Estimation of Discrete Probablility Distribution with a Pareto Tail

University Of North Carolina At Charlotte, Charlotte NC

Investigators

Abstract

The central theme of statistical sciences may be described as making inferences about the probability distribution of a random variable based on a sample of the random variable driven by the underlying distribution. Over the central region of the data range, the higher data frequency often enables more reliable estimation of the distribution. However, regardless how large a sample may be, there are always regions in the range of the random variable where few or no sample observations are available, for example, the tail region. Ironically it is on the tail regions where statistical inferences are often most important. The investigator studies the problem of estimating a parametric tail with power decay in, and only in, the extremely distant tail of a discrete probability distribution. The investigator proposes to develop a consistent estimator of the tail probability distribution via a perspective offered by Turing?s formula. Such an estimator, if established, would represent a previously unknown methodology which could shed light on an array of statistical problems involving tails of discrete probability distributions across a range of research disciplines. The proposed project is motivated by, in addition to its theoretical merits, many practically important problems. The central focus of the project is to provide a methodology to quantify the likelihood of an extremely rare event, so rare that it may not have been previously observed. For example, in finance, the assessment of value at risk may involve quantification of an extremely unlikely event that would cause a huge loss of value during a short time period in a portfolio; the scenarios of stress test for financial industry may also be beneficially considered under this methodology. In insurance, it may be of interest to assess the likelihood of a natural or personal disaster of extreme magnitude. In environmental biology, it may be of interest to assess bio-diversity in a population accounting for those super small minority species that are not represented in a sample. In homeland security, it may be of interest to assess the likelihood of a terrorist attack whose type is previously unobserved or unaware of. The proposed project provides an opportunity to enhance the ability to find solutions to all the above mentioned problems and beyond.