Showing papers on "K-distribution published in 2016"

Solutions to some functional equations and their applications to characterization of probability distributions

Abstract: involving an unknown function ?5 of a single variable t Conditions under which the unknown functions in these two types of equations are polynomials of an assigned degree are given The third, on the characterization of normal and gamma distributions, extends the earlier work of the authors (Rao, 1967 and Khatri and Rao, 1968*) We consider two sets of functions Li9 , Lq and Mlf , Mp of independent random variables Xl9 Xn with the condition

Topp–Leone Family of Distributions: Some Properties and Application

Abstract: In this paper we have proposed a new family of distributions; the Topp–Leone family of distributions. We have given general expression for density and distribution function of the new family. Expression for moments and hazard rate has also been given. We have also given an example of the proposed family.

The Metalog Distributions

Abstract: The metalog distributions constitute a new system of continuous univariate probability distributions designed for flexibility, simplicity, and ease/speed of use in practice. The system is comprised of unbounded, semibounded, and bounded distributions, each of which offers nearly unlimited shape flexibility compared to previous systems of distributions. Explicit shape-flexibility comparisons are provided. Unlike other distributions that require nonlinear optimization for parameter estimation, the metalog quantile functions and probability density functions have simple closed-form expressions that are quantile parameterized linearly by cumulative-distribution-function data. Applications in fish biology and hydrology show how metalogs may aid data and distribution research by imposing fewer shape constraints than other commonly used distributions. Applications in decision analysis show how the metalog system can be specified with three assessed quantiles, how it facilities Monte Carlo simulation, and how applying it aided an actual decision that would have been made wrongly based on commonly used discrete methods.

Proximity of probability distributions in terms of Fourier-Stieltjes transforms

Abstract: A survey is given of some results on smoothing inequalities for various probability metrics (in particular, for the Kolmogorov distance), and some analogues of these results in the class of functions of bounded variation are presented. Bibliography: 61 titles.

Garhy-Generated Family of Distributions with Application

Abstract: This paper introduces a new family of continuous distributions called a Garhy generated family of distributions. Some mathematical properties of this family are discussed. The derived properties are hold to any proper distribution in this family. Some special sub-models in the new family are derived. General explicit expressions for the quantile function, ordinary and incomplete moments, generating function and order statistics are obtained. The estimation of the model parameters is discussed by using maximum likelihood and the potentiality of the extended family is illustrated with one application to real data. Keywords : Kumaraswamy distribution; Exponetiated distribution; Moments; quantile function, Maximum likelihood estimation.

Probability density function formalism for optical coherence tomography signal analysis: a controlled phantom study.

Abstract: The distribution of backscattered intensities as described by the probability density function (PDF) of tissue-scattered light contains information that may be useful for tissue assessment and diagnosis, including characterization of its pathology. In this Letter, we examine the PDF description of the light scattering statistics in a well characterized tissue-like particulate medium using optical coherence tomography (OCT). It is shown that for low scatterer density, the governing statistics depart considerably from a Gaussian description and follow the K distribution for both OCT amplitude and intensity. The PDF formalism is shown to be independent of the scatterer flow conditions; this is expected from theory, and suggests robustness and motion independence of the OCT amplitude (and OCT intensity) PDF metrics in the context of potential biomedical applications.

Inverse Weibull power series distributions: properties and applications

Abstract: Inverse Weibull (IW) distribution is one of the widely used probability distributions for nonnegative data modelling, specifically, for describing degradation phenomena of mechanical components. In this paper, by compounding IW and power series distributions we introduce a new lifetime distribution. The compounding procedure follows the same set-up carried out by Adamidis and Loukas [A lifetime distribution with decreasing failure rate. Stat Probab Lett. 1998;39:35–42]. We provide mathematical properties of this new distribution such as moments, estimation by maximum likelihood with censored data, inference for a large sample and the EM algorithm to determine the maximum likelihood estimates of the parameters. Furthermore, we characterize the proposed distributions using a simple relationship between two truncated moments and maximum entropy principle under suitable constraints. Finally, to show the flexibility of this type of distributions, we demonstrate applications of two real data sets.

Distributions to model overdispersed count data

Abstract: In the early twentieth century, only a few count distributions (binomial and Poisson distributions) were commonly used in modeling. These distributions fail to model bimodal or overdispersed data, especially data related to phenomena for which the occurrence of a given event increases the chance of additional events occurring. New count distributions have since been introduced to address such phenomena; they are named "contagious" distributions. This group of distributions, which includes the negative binomial, Neyman, Thomas and Polya-Aeppli distributions, can be expressed as mixture distributions or as stopped-sum distributions. They take into account bimodality and overdispersion, and show a greater flexibility with regards to value distributions. The aim of this literature review is to 1) explain the introduction of these distributions, 2) describe each of these overdispersed distributions, focusing in particular on their definitions, their basic properties, and their practical utility, and 3) compare their strengths and weaknesses by modeling overdispersed real count data (bovine tuberculosis cases).

Improved Shape Parameter Estimation in K Clutter with Neural Networks and Deep Learning

Abstract: The discrimination of the clutter interfering signal is a current problem in modern radars� design, especially in coastal or offshore environments where the histogram of the background signal often displays heavy tails. The statistical characterization of this signal is very important for the cancellation of sea clutter, whose behavior obeys a K distribution according to the commonly accepted criterion. By using neural networks, the authors propose a new method for estimating the K shape parameter, demonstrating its superiority over the classic alternative based on the Method of Moments. Whereas both solutions have a similar performance when the entire range of possible values of the shape parameter is evaluated, the neuronal alternative achieves a much more accurate estimation for the lower Fig.s of the parameter. This is exactly the desired behavior because the best estimate occurs for the most aggressive states of sea clutter. The final design, reached by processing three different sets of computer generated K samples, used a total of nine neural networks whose contribution is synthesized in the final estimate, thus the solution can be interpreted as a deep learning approximation. The results are to be applied in the improvement of radar detectors, particularly for maintaining the operational false alarm probability close to the one conceived in the design.

Gompertz-power series distributions

Abstract: In this article, we introduce the Gompertz power series (GPS) class of distributions which is obtained by compounding Gompertz and power series distributions. This distribution contains several lifetime models such as Gompertz-geometric (GG), Gompertz-Poisson (GP), Gompertz-binomial (GB), and Gompertz-logarithmic (GL) distributions as special cases. Sub-models of the GPS distribution are studied in details. The hazard rate function of the GPS distribution can be increasing, decreasing, and bathtub-shaped. We obtain several properties of the GPS distribution such as its probability density function, and failure rate function, Shannon entropy, mean residual life function, quantiles, and moments. The maximum likelihood estimation procedure via a EM-algorithm is presented, and simulation studies are performed for evaluation of this estimation for complete data, and the MLE of parameters for censored data. At the end, a real example is given.

Evaluating Choquet Integrals Whose Arguments are Probability Distributions

Abstract: We describe the use of the Choquet integral for finding a mean-like aggregated value of a collection of arguments with respect to a fuzzy measure. We observe the need for ordering the arguments in using this integral. We consider the case where the arguments being aggregated are random variables, probability distributions. In this case, we are faced with the problem of having to order probability distributions. Given the difficulty of obtaining a linear ordering over a collection of probability distributions, we must search for other methods for obtaining a Choquet type aggregation of a collection of probability distributions that does not require a linear ordering; we refer to these as surrogates. Here, we provide one surrogate for calculating the Choquet integral in the case where the objects being aggregated are probability distributions called the probabilistic exceedance method.

Consistency of the penalized MLE for two-parameter gamma mixture models

Abstract: Two-parameter gamma distributions are widely used in liability theory, lifetime data analysis, financial statistics, and other areas. Finite mixtures of gamma distributions are their natural extensions, and they are particularly useful when the population is suspected of heterogeneity. These distributions are successfully employed in various applications, but many researchers falsely believe that the maximum likelihood estimator of the mixing distribution is consistent. Similarly to finite mixtures of normal distributions, the likelihood function under finite gamma mixtures is unbounded. Because of this, each observed value leads to a global maximum that is irrelevant to the true distribution. We apply a seemingly negligible penalty to the likelihood according to the shape parameters in the fitted model. We show that this penalty restores the consistency of the likelihoodbased estimator of the mixing distribution under finite gamma mixture models. We present simulation results to validate the consistency conclusion, and we give an example to illustrate the key points.

Some families of generalized Mathieu-type power series, associated probability distributions and related functional inequalities involving complete monotonicity and log-convexity

Abstract: By making use of the familiar Mathieu series and its generalizations, the authors derive a number of new integral representations and present a systematic study of probability density functions and probability distributions associated with some generalizations of the Mathieu series. In particular, the mathematical expectation, variance and the characteristic functions, related to the probability density functions of the considered probability distributions are derived. As a consequence, some interesting inequalities involving complete monotonicity and log-convexity are derived.

Exact distributions of order statistics of dependent random variables from ln,p-symmetric sample distributions, n ∈ {3,4}

Abstract: Integral representations of the exact distributions of order statistics are derived in a geometric way when three or four random variables depend on each other as the components of continuous ln,psymmetrically distributed random vectors do, n ∈ {3,4}, p > 0. Once the representations are implemented in a computer program, it is easy to change the density generator of the ln,p-symmetric distribution with another one for newly evaluating the distribution of interest. For two groups of stock exchange index residuals, maximum distributions are compared under dependence and independence modeling.

Mixture of compound-Gaussian distributions for radar sea-clutter modeling

Abstract: Compound Gaussian models are mostly considered for describing radar sea-clutter returns and are the basis of adaptive target detection with false alarm rate regulation When high resolution radars operate at small grazing angles, the existing compound Gaussian distributions with additive thermal noise could not fit accurately the empirical data in some cases This communication emphasises on the statistical description of the sea clutter using a mixture of compound inverse Gaussian (CIG) distribution, K distribution and generalized Pareto distribution (GP) with additive thermal noise Non-linear least squares curve fitting technique based on the Nelder-Mead algorithm is used to find simultaneously optimal parameters values of the mixture model Experiments comparisons are conducted to show a goodness of fit of the proposed mixture model for modelling the McMaster IPIX backscatter

Statistical Analysis of Phase-Only Correlation Functions with Phase-Spectrum Differences Following Wrapped Distributions

Abstract: This paper proposes statistical analysis of phase-only correlation functions with phase-spectrum differences following wrapped distributions. We first assume phase-spectrum differences between two signals to be random variables following a linear distribution. Next, based on directional statistics, we convert the linear distribution into a wrapped distribution by wrapping the linear distribution around the circumference of the unit circle. Finally, we derive general expressions of the expectation and variance of the POC functions with phase-spectrum differences following wrapped distributions. We obtain exactly the same expressions between a linear distribution and its corresponding wrapped distribution. key words: phase-only correlation functions, wrapped distributions, directional statistics, circular probability distributions

Some families of count distributions for modelling zero-inflation and dispersion / Low Yeh Ching

Abstract: A popular distribution for the modelling of discrete count data is the Poisson distribution However, count data usually exhibit over dispersion or under dispersion when modelled by a Poisson distribution in empirical modelling The presence of excess zeros is also closely related to over dispersion Two new mixed Poisson distributions, namely a three-parameter Poisson-exponentiated Weibull distribution and a fourparameter generalized Sichel distribution is introduced to model over dispersed, zeroinflated and long-tailed count data Some of the theoretical properties of the distributions are derived and the distributions' characteristics are studied A Monte Carlo simulation technique is examined and employed to overcome the computational issues arising from the intractability of the probability mass function of some mixed Poisson distributions For parameter estimation, the simulated annealing global optimization routine and an EM-algorithm type approach for maximum likelihood estimation are studied Examples are provided to compare the proposed distributions with several other existing mixed Poisson models Another approach to modelling count data is by examining the relationship between the counts of number of events which has occurred up to a fixed time t and the inter-arrival times between the events in a renewal process A family of count distributions, which is able to model under- and over dispersion, is presented by considering the inverse Gaussian distribution, the convolution of two gamma distributions and a finite mixture of exponential distributions as the distribution of the inter-arrival times The probability function of the counts is often complicated thus a method using numerical Laplace transform inversion for computing the probabilities and the renewal function is proposed Parameter estimation with maximum likelihood estimation is considered with applications of the count distributions to under dispersed and over dispersed count data from the literature

Distributions, Moments, and Statistics

Abstract: The moments of probability distributions represent the link between theory and observations since they are readily accessible to measurement. Rather abstract-looking generating functions have become important as highly versatile concepts and tools for solving specific problems. The probability distributions which are most important in applications are reviewed. Then the central limit theorem and the law of large numbers are presented. The chapter is closed by a brief digression into mathematical statistics and shows how to handle real world samples that cover a part, sometimes only a small part, of sample space.

Image retrieval method based on Bessel statistic model

Abstract: The invention discloses an image retrieval method based on a Bessel statistic model. Similarity calculation time can be reduced, and retrieval efficiency can be improved. The image retrieval method includes the following steps that one image is decomposed through non-subsample Shearlet shear wave transformation, and a low-frequency sub-band and multiple high-frequency sub-bands are obtained; a Bessel K distribution probability density function is used for conducting statistical modeling on each high-frequency sub-band, and shape parameters p and scale parameters c are estimated; (c,p) values corresponding to all the high-frequency sub-bands of each image serve as image features, and the image features form mapping with the original images in a one-to-one correspondence mode to serve as an image feature library for retrieval use; the Euclidean distance is used as a method for calculating the similarity between the images to calculate the similarity between different images, and results are sorted and output according to the similarity from large to small.

A New Information Theoretical Concept: Information-Weighted Heavy-tailed Distributions.

Abstract: Given an arbitrary continuous probability density function, it is introduced a conjugated probability density, which is defined through the Shannon information associated with its cumulative distribution function. These new densities are computed from a number of standard distributions, including uniform, normal, exponential, Pareto, logistic, Kumaraswamy, Rayleigh, Cauchy, Weibull, and Maxwell-Boltzmann. The case of joint information-weighted probability distribution is assessed. An additive property is derived in the case of independent variables. One-sided and two-sided information-weighting are considered. The asymptotic behavior of the tail of the new distributions is examined. It is proved that all probability densities proposed here define heavy-tailed distributions. It is shown that the weighting of distributions regularly varying with extreme-value index $\alpha > 0$ still results in a regular variation distribution with the same index. This approach can be particularly valuable in applications where the tails of the distribution play a major role.

Using Card Games for Conditional Probability, Explaining Gamma vs. Poisson Distributions, and Weighing Central Limit Theory

Abstract: Students taking probability course for the first time are often struggling with conditional probability. To help explain the concept better, card games are used to explain especially differences of conditional probabilities of sequential and simultaneous card draws. Several card game experiments are discussed and typical probability results are shown and compared with predictions. These simple experiments can be demonstrated in classroom and students can use them to test the predictions. Another concept students often struggle is distinguishing gamma from Poisson distribution. An identity connecting them, generalizing the connection between exponential and Poisson distribution, will be used to discuss their differences and to point out nuances in the wording of some probability problems that yield different answers when both distributions are used. Lastly, a teaching tool for explaining central limit theorem is discussed based on guessing weights of books. This guessing game proves useful to explain sampling distribution.

Modeling of Insurance Data through Two Heavy Tailed Distributions: Computations of Some of Their Actuarial Quantities through Simulation from Their Equilibrium Distributions and the Use of Their Convolutions

Abstract: In this paper, we have fitted two heavy tailed distributions viz the Weibull distribution and the Burr XII distribution to a set of Motor insurance claim data. As it is known, the probability of ruin is obtained as a solution to an integro differential equation, general solution of which leads to what is known as the Pollaczek-Khinchin Formula for the probability of ultimate ruin. In case, the claim severity is distributed as the above two mentioned distributions, and Pollaczek-Khinchin formula cannot be used to evaluate the probability of ruin through inversion of their Laplace transform since the Laplace Transforms themselves don’t have closed form expression. However, an approximation to the probability of ultimate ruin in such cases can be obtained by the Pollaczek-Khinchin formula through simulation and one crucial step in this simulation is to simulate from the corresponding Equilibrium distribution of the claim severity distribution. The paper lays down methodologies to simulate from the Equilibrium distribution of Burr XII distribution and Weibull distribution and has used them to obtain an approximation to the probability of ultimate ruin through Pollaczek-Khinchin formula by Monte Carlo simulation. An attempt has also been made to obtain numerical values to the probability function for the number of claims until ruin in case of zero initial surplus under these claim severity distributions and this in turn necessitates the computation of the convolutions of these distributions. The paper makes a preliminary effort to address this issue. All the computations are done under the assumption of the Classical Risk Model.

Transformation of distributions into heavy tailed

Abstract: We consider the transformation of the Raleigh distribution into a new distribution so that the new distribution behaves approximately the same as the Rayleigh for small values of the argument but becomes heavy tailed for large values