Inference, information theory and segmentation based on an extended Cauchy-Rayleigh distribution : applications to heavy tailed data

Abstract

The Cauchy-Rayleigh (CR) distribution has been successfully used to describe asymmetrical and heavy-tail events from radar imagery. Employing such model to describe lifetime data may then seem attractive, but drawbacks arise: its probability density function does not cover non-modal behavior as well as its hazard rate function (hrf) assumes only one form. To outperform this difficulty, it is investigated the exponentiated Cauchy-Rayleigh (ECR) distribution. This byparameteric model is flexible enough to accommodate hrf with decreasing, decreasing-increasing-decreasing and upside-down bathtub forms. Several closed-form mathematical expressions for the ECR model are obtained: median, mode, some moments, (h;Φ)-entropies and Fisher information matrix. Their non-existence respective cases are also determined. It is proposed two estimators for the ECR parameters: maximum likelihood (ML) and percentile-based methods. Both of this methods may be biased for small and moderate sample sizes. To overcome it we furnish a expression for its second-order bias according to Cox and Snell (1968) and propose a third bias-corrected ML estimator. Further discussions about hypotheses-based inference and estimation formulas on censored-data are furnished as well. A simulation study is done to assess the estimators performance. The estimates existence and uniqueness are not guaranteed, thus procedures to constrained estimation are developed to overcome this trouble. Notes about hypothesis tests are given under censored and uncensored schemes. An application in a survival dataset illustrates the proposed model usefulness. Results point out that the ECR distribution may outperform classical lifetime biparametric models, such as the gamma, Birnbaum-Saunders, Weibull and log-normal laws, before heavy-tail data. The likelihood ratio test are compared against entropy-based tests as urban texture detector using a San Francisco synthetic aperture radar imagery. This same image is also the final application target which consist of compare segmentation algorithms based on CR and ECR entropies densities finite mixture. It is recurred to a result due Pardo et al. (1997) which provides the (h;Φ)-entropies asymptotic distribution. The finite mixture log-likelihood function is maximized using the Expectation-Maximization algorithm.