Mixture Of Beta Distributions Research Articles

Maximum approximate likelihood estimation in accelerated failure time model for interval-censored data.

The approximate Bernstein polynomial model, a mixture of beta distributions, is applied to obtain maximum likelihood estimates of the regression coefficients, the baseline density and the survival functions in an accelerated failure time model based on interval censored data including current status data. The estimators of the regression coefficients and the underlying baseline density function are shown to be consistent with almost parametric rates of convergence under some conditions for uncensored and/or interval censored data. Simulation shows that the proposed method is better than its competitors. The proposed method is illustrated by fitting the Breast Cosmetic and the HIV infection time data using the accelerated failure time model.

MANAGEMENT OF TESTS OF COMPLEX TECHNICAL SYSTEMS BASED ON A CONSISTENT RELIABILITY ASSESSMENT AND TAKING INTO ACCOUNT A PRIORI DATA ON THEIR ELEMENTS

The article deals with the problem of reducing the volume of tests of complex systems by using a priori data on the reliability of their elements. At the preliminary stage, the a priori distribution of the probability of failure of the system as a whole is determined. To do this, the results of element tests are processed and the parameters of the a posteriori probability distribution of element failure are determined based on the Bayesian procedure. The type of distribution law (beta distribution) is chosen from the conjugacy condition. Statistical modeling of the system failure probability of a known structural-logical reliability scheme is performed for random values of the failure probabilities of each element, set in accordance with the obtained distribution law. The system failure probability distribution law is formed as a mixture of beta distributions; the advantage of this distribution law is a fairly high accuracy of the simulation data description and conjugacy to the binomial distribution. The parameters of a mixture of beta distributions are determined using the EM (Expectation-Maximization) algorithm. The quality of selection of the desired distribution density is checked using the nonparametric Kolmogorov criterion. When testing the system, after each experiment, the a posteriori density of the probability distribution is recalculated; it is represented as a mixture of beta distributions with a constant proportion of components. The parameters of each element of the mixture are easily determined by the results of the experiment. As a point Bayesian estimate, the average value calculated from the a posteriori distribution is taken, the confidence interval for a given confidence probability is found as the central interval. An example is given and the possibility of minimizing the number of tests is shown.

Assessment of uncertainty in bid arrival times: A Bayesian mixture model

In this paper, we propose a Bayesian approach to model uncertainty in the bid arrival time by focusing on the time of the first bid in secondary (retail) market online business-to-business auctions. The proposed model is based on a Bayesian finite mixture of beta distributions. Our main objectives is to study potential heterogeneity of different auctions. In doing so, we incorporate some auction-specific features into the model and analyze their effect on the first bid time. We consider multiple competing models both in terms of fit and predictive performance. We also discuss managerial implications of the study and suggest how auctioneers can benefit from both the explanatory and predictive aspects of the model.

Measuring Segregation When Units are Small: A Parametric Approach

This article considers the issue of measuring segregation in a population of units that contain few individuals (e.g., establishments, classrooms). When units are small, the usual segregation indices, which are based on sample proportions, are biased. We propose a parametric solution: the probability that an individual within a given unit belongs to the minority is assumed to be distributed as a mixture of Beta distributions. The model can be estimated and indices deduced. Simulations show that this new method performs well compared to existing ones, even in the case of misspecification. An application to residential segregation in France according to parents’ nationalities is then undertaken. This article has online supplementary materials.

Dot Detection of Braille Images Using A Mixture of Beta Distributions

Problem statement: Braille is a tactile format of written communication for people with low vision and blindness worldwide. Optical Braille Recognition (OBR) offers many benefits to Braille users and people who work with them. Approach: This study presents an algorithm for detecting dots composing braille characters in an image of embossed braille material obtained by an optical scanner. We assumed that a mixture of Beta distributions could model the histogram of a scanned braille document. The core of the proposed method was the use of stability of thresholding with Beta distribution to initiate the process of thresholds estimation. Segmented Braille image was used to form a grid that contains recto dots and another one that contains verso dots. Results: Braille dots composing characters on both singlesided and double-sided documents were automatically identified from those grids with excellent accuracy. Conclusion: The experiment showed that the proposed method obtained very good results but it requires more testing on different scanned Braille document images.

High-dimensional generation of Bernoulli random vectors

The objective of this paper is to explore different modeling strategies to generate high-dimensional Bernoulli vectors. We discuss the multivariate Bernoulli (MB) distribution, probe its properties and examine three models for generating random vectors. A latent multivariate normal model whose bivariate distributions are approximated with Plackett distributions with univariate normal distributions is presented. A conditional mean model is examined where the conditional probability of success depends on previous history of successes. A mixture of beta distributions is also presented that expresses the probability of the MB vector as a product of correlated binary random variables. Each method has a domain of effectiveness. The latent model offers unpatterned correlation structures while the conditional mean and the mixture model provide computational feasibility for high-dimensional generation of MB vectors.

Bayesian Estimation Using Warner's Randomized Response Model through Simple and Mixture Prior Distributions

Bayesian estimation of population proportion of a sensitive characteristic is proposed by adopting a simple beta distribution and a mixture of Beta distributions as quantification of prior information using simple random sampling with replacement. In the sequel application of the stratified random sampling is also studied in Bayesian scenario. It is assumed that data are collected through Warner (1965) randomized response technique. To study the performance of Bayesian estimators we have used Mean Squared Error (MSE) and/or Relative Efficiency (RE) as performance criterion. Further, comparison of the suggested estimator is made with Kim et al. (2006) stratified estimator and usual maximum likelihood estimator in case of stratified random sampling. It is observed that unlike the moment and maximum likelihood methods, proposed Bayesian estimation method is free of the problems of having an estimate of population proportion outside the interval (0, 1) and large variance when the sample proportion of yes responses is very low or very high.

A Note on Approximating Continuous Distribution Functions

Abtsrcat This paper deals with the approximation of distribution func- tions of complicated statistics using Beta distribution. The problem is motivated and the performance of our method is evaluated by several examples. Two methods are suggested to estimate the parameters of Beta distribution. We approximate distributions of cusum statistic and Hsu's G test statistic by this method. Error analysis is given. Fitting the mixture of Beta distributions are examined. Finally, a discussion Section is given.

Skin Images Segmentation

Problem statement: Image segmentation is a fundamental step in many a pplications of image processing. Skin cancer has been the most common of all new cancers detected each year. At early stage detection of skin cancer, simple and ec onomic treatment can cure it mostly. An accurate segmentation of skin images can help the diagnosis to define well the region of the cancer. The principal approach of segmentation is based on thre sholding (classification) that is lied to the probl em of the thresholds estimation. Approach: The objective of this study is to develop a method to segment the skin images based on a mixture of Beta distribu tions. We assume that the data in skin images can be modeled by a mixture of Beta distributions. We u sed an unsupervised learning technique with Beta distribution to estimate the statistical parameters of the data in skin image and then estimate the thresholds for segmentation. Results: The proposed method of skin images segmentation was implemented and tested on different skin images. We obtained very good results in comparing with the same techniques with Gamma distribution. Conclusion: The experiment showed that the proposed method obtained very good results but it requires m ore testing on different types of skin images.

Mean functions based on meta-mixtures in nonhomogeneous Poisson processes

This paper deals with the software reliability model based on a nonhomogeneous Poisson process. We introduce new types of mean functions which can be either NHPP-I or NHPP-II according to the choice of the distribution function. The proposed mean function is motivated by the fact that a strictly monotone increasing function can be modeled by a distribution function and an unknown distribution function approximated by a mixture of beta distributions. Some existing mean functions can be regarded as special cases of the proposed mean functions. The EM algorithm is used to obtain maximum likelihood estimates of the parameters in the proposed model.

Model-based clustering of DNA methylation array data: a recursive-partitioning algorithm for high-dimensional data arising as a mixture of beta distributions

BackgroundEpigenetics is the study of heritable changes in gene function that cannot be explained by changes in DNA sequence. One of the most commonly studied epigenetic alterations is cytosine methylation, which is a well recognized mechanism of epigenetic gene silencing and often occurs at tumor suppressor gene loci in human cancer. Arrays are now being used to study DNA methylation at a large number of loci; for example, the Illumina GoldenGate platform assesses DNA methylation at 1505 loci associated with over 800 cancer-related genes. Model-based cluster analysis is often used to identify DNA methylation subgroups in data, but it is unclear how to cluster DNA methylation data from arrays in a scalable and reliable manner.ResultsWe propose a novel model-based recursive-partitioning algorithm to navigate clusters in a beta mixture model. We present simulations that show that the method is more reliable than competing nonparametric clustering approaches, and is at least as reliable as conventional mixture model methods. We also show that our proposed method is more computationally efficient than conventional mixture model approaches. We demonstrate our method on the normal tissue samples and show that the clusters are associated with tissue type as well as age.ConclusionOur proposed recursively-partitioned mixture model is an effective and computationally efficient method for clustering DNA methylation data.

Estimating potential environmental loadings of Cryptosporidium spp. and Campylobacter spp. from livestock in the Grand River Watershed, Ontario, Canada.

Exposure to waterborne pathogens in recreational or drinking water is a serious public health concern. Thus, it is important to determine the sources of pathogens in a watershed and to quantify their environmental loadings. The natural variability of potentially pathogenic microorganisms in the environment from anthropogenic, natural, and livestock sources is large and has been difficult to quantify. A first step in characterizing the risk of nonpoint source contamination from pathogens of livestock origin is to determine the potential environmental loading based on animal prevalence and fecal shedding intensity. This study developed a probabilistic model for estimating the production of Cryptosporidium spp. and Campylobacter spp. from livestock sources within a watershed. Probability density functions representing daily pathogen production rates from livestock were simulated for the Grand River Watershed in southwestern Ontario. The prevalence of pathogenic microorganisms in animals was modeled as a mixture of beta-distributions with parameters drawn from published studies. Similarly, gamma-distributions were generated to describe animal pathogen shedding intensity. Results demonstrate that although cattle are responsible for the largest amount of manure produced, other domesticated farm animals contribute large numbers of the two pathogenic microorganisms studied. Daily pathogen production rates are highly sensitive to the parameters of the gamma-distributions, illustrating the need for reliable data on animal shedding intensity. The methodology may be used for identifying source terms for pathogen fate and transport modeling and for defining and targeting regions that are most vulnerable to water contamination from pathogenic sources.

DENSITY FUNCTION APPROXIMATION WITH PARTICULAR REFERENCE TO THE DISTRIBUTION OF OVERLAP OF CIRCLES

summaryGiven that two circles overlap, the area in common is a function of the distance between their centres. This paper adopts a suitable random distribution for the intercentre distance and then derives the distribution of the area of overlap. An approximation is sought for the density function using a criterion which enables bounds to be placed on the difference between the moments of the density function and those of the approximation. This is an approach of general applicability. The importance of matching the end‐point behaviour of the density and the approximation is stressed. It is shown that the distribution of the area of overlap may be well approximated by a mixture of beta distributions in which the parameters change smoothly with the ratio of radii.

On the Exact Distribution of a Normalized Ratio of the Weighted Geometric Mean to the Unweighted Arithmetic Mean in Samples from Gamma Distributions

Abstract The distribution of the product of several independent beta random variables as a mixture of beta distributions is derived by using a solution of Wilks's type B integral equation. This distribution is applied to finding the distribution of a normalized ratio of the weighted geometric mean (GM) to the unweighted arithmetic mean (AM) in random samples consisting of observations from gamma distributions, one observation being taken from each distribution. Two upper bounds for the truncation error related to the mixture representation are obtained. Application of these results to problems in testing statistical hypotheses is indicated.

Central and Noncentral Distributions of Wilks' Statistic in Manova as Mixtures of Incomplete Beta Functions

In this paper it is shown that the distribution of the product of independently distributed beta random variables can be represented as a mixture of beta distributions. This result is then applied to derive the distributions of Wilks' statistic in the central and the noncentral linear cases. In the noncentral linear case the results of computations agree with those of Gupta (1971).

The distribution of the product of two noncentral beta variates

In this paper the exact distribution of the product of two noncentral beta variates is derived using Mellin integral transform. The density function of the product is represented as a mixture of Beta distributions and the distribution function as a mixture of Incomplete Beta Functions.