this article we consider density estimator is given or already estimated but it may not satisfy requirement of non-increasingness property from its known modal point. Without loss of generality we assume mode is equal to zero. Hence, we provide an algorithm to obtain closest non-increasing function to a given density estimator under given measures of closeness.
We have described Pool Maximum Violation Region Algorithm (PMVRA) for obtaining closest non-increasing function to a given one. First we obtain maximum violation region of a given function, if violation of non-increasingness exists. After doing so, we give a closest constant value on the superset of maximum violation region when measure of closeness is Sup-norm, L1-norm and L2-norm.
In case of Sup-norm, we found closest constant on the superset of maximum violation region as arithmetic mean of smallest and largest values of the function on this superset. When measure of closeness is L1-norm, we found closest constant on the superset of maximum violation region as median of the function on this superset. Whereas, in case of measure of closeness is L2-norm, we found closest constant on the superset of maximum violation region as an arithmetic mean of the function on this superset.
Keywords :
Non-increasingness, maximum violation region, measure of closeness, sup-norm, l1-norm, l2-norm, pool maximum violation region algorithm (PMVRA)
