Simulation techniques for generalized Gaussian densities

This contribution deals with the Monte Carlo simulation of generalized Gaussian random variables. Such a parametric family of distributions has been proposed in many applications in science to describe physical phenomena and in engineering, and it seems to be also useful in modelling economic and financial data. For values of the shape parameter α within a certain range, the distribution presents heavy tails. In particular, the cases α=1/3 and α=1/2 are considered. For such values of the shape parameter, different simulation methods are assessed.     

Bayesian analysis for mixtures of discrete distributions with a non-parametric component

Bayesian finite mixture modelling is a flexible parametric modelling approach for classification and density fitting. Many areas of application require distinguishing a signal from a noise component. In practice, it is often difficult to justify a specific distribution for the signal component; therefore, the signal distribution is usually further modelled via a mixture of distributions. However, modelling the signal as a mixture of distributions is computationally non-trivial due to the difficulties in justifying the exact number of components to be used and due to the label switching problem. This paper proposes the use of a non-parametric distribution to model the signal component. We consider the case of discrete data and show how this new methodology leads to more accurate parameter estimation and smaller false non-discovery rate. Moreover, it does not incur the label switching problem. We show an application of the method to data generated by ChIP-sequencing experiments.     

Ecological regression analysis of environmental benzene exposure and childhood leukaemia: sensitivity to data inaccuracies, geographical scale and ecological bias

Benzene is classified as a group 1 human carcinogen by the International Agency for Research on Cancer, and it is now accepted that occupational exposure is associated with an increased risk of various leukaemias. However, occupational exposure accounts for less than 1% of all benzene exposures, the major sources being cigarette smoking and vehicle exhaust emissions. Whether such low level exposures to environmental benzene are also associated with the risk of leukaemia is currently not known. In this study, we investigate the relationship between benzene emissions arising from outdoor sources (predominantly road traffic and petrol stations) and the incidence of childhood leukaemia in Greater London. An ecological design was used because of the rarity of the disease, the difficulty of obtaining individual level measurements of benzene exposure and the availability of data. However, some methodological difficulties were encountered, including problems of case registration errors, the choice of geographical areas for analysis, exposure measurement errors and ecological bias. We use a Bayesian hierarchical modelling framework to address these issues, and we investigate the sensitivity of our inference to various modelling assumptions.     

Case–control genome-wide joint association study using semiparametric empirical model and approximate Bayes factor

We propose a semiparametric approach for the analysis of case–control genome-wide association study. Parametric components are used to model both the conditional distribution of the case status given the covariates and the distribution of genotype counts, whereas the distribution of the covariates are modelled nonparametrically. This yields a direct and joint modelling of the case status, covariates and genotype counts, and gives a better understanding of the disease mechanism and results in more reliable conclusions. Side information, such as the disease prevalence, can be conveniently incorporated into the model by an empirical likelihood approach and leads to more efficient estimates and a powerful test in the detection of disease-associated SNPs. Profiling is used to eliminate a nuisance nonparametric component, and the resulting profile empirical likelihood estimates are shown to be consistent and asymptotically normal. For the hypothesis test on disease association, we apply the approximate Bayes factor (ABF) which is computationally simple and most desirable in genome-wide association studies where hundreds of thousands to a million genetic markers are tested. We treat the approximate Bayes factor as a hybrid Bayes factor which replaces the full data by the maximum likelihood estimates of the parameters of interest in the full model and derive it under a general setting. The deviation from Hardy–Weinberg Equilibrium (HWE) is also taken into account and the ABF for HWE using cases is shown to provide evidence of association between a disease and a genetic marker. Simulation studies and an application are further provided to illustrate the utility of the proposed methodology.     

Subsampling,symmetrization, and robust interpolation

The recently developed subsampling methodology has been shown to be valid for the construction of large-sample confidence regions for a general unknown parameter 9 under very minimal conditions. Nevertheless, in some specific cases—e.g. in the case of the sample mean of i.i.d. data—it has been noted that the subsampling distribution estimator underperforms as compared to alternative estimators such as the bootstrap or the asymptotic normal distribution (with estimated variance). In the present report we introduce a (partially) symmetrized.     

Norm-preserving constraint in the Fisher–Rao registration and its application in signal estimation

Registration of temporal observations is a fundamental problem in functional data analysis. Various frameworks have been developed over the past two decades where registrations are conducted based on optimal time warping between functions. Comparison of functions solely based on time warping, however, may have limited application, in particular when certain constraints are desired in the registration. In this paper, we study registration with norm-preserving constraint. A closely related problem is on signal estimation, where the goal is to estimate the ground-truth template given random observations with both compositional and additive noises. We propose to adopt the Fisher–Rao framework to compute the underlying template, and mathematically prove that such framework leads to a consistent estimator. We then illustrate the constrained Fisher–Rao registration using simulations as well as two real data sets. It is found that the constrained method is robust with respect to additive noise and has superior alignment and classification performance to conventional, unconstrained registration methods.     

A weighted likelihood ratio of two related negative hypergeometric distributions

In this paper we consider some related negative hypergeometric distributions arising from the problem of sampling without replacement from an urn containing balls of different colours and in different proportions but stopping only after some specific number of balls of different colours have been obtained. With the aid of some simple recurrence relations and identities we obtain in the case of two colours the moments for the maximum negative hypergeometric distribution, the minimum negative hypergeometric distribution, the likelihood ratio negative hypergeometric distribution and consequently the likelihood proportional negative hypergeometric distribution. To the extent that the sampling scheme is applicable to modelling data as illustrated with a biological example and, in fact, many situations of estimating Bernoulli parameters for binary traits within a finite population, these are important first-step results.     

Retrospective Parametric Tests for Homogeneity of Data

The problem considered in this study is that of detecting a change in the unknown parameters of known distribution on the basis of a finite sequence of independent observations, assuming that if a change in the parameters of a distribution has occurred then it is unique. We examine various methods that have been suggested for this problem and suggest a uniform approach, based on likelihood ratio analysis. For tests derived this way we present approximations for levels of significance based on asymptotic analyses. The suggested tests meet the needs of specific problems (such as a one-sided alternative), for which general parametric case solutions have not been suggested explicitly before. We also find that rate of convergence of our asymptotics is fast, and provide accurate results for a level of significance of the suggested tests for sample sizes commonly observed in practice.     

The distance between random points in rectangles

Since 1943, numerous papers have discussed the problem of the distribution of the distance between random points in rectangles, considering special cases such as two points in the same square, points in adjacent squares, two rectangles sharing a side and others. The problems arise in a variety of settings: operations research, population studies, urban planning, physical chemistry, chemical physics and materials science. Reported results are all of special cases with formulas specific to each case. It is possible to put such problems in a general setting with a single formula that handles all the particular cases. The method is well suited to computing and use of graphics. Now that computers and graphic output are commonplace it seems worthwhile to describe the general method and provide program outlines for computing and plotting the resulting distributions. We do that in this article.     

Generalized gamma parameter estimation and moment evaluation

The generalized gamma distribution includes the exponential distribution, the gamma distribution, and the Weibull distribution as special cases. It also includes the log-normal distribution in the limit as one of its parameters goes to infinity. Prentice (1974) developed an estimation method that is effective even when the underlying distribution is nearly log-normal. He reparameterized the density function so that it achieved the limiting case in a smooth fashion relative to the new parameters. He also gave formulas for the second partial derivatives of the log-density function to be used in the nearly log-normal case. His formulas included infinite summations, and he did not estimate the error in approximating these summations.

We derive approximations for the log-density function and moments of the generalized gamma distribution that are smooth in the nearly log-normal case and involve only finite summations. Absolute error bounds for these approximations are included. The approximation for the first moment is applied to the problem of estimating the parameters of a generalized gamma distribution under the constraint that the distribution have mean one. This enables the development of a correspondence between the parameters in a mean one generalized gamma distribution and certain parameters in acoustic scattering theory.     

A flexible parametric survival model which allows a bathtub-shaped hazard rate function

A new parametric (three-parameter) survival distribution, the lognormal–power function distribution, with flexible behaviour is introduced. Its hazard rate function can be either unimodal, monotonically decreasing or can exhibit a bathtub shape. Special cases include the lognormal distribution and the power function distribution, with finite support. Regions of parameter space where the various forms of the hazard-rate function prevail are established analytically. The distribution lends itself readily to accelerated life regression modelling. Applications to five data sets taken from the literature are given. Also it is shown how the distribution can behave like a Weibull distribution (with negative aging) for certain parameter values.     

Analysis of accelerated life tests with weibull failure distribution via generalized linear models

Efficient industrial experiments for reliability analysis of manufactured goods may consist in subjecting the units to higher stress levels than those of the usual working conditions. This results in the so called "accelerated life tests" where, for each pre-fixed stress level, the experiment ends after the failure of a certain pre-fixed proportion of units or a certain test time is reached. The aim of this paper is to determine estimates of the mean lifetime of the units under usual working conditions from censored failure data obtained under stress conditions. This problem is approached through generalized linear modelling and related inferential techniques, considering a Weibull failure distribution and a log-linear stress-response relationship. The general framework considered has as particular cases, the Inverse Power Law model, the Eyring model, the Arrhenius model and the generalized Eyring model. In order to illustrate the proposed methodology, a numerical example is provided.     

Bayesian multivariate Poisson mixtures with an unknown number of components

In this paper we present Bayesian analysis of finite mixtures of multivariate Poisson distributions with an unknown number of components. The multivariate Poisson distribution can be regarded as the discrete counterpart of the multivariate normal distribution, which is suitable for modelling multivariate count data. Mixtures of multivariate Poisson distributions allow for overdispersion and for negative correlations between variables. To perform Bayesian analysis of these models we adopt a reversible jump Markov chain Monte Carlo (MCMC) algorithm with birth and death moves for updating the number of components. We present results obtained from applying our modelling approach to simulated and real data. Furthermore, we apply our approach to a problem in multivariate disease mapping, namely joint modelling of diseases with correlated counts.     

Dynamic reliability models with conditional proportional hazards

A dynamic approach to the stochastic modelling of reliability systems is further explored. This modelling approach is particularly appropriate for load-sharing, software reliability, and multivariate failure-time models, where component failure characteristics are affected by their degree of use, amount of load, or extent of stresses experienced. This approach incorporates the intuitive notion that when a set of components in a coherent system fail at a certain time, there is a jump from one structure function to another which governs the residual lifetimes of the remaining functioning components, and since the component lifetimes are intrinsically affected by the structure function which they constitute, then at such a failure time there should also be a jump in the stochastic structure of the lifetimes of the remaining components. For such dynamically-modelled systems, the stochastic characteristics of their jump times are studied. These properties of the jump times allow us to obtain the properties of the lifetime of the system. In particular, for a Markov dynamic model, specific expressions for the exact distribution functions of the jump times are obtained for a general coherent system, a parallel system, and a series-parallel system. We derive a new family of distribution functions which describes the distributions of the jump times for a dynamically-modelled system.     

Applications of the Bootstrap in ROC Analysis

The problem of estimating standard errors for diagnostic accuracy measures might be challenging for many complicated models. We can address such a problem by using the Bootstrap methods to blunt its technical edge with resampled empirical distributions. We consider two cases where bootstrap methods can successfully improve our knowledge of the sampling variability of the diagnostic accuracy estimators. The first application is to make inference for the area under the ROC curve resulted from a functional logistic regression model which is a sophisticated modelling device to describe the relationship between a dichotomous response and multiple covariates. We consider using this regression method to model the predictive effects of multiple independent variables on the occurrence of a disease. The accuracy measures, such as the area under the ROC curve (AUC) are developed from the functional regression. Asymptotical results for the empirical estimators are provided to facilitate inferences. The second application is to test the difference of two weighted areas under the ROC curve (WAUC) from a paired two sample study. The correlation between the two WAUC complicates the asymptotic distribution of the test statistic. We then employ the bootstrap methods to gain satisfactory inference results. Simulations and examples are supplied in this article to confirm the merits of the bootstrap methods.     

Estimation of reliability from a bivariate log-normal data

It has been established that the bivariate log-normal distribution is appropriate for modelling certain paired observations. In this paper, we have developed large-sample confidence intervals of the dependence and reliability R=P(X>Y) parameters from a bivariate log-normal distribution with equal log-normal means. The parameter R provides a general measure of difference between the two populations and has applications in many areas. The performance of these confidence intervals has been examined by extensive simulation studies. The results are illustrated with an example dealing with a quantitative assay problem.     

Monotony relations between distribution functions and their application to experimental design

In a linear regression model an estimator of the unknown coefficients is considered which, in special cases, includes the least squares estimator. In the ease of stable symmetric error distribution and by means of a certain monotony relation between distribution functions optimality of this estimator is proved and the designing problem is investigated. A robustness property of optimal designs against the designing criterion and some conclusions are given concerning the least squares estimator in the case of G- and C-optimality.     

Handling the Label Switching Problem in Latent Class Models Via the ECR Algorithm

Latent class models (LCMs) are specific cases of mixture models. Under a Bayesian setup, the symmetric posterior distribution of these models leads Markov chain Monte Carlo (MCMC) methods to suffer from the so-called label switching problem. In this article, we treat the corresponding MCMC outputs using a recent approach, namely, the Equivalence Classes Representative (ECR) algorithm and conclude that it can effectively solve the label switching problem by considering several examples of LCMs, such as mixtures of regressions, hidden Markov models, and Markov random fields. Moreover, the superiority of this method over other approaches becomes apparent.     

Study of a Markov model for a high-quality dependent process

For high-quality processes, non-conforming items are seldom observed and the traditional p (or np) charts are not suitable for monitoring the state of the process. A type of chart based on the count of cumulative conforming items has recently been introduced and it is especially useful for automatically collected one-at-a-time data. However, in such a case, it is common that the process characteristics become dependent as items produced one after another are inspected. In this paper, we study the problem of process monitoring when the process is of high quality and measurement values possess a certain serial dependence. The problem of assuming independence is examined and a Markov model for this type of process is studied, upon which suitable control procedures can be developed.