Research Letter Generalized Cumulative Residual Entropy for Distributions with Unrestricted Supports

We consider the cumulative residual entropy (CRE) a recently introduced measure of entropy. While in previous works distributions with positive support are considered, we generalize the definition of CRE to the case of distributions with general support. We show that several interesting properties of the earlier CRE remain valid and supply further properties and insight to problems such as maximum CRE power moment problems. In addition, we show that this generalized CRE can be used as an alternative to differential entropy to derive information-based optimization criteria for system identification purpose

Generalized Weibull and Inverse Weibull Distributions with Applications

In this thesis, new classes of Weibull and inverse Weibull distributions including the generalized new modified Weibull (GNMW), gamma-generalized inverse Weibull (GGIW), the weighted proportional inverse Weibull (WPIW) and inverse new modified Weibull (INMW) distributions are introduced. The GNMW contains several sub-models including the new modified Weibull (NMW), generalized modified Weibull (GMW), modified Weibull (MW), Weibull (W) and exponential (E) distributions, just to mention a few. The class of WPIW distributions contains several models such as: length-biased, hazard and reverse hazard proportional inverse Weibull, proportional inverse Weibull, inverse Weibull, inverse exponential, inverse Rayleigh, and Frechet distributions as special cases. Included in the GGIW distribution are the sub-models: gamma-generalized inverse Weibull, gamma-generalized Frechet, gamma-generalized inverse Rayleigh, gamma-generalized inverse exponential, inverse Weibull, inverse Rayleigh, inverse exponential, Frechet distributions. The INMW distribution contains several sub-models including inverse Weibull, inverse new modified exponential, inverse new modified Rayleigh, new modified Frechet, inverse modified Weibull, inverse Rayleigh and inverse exponential distributions as special cases. Properties of these distributions including the behavior of the hazard function, moments, coefficients of variation, skewness, and kurtosis, s-entropy, distribution of order statistic and Fisher information are presented. Estimates of the parameters of the models via method of maximum likelihood (ML) are presented. Extensive simulation study is conducted and numerical examples are given

Residual Entropy Function and Its Applications

The word ‗information‘ is very common word used in everyday language. Information transmission usually occurs through human voice (as in telephone, radio, television, etc.), books, newspapers, letters, etc. In all these cases a piece of information is transmitted from one place to another. However, one might like to quantitatively assess the quality of information contained in a piece of information

The Gamma-Generalized Inverse Weibull Distribution with Applications to Pricing and Lifetime Data

A new distribution called the gamma-generalized inverse Weibull distribution which includes inverse exponential, inverse Rayleigh, inverse Weibull, Frechet, generalized inverse Weibull, gamma-exponentiated inverse exponential, exponentiated inverse exponential, Zografos and Balakrishnan-generalized inverse Weibull, Zografos and Balakrishnan-inverse Weibull, Zografos and Balakrishnan-generalized inverse exponential, Zografos and Balakrishnan-inverse exponential, Zografos and Balakrishnan-generalized inverse Rayleigh, Zografos and Balakrishnan-inverse Rayleigh, and Zografos and Balakrishnan-Fr\u27echet distributions as special cases is proposed and studied in detail. Some structural properties of this new distribution including density expansion, moments, Renyi entropy, distribution of the order statistics, moments of the order statistics and L-moments are presented. Maximum likelihood estimation technique is used to estimate the model parameters and applications to a real datasets to illustrate its usefulness are presented

Neural Systems with Numerically Matched Input-Output Statistic: Isotonic Bivariate Statistical Modeling

Bivariate statistical modeling from incomplete data is a useful statistical tool that allows to discover the model underlying two data sets when the data in the two sets do not correspond in size nor in ordering. Such situation may occur when the sizes of the two data sets do not match (i.e., there are “holes” in the data) or when the data sets have been acquired independently. Also, statistical modeling is useful when the amount of available data is enough to show relevant statistical features of the phenomenon underlying the data. We propose to tackle the problem of statistical modeling via a neural (nonlinear) system that is able to match its input-output statistic to the statistic of the available data sets. A key point of the new implementation proposed here is that it is based on look-up-table (LUT) neural systems, which guarantee a computationally advantageous way of implementing neural systems. A number of numerical experiments, performed on both synthetic and real-world data sets, illustrate the features of the proposed modeling procedure

USING GENERATIVE ADVERSARIAL NETWORK AS A VALUE-AT-RISK ESTIMATOR

Value-at-risk (VaR) estimation is a critical task for modern financial institution. Most methods to estimate VaR rely on classical statistical methods. They produce reliable estimates but there is demand for ever more accurate estimates. Recently there has been major breakthroughs for machine learning models in other fields. This has led to increasing interest in applying machine learning for financial applications. This thesis applies new data-driven machine learning method, generative adversarial network (GAN), for (VaR) estimation. GAN was proposed for fake image generation. Since then it has found applications in multiple domains, such as finance. Estimating the true underlying distribution of financial time series is notoriously difficult task. GAN doesn’t explicitly estimate the underlying distribution but tries to generate new samples from the distribution. This thesis applies a basic GAN model to simulate stock market returns and then estimate the VaR from these. The experiments are conducted on S&P500-index. The GAN model is compared to a simple historical simulation baseline. In the experiments it becomes evident that the GAN model lacks robustness and responds poorly to changes in market. The GAN is unable to fully capture the statistical properties of stock market returns. It can replicate a little of the excess kurtosis present in stock market returns and some of the volatility clustering. The results show that the GAN model has tendency to estimate the VaR between a fairly narrow range. This is in contrast to historical simulation, which can respond to changes in the stock market. Machine learning models, especially neural networks like GANs, present challenges to financial practitioners. Although they provide sometimes more accurate estimates than traditional methods, they lack transparency. GANs have shown promise in the literature but suffer from being unstable to train. It is difficult to guess will a trained GAN work as it is meant to work. Regardless of these shortcomings, it is worthwhile to study GANs and other neural networks in finance. They have performed exceptionally in other fields. Researchers must try to open the black-box nature of the models. Interpretability of the models will allow their use in the financial industry. This thesis shows that more research is needed to provide robust estimates that can be relied on

Evolution and entropy in the organization of urban street patterns

The street patterns of cities are the result of long-term evolution and interaction between various internal, social and economic, and external, environmental and landscape, processes and factors. In this article, we use entropy as a measure of dispersion to study the effects of landscapes on the evolution and associated street patterns of two cities: Dundee in Eastern Scotland and Khorramabad in Western Iran, cities which have strong similarities in terms of the size of their street systems and populations but considerable differences in terms of their evolution within the landscape. Landscape features have strong effects on the city shape and street patterns of Dundee, which is primarily a shoreline city, while Khorramabad is primarily located within mountainous and valley terrain. We show how cumulative distributions of street lengths when graphed as log-log plots show abrupt changes in their straight-line slopes at lengths of about 120 m, indicating a change in street functionality across scale: streets shorter than 120 m are primarily local streets, whereas longer streets are mainly collectors and arterials. The entropy of a street-length population varies positively over its average length and length range which is the difference between the longest and the shortest streets in a population. Similarly, the entropies of the power law tails of the street populations of both cities have increased during their growth, indicating that the distribution of street lengths has gradually become more dispersed as these cities have expanded. © 2013 Copyright Taylor and Francis Group, LLC