Concavity and sigmoidicity hypotheses are developed as a natural extension of the simple ordered hypothesis in normal means. Those hypotheses give reasonable shape constraints for obtaining a smooth response curve in the non-parametric input±output analysis. The slope change and in¯ection point models are introduced correspondingly as the corners of the polyhedral cones de®ned by those isotonic hypotheses. Then a maximal contrast type test is derived systematically as the likelihood ratio test for each of those changepoint hypotheses. The test is also justi®ed for the original isotonic hypothesis by a complete class lemma. The component variables of the resulting test statistic have second or third order Markov property which, together with an appropriate nonlinear transformation, leads to an exact and very ef®cient algorithm for the probability calculation. Some considerations on the power of the test are given showing this to be a very promising way of approaching to the isotonic inference.
We discuss the application of orthogonal polynomial to estimation of probability density functions, particularly for accessing features of a portfolio's profit/loss distribution. Such expansions are given by the sum of known orthogonal polynomials multiplied by an associated weight function.However, naïve applications of expansion methods are flawed. The shape of the estimator's tail can undulate, under the influence of the constituent polynomials in the expansion, and can even exhibit regions of negative density.This paper presents techniques to redeem these flaws and to improve quality of risk estimation. We show that by targeting a smooth density which is sufficiently close to the target density, we can obtain expansionbased estimators which do not have the shortcomings of equivalent naïve estimators. In particular, we apply optimisation and smoothing techniques which place greater weight on the tails than the body of the distribution.Numerical examples using both real and simulated data illustrate our approach. We further outline how our techniques can apply to a wide class of expansion methods, and indicate opportunities to extend to the multivariate case, where distributions of individual component risk factors in a portfolio can be accessed for the purpose of risk management.
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