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In recent years, modifications of the classical Lindley distribution have been considered by many authors. In this paper, we introduce a new generalization of the Lindley distribution based on a mixture of exponential and gamma distributions with different mixing proportions and compare its performance with its sub-models. The new distribution accommodates the classical Lindley, Quasi Lindley, Two-parameter Lindley, Shanker, Lindley distribution with location parameter, and Three-parameter Lindley distributions as special cases. Various structural properties of the new distribution are discussed and the size-biased and the lengthbiased are derived. A simulation study is conducted to examine the mean square error for the parameters by means of the method of maximum likelihood. Finally, simulation studies and some real-world data sets are used to illustrate its flexibility in terms of its location, scale and shape parameters. [ABSTRACT FROM AUTHOR]

The first-ever real data application of a two-component Burr mixture distribution is provided. It is fitted to three loss data sets: fire loss claims in Denmark, fire loss claims for three building categories in Belgium and fire loss data in Norway. Each of these data sets exhibits significant bimodality. The fits of the two-component Burr mixture distribution are compared to those of five other two-component mixture distributions: the two-component Weibull mixture, two-component gamma mixture, two-component Pareto mixture, two-component lognormal mixture and the two-component exponential mixture distributions. The Burr mixture distribution is shown to give the best fit for each data set. The relative performances of the fitted distributions were assessed in terms of Akaike information criterion values, Bayesian information criterion values, consistent Akaike information criterion values, corrected Akaike information criterion values, Hannan-Quinn criterion values, density plots and probability-probability plots. [ABSTRACT FROM AUTHOR]

A new probability distribution function (PDF) called hyper-Exponentially Modified Gaussian (hyper-EMG) is introduced for the analysis of high-resolution spectra from multiple-reflection time-of-flight mass spectrometers. The hyper-EMG consists of a central Gaussian distribution modified by multiple exponential tails with different strengths at one or both sides. The basic statistical properties of the new PDF are given and the analysis of mass spectra containing separated and overlapping peaks is presented. The main requirement is to accurately determine the positions and areas of the individual mass peaks. From the distances of positions the mass values can be determined, from the areas the population of different ground and isomeric states can be obtained. The hyper-EMG has been applied to high-resolution time and mass spectra characterized by mass resolving powers of 140,000 and 520,000 obtained with Cs + 133 and K + 39 ions, respectively. From the measured mass distribution of K + 39 ions, an overlapping distribution of two peaks with an area ratio of 1:10 and a mass difference of 2.6 ppm (parts-per-million) is generated and analyzed. The results reveal significant advantages of the new PDF for the evaluation of overlapping distributions for accurate mass and area determinations compared with commonly used PDFs which are more than one order of magnitude less accurate. It is obvious that the hyper-EMG can be favorably applied also to other fields. [ABSTRACT FROM AUTHOR]

The accelerated failure time (AFT) model is a well-known alternative to the Cox proportional hazard model for analyzing time-to-event data. In this paper we consider fitting an AFT model to right censored data when a predictor variable is subject to measurement errors. First, without measurement errors, estimation of the model parameters in the AFT model is a challenging task due to the presence of censoring, especially when no specific assumption is made regarding the distribution of the logarithm of the time-to-event. The model complexity increases when a predictor is measured with error. We propose a non-parametric Bayesian method for analyzing such data. The novel component of our approach is to model (1) the distribution of the time-to-event, (2) the distribution of the unobserved true predictor, and (3) the distribution of the measurement errors all non-parametrically using mixtures of the Dirichlet process priors. Along with the parameter estimation we also prescribe how to estimate survival probabilities of the time-to-event. Some operating characteristics of the proposed approach are judged via finite sample simulation studies. We illustrate the proposed method by analyzing a data set from an AIDS clinical trial study. [ABSTRACT FROM AUTHOR]

We show that if the density of the absolutely continuous part of a copula is bounded away from zero on a set of Lebesgue measure 1, then that copula generates “lower ψ -mixing” stationary Markov chains. This conclusion implies ϕ -mixing, ρ -mixing, β -mixing and “interlaced ρ -mixing”. We also provide some new results on the mixing structure of Markov chains generated by mixtures of copulas. [ABSTRACT FROM AUTHOR]

Mixture probability density functions have recently been proposed to describe some fertility patterns characterized by a bimodal shape. These functions are adequate when the fertility pattern is actually bimodal, but they appear to be less useful when the shape of age-specific fertility rates is unimodal. A further model is proposed, based on flexible skew symmetric probability density functions. This model can fit symmetric and skew patterns as well as reflect humps that are observed in some fertility patterns. It is more parsimonious than mixture distributions and more flexible, showing a good fit with several shapes (bimodal or unimodal) of fertility patterns. Empirical evaluation of the model proposed and comparisons with other functions for Italian data from 1952 to 2003 and US data from 1933 to 2006 are also discussed. [ABSTRACT FROM AUTHOR]

Simple, fast methods for modeling the portfolio distribution corresponding to a non-elliptical, leptokurtic, asymmetric, and conditionally heteroskedastic set of asset returns are entertained. Portfolio optimization via simulation is demonstrated, and its benefits are discussed. An augmented mixture of normals model is shown to be superior to both standard (no short selling) Markowitz and the equally weighted portfolio in terms of out of sample returns and Sharpe ratio performance. [ABSTRACT FROM AUTHOR]

Chromatin ImmunoPrecipitation-sequencing (ChIP-seq) experiments have now become routine in biology for the detection of protein-binding sites. In this paper, we present a Markov random field model for the joint analysis of multiple ChIP-seq experiments. The proposed model naturally accounts for spatial dependencies in the data, by assuming first-order Markov dependence and, for the large proportion of zero counts, by using zero-inflated mixture distributions. In contrast to all other available implementations, the model allows for the joint modeling of multiple experiments, by incorporating key aspects of the experimental design. In particular, the model uses the information about replicates and about the different antibodies used in the experiments. An extensive simulation study shows a lower false non-discovery rate for the proposed method, compared with existing methods, at the same false discovery rate. Finally, we present an analysis on real data for the detection of histone modifications of two chromatin modifiers from eight ChIP-seq experiments, including technical replicates with different IP efficiencies. [ABSTRACT FROM AUTHOR]

The exclusion restriction is usually assumed for identifying causal effects in true or only natural randomized experiments with noncompliance. It requires that the assignment to treatment does not have a direct causal effect on the outcome. Despite its importance, the restriction can often be unrealistic, especially in situations of natural experiments. It is shown that, without the exclusion restriction, the parametric model is identified if the outcome distributions of various compliance statuses are in the same parametric class and that class is a linearly independent set over the field of real numbers. However, the relaxation of the exclusion restriction yields a parametric model that is characterized by the presence of mixtures of distributions. This scenario complicates the likelihood-based estimation procedures because it implies more than one maximum likelihood point. A two-step estimation procedure based on detecting the root that is closest to the method of moments estimate of the parameter vector is then proposed and analyzed in detail, under normally distributed outcomes. An economic example with real data concerning returns to schooling concludes the paper. [ABSTRACT FROM AUTHOR]

Mixture models have recently received a considerable attention of the analysts. These can be regarded as finite mixture of various components of the distributions, each component describing the different causes of failure. This paper considers the Bayesian analysis of the two-component mixture of Topp Leone distribution based on type I censored samples. The statistical properties of point and interval estimators have been discussed under a class of priors using four loss functions. As some of the results are not in closed form, we have used the iterative methods to obtain the numerical results. The performance of different point and interval estimators has been compared under an extensive simulation study. The applicability of the results has been illustrated by a real life example. [ABSTRACT FROM AUTHOR]

Given two independent non-degenerate positive random variables X and Y, Lukacs (Ann Math Stat 26:319-324, ) proved that X/( X + Y) and X + Y are independent if and only if X and Y are gamma distributed with the same scale parameter. In this work, under the assumption X/ U and U are independent, and X/ U has a $${{\mathcal Be}(p,\,q)}$$ distribution, we characterize the distribution of ( U, X) by the condition E( h( U − X)| X) = b, where h is allowed to be a linear combination of exponential functions. Since the assumption for X and U above is equivalent to X| U being $${\mathcal{B}e(p,\,1)}$$ scaled by U, hence our results can be viewed as identification of a power distribution mixture. [ABSTRACT FROM AUTHOR]

An important issue in both welfare and development economics is the interaction between institutions and economic outcomes. While welfarists are typically concerned with how these variables contribute to overall wellbeing, empirical assessments of their joint contribution are limited. Development economists, on the other hand, have focused extensively on whether institutions cause or are caused by growth yet the relevant literature is still rife with debate. In this article, we use a notion of distributional dominance to tackle both the measurement of multivariate welfare and the evaluation of inter-temporal dependence without hindrance from the mix of discrete (political) and continuous (economic) variables in our data set. On the causality front, our results support the view that institutions promote growth more than growth promotes institutions. On the welfare front, we find that economic growth had a positive impact from 1960 to 2000 but declines in institutional quality over the earlier part of this period were sufficient to produce a decline in overall wellbeing until the mid-1970s. Subsequent improvements in institutions then reversed the trend and, ultimately, wellbeing in 2000 was higher than that in 1960. [ABSTRACT FROM AUTHOR]

Randomized experiments are the gold standard for evaluating proposed treatments. The intent to treat estimand measures the effect of treatment assignment, but not the effect of treatment if subjects take treatments to which they are not assigned. The desire to estimate the efficacy of the treatment in this case has been the impetus for a substantial literature on compliance over the last 15 years. In papers dealing with this issue, it is typically assumed there are different types of subjects, for example, those who will follow treatment assignment (compliers), and those who will always take a particular treatment irrespective of treatment assignment. The estimands of primary interest are the complier proportion and the complier average treatment effect (CACE). To estimate CACE, researchers have used various methods, for example, instrumental variables and parametric mixture models, treating compliers as a single class. However, it is often unreasonable to believe all compliers will be affected. This article therefore treats compliers as a mixture of two types, those belonging to a zero-effect class, others to an effect class. Second, in most experiments, some subjects drop out or simply do not report the value of the outcome variable, and the failure to take into account missing data can lead to biased estimates of treatment effects. Recent work on compliance in randomized experiments has addressed this issue by assuming missing data are missing at random or latently ignorable. We extend this work to the case where compliers are a mixture of types and also examine alternative types of nonignorable missing data assumptions. [ABSTRACT FROM AUTHOR]

We resolve a useful formulation of the question how a statistician can coherently incorporate the information in a consulted expert's probability assessment for an event into a personal posterior probability assertion. Using a framework that recognises the total information available as composed of units available only to each of them along with units available to both, we show: that a sufficient statistic for all the information available to both the expert and the statistician is the product of their odds ratios in favour of the event; that the geometric mean of their two probabilities specifies a contour of pairs of assertions in the unit-square that yield the same posterior probability; that the information-combining function is parameterised by an unknown probability for the event conditioned only on the unspecified information common to both the statistician and the expert; and that an assessable mixing distribution over this unspecified probability allows an integrable mixture distribution to represent a computable posterior probability. The exact results allow the identification of the subclass of coherent probabilities that are externally Bayesian operators. This subclass is equivalent to the class of combining functions that honour the principles of uniformity and compromise. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of mixture models based on the inverse Gaussian distribution, which is highly flexible and contains several well-known probability models. The new class of models is generated from symmetric distributions around zero by using the connection between the inverse Gaussian and standard normal distributions. We illustrate the obtained results by means of two real data sets through likelihood, goodness-of-fit and diagnostic methods. This illustration indicates the adequacy of the new model. [ABSTRACT FROM AUTHOR]

Abstract: Risk analyses made in the area of seismic activity are going to be of great importance in determining the earthquake interoccurence times. Several statistical methods have been developed for this purpose. Recently, Exponential, Gamma and Weibull distributions are the frequently used methods in this regard. In this study, we investigate the interoccurence time statistics of earthquakes which occurred in the area coordinated 39º–42º N latitude and 30º–40º E longitude in the North Anatolian Fault Zone (NAFZ) between the years 1960–2008, with a mixture of two different distributions of Exponential, Gamma and Weibull and a mixture of the same kind of distribution. We found that the mixture distributions are more suitable than the other examined distribution models for small magnitudes (m c ≥ 3). Also Weibull-Gamma and Weibull-Exponential distributions are agreeable for large magnitudes (m c ≥ 5). [Copyright &y& Elsevier]

Abstract: In this paper we study the robustness of the directional mean (a.k.a. circular mean) for different families of circular distributions. We show that the directional mean is robust in the sense of finite standardized gross error sensitivity (SB-robust) for the following families: (1) mixture of two circular normal distributions, (2) mixture of wrapped normal and circular normal distributions and (3) mixture of two wrapped normal distributions. We also show that the directional mean is not SB-robust for the family of all circular normal distributions with varying concentration parameter. We define the circular trimmed mean and prove that it is SB-robust for this family. In general the property of SB-robustness of an estimator at a family of probability distributions is dependent on the choice of the dispersion measure. We introduce the concept of equivalent dispersion measures and prove that if an estimator is SB-robust for one dispersion measure then it is SB-robust for all equivalent dispersion measures. Three different dispersion measures for circular distributions are considered and their equivalence studied. [ABSTRACT FROM AUTHOR]

Progressive encoding of a signal generally involves an estimation step, designed to reduce the entropy of the residual of an observation over the entropy of the observation itself. Oftentimes the conditional distributions of an observation, given already-encoded observations, are well fit within a class of symmetric and unimodal distributions (e.g., the two-sided geometric distributions in images of natural scenes, or symmetric Paretian distributions in models of financial data). It is common practice to choose an estimator that centers, or aligns, the modes of the conditional distributions, since it is common sense that this will minimize the entropy, and hence the coding cost of the residuals. But with the exception of a special case, there has been no rigorous proof. Here we prove that the entropy of an arbitrary mixture of symmetric and unimodal distributions is minimized by aligning the modes. The result generalizes to unimodal and rotation-invariant distributions in Rn. We illustrate the result through some experiments with natural images. [ABSTRACT FROM AUTHOR]

The aim of the paper is to go beyond the detection of outliers in multivariate time series, and to find regularities in the effect of special events on the series. The tool is a factor model in which the direction of every column of the loading matrix is identified, in contrast with Gaussian factor models, where only the span of the whole loading matrix is identified. Under asymptotics for rare and influential stochastic outliers, it is shown that the outliers' location is estimated consistently, and outliers are consistently classified into the factor components that have generated them. The direction, but not the length, of every column of the loading matrix is also estimated consistently. Inference on the directions, which underlies the interpretation of the factor structure, is asymptotically Gaussian under conditions that include Gaussianity of the innovations after accounting for the outliers. The model, augmented with a VAR specification for the conditional mean, provides a statistically acceptable and historically meaningful description of bond rates series for Denmark, Germany and the Netherlands. [ABSTRACT FROM AUTHOR]

We consider the problem of learning mixtures of product distributions over discrete domains in the distribution learning framework introduced by Kearns et al. [Proceedings of the 26th Annual Symposium on Theory of Computing (STOC), Montréal, QC, 1994, ACM, New York, pp. 273-282]. We give a poly(n/ ϵ)-time algorithm for learning a mixture of κ arbitrary product distributions over the n-dimensional Boolean cube {0, 1}n to accuracy ϵ, for any constant κ. Previous polynomial-time algorithms could achieve this only for κ = 2 product distributions; our result answers an open question stated independently in [M. Cryan, Learning and Approximation Algorithms for Problems Motivated by Evolutionary Trees, Ph.D. thesis, University of Warwick, Warwick, UK, 1999] and [Y. Freund and Y. Mansour, Proceedings of the 12th Annual Conference on Computational Learning Theory, 1999, pp. 183-192]. We further give evidence that no polynomial-time algorithm can succeed when κ is superconstant, by reduction from a difficult open problem in PAC (probably approximately correct) learning. Finally, we generalize our poly(n/ ϵ)-time algorithm to learn any mixture of κ = O(1) product distributions over {0, 1, . . . , b - 1}n, for any b = O(1). [ABSTRACT FROM AUTHOR]

This paper proposes a simple and effective method for selecting the number of components in density mixtures. The proposed method is based on a combination of Stein's unbiased risk estimation and the plug-in principle. Also provided by this paper are numerical results comparing several popular methods for Gaussian mixture estimation. These results demonstrate the superior performance of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract: Following Gupta and Wesolowski [1997. Uniform mixtures via posterior means. Ann. Inst. Statist. Math. 49, 171–180], in this work, under the condition and U are independent, has a distribution, and given X the conditional expectation of a certain function of is constant, we characterize the distribution of . This problem is related to Lukacs type characterization, where both X and Y have to be gamma distributed with the same scale parameter, if both X and Y, and and are independent. Among others, we prove if , and for some integer , , where , are real constants such that and , or for some real number , , where is a constant, then the distribution of can be determined. [Copyright &y& Elsevier]

The issue of normalization arises whenever two different values for a vector of unknown parameters imply the identical economic model. A normalization implies not just a rule for selecting which among equivalent points to call the maximum likelihood estimate (MLE), but also governs the topography of the set of points that go into a small-sample confidence interval associated with that MLE. A poor normalization can lead to multimodal distributions, disjoint confidence intervals, and very misleading characterizations of the true statistical uncertainty. This paper introduces an identification principle as a framework upon which a normalization should be imposed, according to which the boundaries of the allowable parameter space should correspond to loci along which the model is locally unidentified. We illustrate these issues with examples taken from mixture models, structural vector autoregressions, and cointegration models. [ABSTRACT FROM AUTHOR]

Abstract: The stochastic nature of high-throughput screening (HTS) data indicates that information may be gleaned by applying statistical methods to HTS data. A foundation of parametric statistics is the study and elucidation of population distributions, which can be modeled using modern spreadsheet software. The methods and results described here use fundamental concepts of statistical population distributions analyzed using a spreadsheet to provide tools in a developing armamentarium for extracting information from HTS data. Specific examples using two HTS kinase assays are analyzed. The analyses use normal and gamma distributions, which combine to form mixture distributions. HTS data were found to be described well using such mixture distributions, and deconvolution of the mixtures to the constituent gamma and normal parts provided insight into how the assays performed. In particular, the proportion of hits confirmed was predicted from the original HTS data and used to assess screening assay performance. The analyses also provide a method for determining how hit thresholds—values used to separate active from inactive compounds—affect the proportion of compounds verified as active and how the threshold can be chosen to optimize the selection process. [Copyright &y& Elsevier]

Absentee data is analyzed using a worker environmental survey carried out in 1995. In a model which distinguishes between discretionary and involuntary absences individuals are assumed to maximize expected utility. This generates a probability distribution of days absent per year which is a mixture of two negative binomial distributions representing the behaviour of two unknown types of agent. This distribution is estimated separately for men and women. For each gender the number of days absent is quite small relative to what researchers have found in other countries. In the empirical model we identify different effects of the explanatory variables for workers with low rates of absenteeism and workers with high rates of absenteeism. [ABSTRACT FROM AUTHOR]

Many traditional econometric methods forecast the conditional distribution of asset returns by a point prediction of volatility. Alternatively, forecasts of this distribution may be generated from a mixture of distributions. This paper proposes a method by which information extracted from the estimation of a standard stochastic volatility model (using non-linear filtering) can be used to generate mixture distribution forecasts. In general, it is found that forecasts based on mixture distributions are superior to those simply using point predictions of volatility. In terms of mixture distribution forecasts, the method proposed in this paper is found to be superior to a number of competing approaches. Copyright © 2006 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we obtain a new mixture distribution, which generalizes some other mixture distributions established earlier. Some basic functions associated with the probability density function of the mixture distribution, such as kth moments, characteristic function and factorial moments are computed. Further we obtain a three-term recurrence relation for the established mixture distribution. [Copyright &y& Elsevier]

Pareto, Zipf and numerous subsequent investigators of inverse power distributions have often represented their findings as though their data conformed to a power law form for all ranges of the variable of interest. I refer to this ideal case as a strong inverse power law (SIPL). However, many of the examples used by Pareto and Zipf, as well as others who have followed them, have been truncated data sets, and if one looks more carefully in the lower range of values that was originally excluded, the power law behavior usually breaks down at some point. This breakdown seems to fall into two broad cases, called here (1) weak and (2) false inverse power laws (WIPL and FIPL, resp.). Case 1 refers to the situation where the sample data fit a distribution that has an approximate inverse power form only in some upper range of values. Case 2 refers to the situation where a highly truncated sample from certain exponential-type (and in particular, "lognormal-like") distributions can convincingly mimic a power law. The main objectives of this paper are (a) to show how the discovery of Pareto-Zipf-type laws is closely associated with truncated data sets; (b) to elaborate on the categories of strong, weak and false inverse power laws; and (c) to analyze FIPLs in some detail. I conclude that many, but not all, Pareto--Zipf examples are likely to be FIPL finite mixture distributions and that there are few genuine instances of SIPLs. [ABSTRACT FROM AUTHOR]

Variability is the heterogeneity of values within a population. Uncertainty refers to lack of knowledge regarding the true value of a quantity. Mixture distributions have the potential to improve the goodness of fit to data sets not adequately described by a single parametric distribution. Uncertainty due to random sampling error in statistics of interests can be estimated based upon bootstrap simulation. In order to evaluate the robustness of using mixture distribution as a basis for estimating both variability and uncertainty, 108 synthetic data sets generated from selected population mixture log-normal distributions were investigated, and properties of variability and uncertainty estimates were evaluated with respect to variation in sample size, mixing weight, and separation between components of mixtures. Furthermore, mixture distributions were compared with single-component distributions. Findings include: (1) mixing weight influences the stability of variability and uncertainty estimates; (2) bootstrap simulation results tend to be more stable for larger sample sizes; (3) when two components are well separated, the stability of bootstrap simulation is improved; however, a larger degree of uncertainty arises regarding the percentiles coinciding with the separated region; (4) when two components are not well separated, a single distribution may often be a better choice because it has fewer parameters and better numerical stability; and (5) dependencies exist in sampling distributions of parameters of mixtures and are influenced by the amount of separation between the components. An emission factor case study based upon NO x emissions from coal-fired tangential boilers is used to illustrate the application of the approach. [ABSTRACT FROM AUTHOR]

A random-effects regression model that allows the random coefficients to have a multivariate normal mixture distribution is described for classifying treatment response in longitudinal clinical trials. The proposed model is capable of dealing with longitudinal data from unknown heterogeneous populations. As applied to longitudinal clinical trials, for example, the model can distinguish subgroups of treatment response. Use of the proposed model is illustrated by analyzing data from two psychiatric clinical trials. The first includes depressed patients assigned to drug treatment who are repeatedly measured in terms of their level of depression. The second trial examined schizophrenic patients longitudinally who were assigned to either a drug or placebo condition. For both, the random-effects mixture model allows an assessment of whether patients comprise distinct populations in terms of their treatment response. Based on parameter estimates of the mixture model, ample evidence for a mixture of response to treatment is observed for both datasets. [ABSTRACT FROM AUTHOR]

Linear discriminant analysis for a two-class case is studied in the presence of misallocation in training samples. A general approach to modeling of misallocation is formulated, and the mean vectors and covariance matrices of the mixture distributions are derived. The asymptotic distribution of the discriminant boundary is obtained, arid the asymptotic first two moments of the error rates are given. Certain numerical results for the error rates are presented by considering the random and two nonrandom misallocation models. [ABSTRACT FROM AUTHOR]