R×C ecological inference: bounds, correlations, flexibility and transparency of assumptions

Summary.  Despite its potential pitfalls, ecological inference is an unavoidable part of some quantitative settings, including US voting rights litigation. In such applications, the analyst will typically encounter two-way tables with more than two rows and columns. Although several ecological inference methods are currently available for 2×2 tables, there are fewer options for analysing general R × C tables, and virtually none that model counts as opposed to fractions. We propose a count R × C method that respects the bounds deterministically, that allows for complex relationships between internal cell quantities, that is easily extensible and that results from transparent assumptions. We study the method via simulation, and then apply it to an example that is drawn from the state of Texas relevant to recent redistricting litigation there.     

Perfect simulation for Reed-Frost epidemic models

The Reed-Frost epidemic model is a simple stochastic process with parameter q that describes the spread of an infectious disease among a closed population. Given data on the final outcome of an epidemic, it is possible to perform Bayesian inference for q using a simple Gibbs sampler algorithm. In this paper it is illustrated that by choosing latent variables appropriately, certain monotonicity properties hold which facilitate the use of a perfect simulation algorithm. The methods are applied to real data.     

半参数纵向模型的惩罚二次推断函数估计

针对纵向数据半参数模型E(y|x,t)=XTβ+f(t),采用惩罚二次推断函数方法同时估计模型中的回归参数β和未知光滑函数f(t)。首先利用截断幂函数基对未知光滑函数进行基函数展开近似,然后利用惩罚样条的思想构造关于回归参数和基函数系数的惩罚二次推断函数,最小化惩罚二次推断函数便可得到回归参数和基函数系数的惩罚二次推断函数估计。理论结果显示,估计结果具有相合性和渐近正态性,通过数值方法也得到了较好的模拟结果。     

Predicting future discoveries of European marine species by using a non-homogeneous renewal process

Summary.  Predicting future rates of species discovery and the number of species remaining are important in efforts to preserve biodiversity, discussions on the rate of species extinction and comparisons on the state of knowledge of animals and plants of different taxa. Data on discovery dates of species in 32 European marine taxa are analysed by using a class of thinned temporal renewal process models. These models allow for both underdispersion and overdispersion with respect to the non-homogeneous Poisson process. An approach for implementing Bayesian inference for these models is described that uses Markov chain Monte Carlo simulation and that is applicable to other types of thinned process. Predictions are made on the number of species remaining to be discovered in each taxon.     

Exact likelihood inference for two populations from two-parameter exponential distributions under joint Type-II censoring

In this article, we develop exact inference for two populations that have a two-parameter exponential distribution with the same location parameter and different scale parameters when Type-II censoring is implemented on the two samples in a combined manner. We obtain the conditional maximum likelihood estimators (MLEs) of the three parameters. We then derive the exact distributions of these MLEs along with their moment generating functions. Based on general entropy loss function, Bayesian study about the parameters is presented. Finally, some simulation results and an illustrative example are presented to illustrate the methods of inference developed here.     

The Poisson–exponential distribution: a Bayesian approach

In this paper, we proposed a new two-parameter lifetime distribution with increasing failure rate. The new distribution arises on a latent complementary risk scenario. The properties of the proposed distribution are discussed, including a formal proof of its density function and an explicit algebraic formulae for its quantiles and survival and hazard functions. Also, we have discussed inference aspects of the model proposed via Bayesian inference by using Markov chain Monte Carlo simulation. A simulation study investigates the frequentist properties of the proposed estimators obtained under the assumptions of non-informative priors. Further, some discussions on models selection criteria are given. The developed methodology is illustrated on a real data set.     

Fiducial inference for Birnbaum-Saunders distribution

In this paper, we consider fiducial inference for the unknown parameters of the Birnbaum-Saunders distribution. Two generalized fiducial distributions of the parameters are obtained. One is based on the inverse of the structural equation, and the fiducial estimates of the parameters are obtained by a simulation method. The other is based on the method of [Hannig J. Generalized fiducial inference via discretization. Stat. Sinica. 2013;23:489–514], then we use adaptive rejection Metropolis sampling to get the fiducial estimates. We compare the fiducial estimates with the maximum likelihood estimates and Bayesian estimates by simulations. Two real data sets are analysed for illustration.     

Causal inference for recurrent events data with all-or-none compliance

ABSTRACT

In this article, causal inference in randomized studies with recurrent events data and all-or-none compliance is considered. We use the counting process to analyze the recurrent events data and propose a causal proportional intensity model. The maximum likelihood approach is adopted to estimate the parameters of the proposed causal model. To overcome the computational difficulties created by the mixture structure of the problem, we develop an expectation-maximization (EM) algorithm. The resulting estimators are shown to be consistent and asymptotically normal. We further estimate the complier average causal effect (CACE), which is defined as the difference of the average numbers of recurrence between treatment and control groups within the complier class. The corresponding inferential procedures are established. Some simulation studies are conducted to assess the finite sample performance of the proposed approach.     

Inference for the common mean of several Birnbaum–Saunders populations

The Birnbaum–Saunders distribution is a widely used distribution in reliability applications to model failure times. For several samples from possible different Birnbaum–Saunders distributions, if their means can be considered as the same, it is of importance to make inference for the common mean. This paper presents procedures for interval estimation and hypothesis testing for the common mean of several Birnbaum–Saunders populations. The proposed approaches are hybrids between the generalized inference method and the large sample theory. Some simulation results are conducted to present the performance of the proposed approaches. The simulation results indicate that our proposed approaches perform well. Finally, the proposed approaches are applied to analyze a real example on the fatigue life of 6061-T6 aluminum coupons for illustration.     

On the Order of Data Reductions by Ancillarity and by Sufficiency

In the context of frequentist inference there are strong arguments in favour of data reduction by both (a) conditioning on the most appropriate ancillary statistic and (b) restricting attention to a minimal sufficient statistic. However, significantly for the study of the foundations of frequentist inference, there are some examples in which the order of application of these data reductions has an important bearing on the statistical inference of interest. This paper presents a new simple example of this kind.     

Accuracy of variational estimates for random graph mixture models

Variational and variational Bayes techniques are popular approaches for statistical inference of complex models but their theoretical properties are still not well known. Because of both unobserved variables and intricate dependency structures, mixture models for random graphs constitute a good case study. We first present four different variational estimates for the parameters of these models. We then compare their accuracy through simulation studies and show that the variational Bayes estimates seem the most accurate for moderate graph size. We finally re-analyse the regulatory network of Escherichia coli with this approach.     

Non-parametric Bayesian inference through MCMC method for Y-linked two-sex branching processes with blind choice

ABSTRACT

Dirichlet-process-based non-parametric Bayesian inference is developed for a Y-linked two-sex branching process with blind choice. This stochastic model is suitable for analysing the evolution of the number of carriers of two alleles of a Y-linked gene in a two-sex monogamous population where each female chooses her partner from among the male population without caring about his type (i.e. the allele he carries). The only data assumed to be available are the total number of females and males (regardless of their types) up to some generation and the numbers of each type of male in the last generation. A simulation method which is based on a Dirichlet process and a Gibbs sampler is developed to estimate the posterior distributions of the model's main parameters. Finally, the computational efficiency of the algorithm is illustrated with example simulations and an application to real data.     

Theoretic and empirical data-inclusive process characterization

Summary.  In process characterization the quality of information that is obtained depends directly on the quality of process model. The current quality revolution is now providing a strong stimulus for rethinking and re-evaluating many statistical ideas. Among these are the role of theoretic knowledge and data in statistical inference and some issues in theoretic–empirical modelling. With this concern the paper takes a broad, pragmatic view of statistical inference to include all aspects of model formulation. The estimation of model parameters traditionally assumes that a model has a prespecified known form and takes no account of possible uncertainty regarding model structure. But in practice model structural uncertainty is a fact of life and is likely to be more serious than other sources of uncertainty which have received far more attention. This is true whether the model is specified on subject-matter grounds or when a model is formulated, fitted and checked on the same data set in an iterative interactive way. For that reason novel modelling techniques have been fashioned for reducing model uncertainty. Using available knowledge for theoretic model elaboration the techniques that have been created approximate the exact unknown process model concurrently by accessible theoretic and polynomial empirical functions. The paper examines the effects of uncertainty for hybrid theoretic–empirical models and, for reducing uncertainty, additive and multiplicative methods of model formulation are fashioned. Such modelling techniques have been successfully applied to perfect a steady flow model for an air gauge sensor. Validation of the models elaborated has revealed that the multiplicative modelling approach allows us to attain a satisfactory model with small discrepancy from empirical evidence.     

Accelerating inference for diffusions observed with measurement error and large sample sizes using approximate Bayesian computation

In recent years, dynamical modelling has been provided with a range of breakthrough methods to perform exact Bayesian inference. However, it is often computationally unfeasible to apply exact statistical methodologies in the context of large data sets and complex models. This paper considers a nonlinear stochastic differential equation model observed with correlated measurement errors and an application to protein folding modelling. An approximate Bayesian computation (ABC)-MCMC algorithm is suggested to allow inference for model parameters within reasonable time constraints. The ABC algorithm uses simulations of ‘subsamples’ from the assumed data-generating model as well as a so-called ‘early-rejection’ strategy to speed up computations in the ABC-MCMC sampler. Using a considerate amount of subsamples does not seem to degrade the quality of the inferential results for the considered applications. A simulation study is conducted to compare our strategy with exact Bayesian inference, the latter resulting two orders of magnitude slower than ABC-MCMC for the considered set-up. Finally, the ABC algorithm is applied to a large size protein data. The suggested methodology is fairly general and not limited to the exemplified model and data.     

Simulating Correlated Marked-point Processes

The area of marked-point processes is well developed but simulation is still a challenging problem when mark correlations are to be included. In this paper we propose the use of simulated annealing to incorporate the spatial mark correlation into the simulations of correlated marked-point processes. Such a simulation has wide applications in areas such as inference and goodness-of-fit investigations of proposed models. The technique is applied to a forest dataset for which the results are extremely encouraging.     

A Bayesian mixture model for differential gene expression

Summary.  We propose model-based inference for differential gene expression, using a nonparametric Bayesian probability model for the distribution of gene intensities under various conditions. The probability model is a mixture of normal distributions. The resulting inference is similar to a popular empirical Bayes approach that is used for the same inference problem. The use of fully model-based inference mitigates some of the necessary limitations of the empirical Bayes method. We argue that inference is no more difficult than posterior simulation in traditional nonparametric mixture-of-normal models. The approach proposed is motivated by a microarray experiment that was carried out to identify genes that are differentially expressed between normal tissue and colon cancer tissue samples. Additionally, we carried out a small simulation study to verify the methods proposed. In the motivating case-studies we show how the nonparametric Bayes approach facilitates the evaluation of posterior expected false discovery rates. We also show how inference can proceed even in the absence of a null sample of known non-differentially expressed scores. This highlights the difference from alternative empirical Bayes approaches that are based on plug-in estimates.     

Diagnostic tools for approximate Bayesian computation using the coverage property

Approximate Bayesian computation (ABC) is an approach to sampling from an approximate posterior distribution in the presence of a computationally intractable likelihood function. A common implementation is based on simulating model, parameter and dataset triples from the prior, and then accepting as samples from the approximate posterior, those model and parameter pairs for which the corresponding dataset, or a summary of that dataset, is ‘close’ to the observed data. Closeness is typically determined though a distance measure and a kernel scale parameter. Appropriate choice of that parameter is important in producing a good quality approximation. This paper proposes diagnostic tools for the choice of the kernel scale parameter based on assessing the coverage property, which asserts that credible intervals have the correct coverage levels in appropriately designed simulation settings. We provide theoretical results on coverage for both model and parameter inference, and adapt these into diagnostics for the ABC context. We re‐analyse a study on human demographic history to determine whether the adopted posterior approximation was appropriate. Code implementing the proposed methodology is freely available in the R package abctools .     

Empirical likelihood inference in the presence of measurement error

Suppose that several different imperfect instruments and one perfect instrument are used independently to measure some characteristic of a population. The authors consider the problem of combining this information to make statistical inference on parameters of interest, in particular the population mean and cumulative distribution function. They develop maximum empirical likelihood estimators and study their asymptotic properties. They also present simulation results on the finite sample efficiency of these estimators.     

The fiducial argument and hacking's principle of irrelevance

In a searching analysis of the fiducial argument Hacking (1965) proposed the Principle of Irrelevance as a condition under which the argument is valid. His statement of the Principle was essentially non-mathematical and this paper presents a mathematical development of the Principle. The relationship with likelihood inference is explored and some of the proposed counter-examples to fiducial theory are considered. It is shown that even with the Principle of Irrelevance examples of non-uniqueness of fiducial distributions exist.     

Single-step and multiple-step forecasting in one-dimensional single chirp signal using MCMC-based Bayesian analysis

Chirp signals are frequently used in different areas of science and engineering. MCMC-based Bayesian inference is done here for the purpose of one-step and multiple-step prediction in the case of the one-dimensional single chirp signal with iid error structure as well as dependent error structure with exponentially decaying covariances. We use Gibbs sampling technique and random walk MCMC to update the parameters. We perform total five simulation studies for the illustration purpose. We also do some real-data analysis to show how the method is working in practice.