Building new probability distributions: the composition method and a computer based method

Abstract

We discuss the creation of new probability distributions for continuous data in two distinct approaches. The first one is, to our knowledge, novelty and consists of using Estimation of Distribution Algorithms (EDAs) to obtain new cumulative distribution functions. This class of algorithms work as follows. A population of solutions for a given problem is randomly selected from a space of candidates, which may contain candidates that are not feasible solutions to the problem. The selection occurs by following a set of probability rules that, initially, assign a uniform distribution to the space of candidates. Each individual is ranked by a fitness criterion. A fraction of the most fit individuals is selected and the probability rules are then adjusted to increase the likelihood of obtaining solutions similar to the most fit in the current population. The algorithm iterates until the set of probability rules are able to provide good solutions to the problem. In our proposal, the algorithm is used to generate cumulative distribution functions to model a given continuous data set. We tried to keep the mathematical expressions of the new functions as simple as possible. The results were satisfactory. We compared the models provided by the algorithm to the ones in already published papers. In every situation, the models proposed by the algorithms had advantages over the ones already published. The main advantage is the relative simplicity of the mathematical expressions obtained. Still in the context of computational tools and algorithms, we show the performance of simple neural networks as a method for parameter estimation in probability distributions. The motivation for this was the need to solve a large number of non linear equations when dealing with SAR images (SAR stands for synthetic aperture radar) in the statistical treatment of such images. The estimation process requires solving, iteratively, a non-linear equation. This is repeated for every pixel and an image usually consists of a large number of pixels. We trained a neural network to approximate an estimator for the parameter of interest. Once trained, the network can be fed the data and it will return an estimate of the parameter of interest without the need of iterative methods. The training of the network can take place even before collecting the data from the radar. The method was tested on simulated and real data sets with satisfactory results. The same method can be applied to different distributions. The second part of this thesis shows two new probability distribution classes obtained from the composition of already existing ones. In each situation, we present the new class and general results such as power series expansions for the probability density functions, expressions for the moments, entropy and alike. The first class is obtained from the composition of the beta-G and Lehmann-type II classes. The second class, from the transmuted-G and Marshall-Olkin-G classes. Distributions in these classes are compared to already existing ones as a way to illustrate the performance of applications to real data sets.