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AbstractEstimating the Hawkes process from a discretely observed sample path is challenging due to the intractability of the likelihood in such cases. To overcome this, we employ a state-space representation of the incomplete data problem and use the sequential Monte Carlo (SMC, aka particle filters) to approximate the likelihood function. The resulting estimator of the likelihood function is unbiased and, therefore, can be used along with the Metropolis-Hastings algorithm to construct Markov Chains to approximate the likelihood distribution and, more generally, the posterior distribution of model parameters. The performance of our methodology is assessed using simulation experiments and compared with other recently published methods. The proposed estimator exhibits a smaller mean square error compared to two benchmark estimators. Furthermore, an advantage of our method compared to existing methods is that confidence intervals for the parameters are readily computable. Finally, we apply the proposed estimator to the analysis of weekly count data on measles cases in Tokyo, Japan, and compare the results to one of the benchmark methods. The online supplementary materials contain a Julia package that implements our methodology, along with the technical proofs for two propositions. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we develop two procedures for identifying the dynamic trends in the parameters of length-biased lognormal distribution based on likelihood ratio and modified information criterion. These methods mainly consider the test of the existence of the change point and provide the maximum likelihood estimation of the change point when the change point exists. In addition, the asymptotic distribution of test statistic based on likelihood ratio is derived as an extreme value distribution, while the asymptotic distribution of test statistic based on modified information criterion is derived as a chi-square distribution. And the consistency of parameter estimation is proved. Simulations are conducted to study the performance of the proposed method in terms of power, coverage probabilities and average sizes of confidence sets. The proposed methods are applied to a real data to illustrate the detecting procedures. [ABSTRACT FROM AUTHOR]

Nonparametric confidence distributions estimate statistical functionals by a distribution function on the parameter space, instead of the classical point or interval estimators. The concept bears analogy to the Bayesian posterior, but is nevertheless a completely frequentist concept. In order to ensure the desired statistical properties, we require that the cumulative distribution function on the parameter space is, evaluated at the true parameter, uniformly distributed over the unit interval. Our main focus lies on developing confidence distributions for the nonparametric relative effect and some natural extensions thereof. We develop asymptotic, range preserving and—especially important in the case of small sample sizes—approximate confidence distributions based on rank and pseudo-rank procedures. Due to the close relationship between point estimators, confidence intervals and p-values, these can all be approached in a unified manner within the framework of confidence distributions. The main goal of our contribution is to make the powerful theory of confidence distributions available in a nonparametric context, that is, for situations where methods relying on parametric assumptions are not justifiable. Application of the proposed methods and interpretation of the results is demonstrated using real data sets, including ordinal, non-metric data. [ABSTRACT FROM AUTHOR]

An attractive feature of using a Bayesian analysis for a clinical trial is that knowledge and uncertainty about the treatment effect is summarized in a posterior probability distribution. Researchers often find probability statements about treatment effects highly intuitive and the fact that this is not accommodated in frequentist inference is a disadvantage. At the same time, the requirement to specify a prior distribution in order to obtain a posterior distribution is sometimes an artificial process that may introduce subjectivity or complexity into the analysis. This paper considers a compromise involving confidence distributions, which are probability distributions that summarize uncertainty about the treatment effect without the need for a prior distribution and in a way that is fully compatible with frequentist inference. The concept of a confidence distribution provides a posterior–like probability distribution that is distinct from, but exists in tandem with, the relative frequency interpretation of probability used in frequentist inference. Although they have been discussed for decades, confidence distributions are not well known among clinical trial statisticians and the goal of this paper is to discuss their use in analyzing treatment effects from randomized trials. As well as providing an introduction to confidence distributions, some illustrative examples relevant to clinical trials are presented, along with various case studies based on real clinical trials. It is recommended that trial statisticians consider presenting confidence distributions for treatment effects when reporting analyses of clinical trials. [ABSTRACT FROM AUTHOR]

Generalized linear mixed models are commonly used to describe relationships between correlated responses and covariates in medical research. In this paper, we propose a simple and easily implementable regularized estimation approach to select both fixed and random effects in generalized linear mixed model. Specifically, we propose to construct and optimize the objective functions using the confidence distributions of model parameters, as opposed to using the observed data likelihood functions, to perform effect selections. Two estimation methods are developed. The first one is to use the joint confidence distribution of model parameters to perform simultaneous fixed and random effect selections. The second method is to use the marginal confidence distributions of model parameters to perform the selections of fixed and random effects separately. With a proper choice of regularization parameters in the adaptive LASSO framework, we show the consistency and oracle properties of the proposed regularized estimators. Simulation studies have been conducted to assess the performance of the proposed estimators and demonstrate computational efficiency. Our method has also been applied to two longitudinal cancer studies to identify demographic and clinical factors associated with patient health outcomes after cancer therapies. [ABSTRACT FROM AUTHOR]

We define confidence to be epistemic if it applies to an observed confidence interval. Epistemic confidence is unavailable—or even denied—in orthodox frequentist inference, as the confidence level is understood to apply to the procedure. Yet there are obvious practical and psychological needs to think about the uncertainty in the observed interval. We extend the Dutch Book argument used in the classical Bayesian justification of subjective probability to a stronger market‐based version, which prevents external agents from exploiting unused information in any relevant subset. We previously showed that confidence is an extended likelihood, and the likelihood principle states that the likelihood contains all the information in the data, hence leaving no relevant subset. Intuitively, this implies that confidence associated with the full likelihood is protected from the Dutch Book, and hence is epistemic. Our goal is to validate this intuitive notion through theoretical backing and practical illustrations. [ABSTRACT FROM AUTHOR]

Skew slash distribution is a distribution which considers both skewness and heavy tail. It is very useful in simulation studies and realistic in representing practical data due to its less peaks, especially in data sets that violate the assumption of normality. In this article, we propose a change-point detection procedure for skew slash distribution based on the modified information criterion (MIC). Meanwhile, we provide an estimation approach based on confidence distribution (CD) to measure the accuracy of change point location estimation. By comparing with the likelihood ratio test, the simulation results show that the MIC-based method performs better in terms of powers, the coverage probabilities and average lengths of confidence sets. In the end, we apply our proposed method to real data and locate the positions of the change points successfully. [ABSTRACT FROM AUTHOR]

A conditioning on the event of having selected one model from a set of possibly misspecified normal linear regression models leads to the construction of uniformly optimal conditional confidence distributions. They can be used for valid postselection inference. The constructed conditional confidence distributions are finite sample exact and encompass all information regarding the focus parameter in the selected model. This includes the construction of optimal postselection confidence intervals at all significance levels and uniformly most powerful hypothesis tests. [ABSTRACT FROM AUTHOR]

Uniformly most powerful confidence distributions are obtained for parameters in selected models of the exponential family. A conditioning on the selection event as well as on the sufficient statistics of nuisance parameters guarantees valid post-selection inference. Optimal confidence intervals are obtained directly from the confidence distribution without requiring an inversion of pivotal quantities. Simulations showcase that the method works also when all models are misspecified. • Valid inference after model selection for parameters in exponential family models. • Confidence curves conditional on the selected model. • Confidence distributions yield correct p-values and best confidence intervals. • Finite sample uniformly most powerful confidence curves for continuous distributions. • Good approximations for discrete distributions. [ABSTRACT FROM AUTHOR]

Fusion learning refers to synthesizing inferences from multiple sources or studies to make a more effective inference and prediction than from any individual source or study alone. Most existing methods for synthesizing inferences rely on parametric model assumptions, such as normality, which often do not hold in practice. We propose a general nonparametric fusion learning framework for synthesizing inferences for multiparameters from different studies. The main tool underlying the proposed framework is the new notion of depth confidence distribution (depth-CD), which is developed by combining data depth and confidence distribution. Broadly speaking, a depth-CD is a data-driven nonparametric summary distribution of the available inferential information for a target parameter. We show that a depth-CD is a powerful inferential tool and, moreover, is an omnibus form of confidence regions, whose contours of level sets shrink toward the true parameter value. The proposed fusion learning approach combines depth-CDs from the individual studies, with each depth-CD constructed by nonparametric bootstrap and data depth. The approach is shown to be efficient, general and robust. Specifically, it achieves high-order accuracy and Bahadur efficiency under suitably chosen combining elements. It allows the model or inference structure to be different among individual studies. And, it readily adapts to heterogeneous studies with a broad range of complex and irregular settings. This last property enables the approach to use indirect evidence from incomplete studies to gain efficiency for the overall inference. We develop the theoretical support for the proposed approach, and we also illustrate the approach in making combined inference for the common mean vector and correlation coefficient from several studies. The numerical results from simulated studies show the approach to be less biased and more efficient than the traditional approaches in nonnormal settings. The advantages of the approach are also demonstrated in a Federal Aviation Administration study of aircraft landing performance. for this article are available online. [ABSTRACT FROM AUTHOR]

Multivariate failure time data are frequently analyzed using the marginal proportional hazards models and the frailty models. When the sample size is extraordinarily large, using either approach could face computational challenges. In this paper, we focus on the marginal model approach and propose a divide‐and‐combine method to analyze large‐scale multivariate failure time data. Our method is motivated by the Myocardial Infarction Data Acquisition System (MIDAS), a New Jersey statewide database that includes 73,725,160 admissions to nonfederal hospitals and emergency rooms (ERs) from 1995 to 2017. We propose to randomly divide the full data into multiple subsets and propose a weighted method to combine these estimators obtained from individual subsets using three weights. Under mild conditions, we show that the combined estimator is asymptotically equivalent to the estimator obtained from the full data as if the data were analyzed all at once. In addition, to screen out risk factors with weak signals, we propose to perform the regularized estimation on the combined estimator using its combined confidence distribution. Theoretical properties, such as consistency, oracle properties, and asymptotic equivalence between the divide‐and‐combine approach and the full data approach are studied. Performance of the proposed method is investigated using simulation studies. Our method is applied to the MIDAS data to identify risk factors related to multivariate cardiovascular‐related health outcomes. [ABSTRACT FROM AUTHOR]

In this article, we propose procedures based on the modified information criterion and the confidence distribution for detecting and estimating changes in a three-parameter Weibull distribution. Corresponding asymptotic results of the test statistic associated the detection procedure are established. Moreover, instead of only providing point estimates of change locations, the proposed estimation procedure provides the confidence sets for change locations at a given significance level through the confidence distribution. In general, the proposed procedures are valid for a large class of parametric distributions under Wald conditions and the certain regularity conditions being satisfied. Simulations are conducted to investigate the performance of the proposed method in terms of powers, coverage probabilities and average lengths of confidence sets with respect to a three-parameter Weibull distribution. Corresponding comparisons are also made with other existing methods to indicate the advantages of the proposed method. Rainfall data is used to illustrate the application of the proposed method. [ABSTRACT FROM AUTHOR]

For testing with paired data (eg, twins randomized between two treatments), a simple test is the sign test, where we test if the distribution of the sign of the differences in responses between the two treatments within pairs is more often positive (favoring one treatment) or negative (favoring the other). When the responses are binary, this reduces to a McNemar-type test, and the calculations are the same. Although it is easy to calculate an exact P-value by conditioning on the total number of discordant pairs, the accompanying confidence interval on a parameter of interest (proportion positive minus proportion negative) is not straightforward. Effect estimates and confidence intervals are important for interpretation because it is possible that the treatment helps a very small proportion of the population yet gives a highly significant effect. We construct a confidence interval that is compatible with an exact sign test, meaning the 100 (1-α)% interval excludes the null hypothesis of equality of proportions if and only if the associated exact sign test rejects at level α . We conjecture that the proposed confidence intervals guarantee nominal coverage, and we support that conjecture with extensive numerical calculations, but we have no mathematical proof to show guaranteed coverage. We have written and made available the function mcnemarExactDP in the exact2x2 R package and the function signTest in the asht R package to perform the methods described in this article. [ABSTRACT FROM AUTHOR]

Massive correlated data with many inputs are often generated from computer experiments to study complex systems. The Gaussian process (GP) model is a widely used tool for the analysis of computer experiments. Although GPs provide a simple and effective approximation to computer experiments, two critical issues remain unresolved. One is the computational issue in GP estimation and prediction where intensive manipulations of a large correlation matrix are required. For a large sample size and with a large number of variables, this task is often unstable or infeasible. The other issue is how to improve the naive plug‐in predictive distribution which is known to underestimate the uncertainty. In this article, we introduce a unified framework that can tackle both issues simultaneously. It consists of a sequential split‐and‐conquer procedure, an information combining technique using confidence distributions (CD), and a frequentist predictive distribution based on the combined CD. It is shown that the proposed method maintains the same asymptotic efficiency as the conventional likelihood inference under mild conditions, but dramatically reduces the computation in both estimation and prediction. The predictive distribution contains comprehensive information for inference and provides a better quantification of predictive uncertainty as compared with the plug‐in approach. Simulations are conducted to compare the estimation and prediction accuracy with some existing methods, and the computational advantage of the proposed method is also illustrated. The proposed method is demonstrated by a real data example based on tens of thousands of computer experiments generated from a computational fluid dynamic simulator. [ABSTRACT FROM AUTHOR]

Inferences from different data sources can often be fused together, a process referred to as "fusion learning," to yield more powerful findings than those from individual data sources alone. Effective fusion learning approaches are in growing demand as increasing number of data sources have become easily available in this big data era. This article proposes a new fusion learning approach, called "iFusion," for drawing efficient individualized inference by fusing learnings from relevant data sources. Specifically, iFusion (i) summarizes inferences from individual data sources as individual confidence distributions (CDs); (ii) forms a clique of individuals that bear relevance to the target individual and then combines the CDs from those relevant individuals; and, finally, (iii) draws inference for the target individual from the combined CD. In essence, iFusion strategically "borrows strength" from relevant individuals to enhance the efficiency of the target individual inference while preserving its validity. This article focuses on the setting where each individual study has a number of observations but its inference can be further improved by incorporating additional information from similar studies that is referred to as its clique. Under the setting, iFusion is shown to achieve oracle property under suitable conditions. It is also shown to be flexible and robust in handling heterogeneity arising from diverse data sources. The development is ideally suited for goal-directed applications. Computationally, iFusion is parallel in nature and scales up easily for big data. An efficient scalable algorithm is provided for implementation. Simulation studies and a real application in financial forecasting are presented. In effect, this article covers methodology, theory, computation, and application for individualized inference by iFusion. [ABSTRACT FROM AUTHOR]

Summary: We study properties of the maximum h‐likelihood estimators for random effects in clustered data. To define optimality in random effects predictions, several foundational concepts of statistics such as likelihood, unbiasedness, consistency, confidence distribution and the Cramer–Rao lower bound are extended. Exact probability statements about interval estimators for random effects can be made asymptotically without a prior assumption. Using the binary‐matched pair example, we illustrated that the use of random effects recover information, leading to the boon on estimating treatment effects. [ABSTRACT FROM AUTHOR]

Confidence sets, p values, maximum likelihood estimates, and other results of non-Bayesian statistical methods may be adjusted to favor sampling distributions that are simple compared to others in the parametric family. The adjustments are derived from a prior likelihood function previously used to adjust posterior distributions. [ABSTRACT FROM AUTHOR]

This paper focuses on inference based on the confidence distributions of the nonparametric regression function and its derivatives, in which dependent inferences are combined by obtaining information about their dependency structure. We first give a motivating example in production operation system to illustrate the necessity of the problems studied in this paper in practical applications. A goodness-of-fit test for polynomial regression model is proposed on the basis of the idea of combined confidence distribution inference, which is the Fisher's combination statistic in some cases. On the basis of this testing results, a combined estimator for the p-order derivative of nonparametric regression function is provided as well as its large sample size properties. Consequently, the performances of the proposed test and estimation method are illustrated by three specific examples. Finally, the motivating example is analyzed in detail. The simulated and real data examples illustrate the good performance and practicability of the proposed methods based on confidence distribution. [ABSTRACT FROM AUTHOR]

We propose randomized inference (RI), a new statistical inference approach. RI may be realized through a randomized estimate (RE) of a parameter vector, which is a random vector that takes values in the parameter space with a probability density function (PDF) that depends on the sample or sufficient statistics, such as the posterior distributions in Bayesian inference. Based on the PDF of an RE of an unknown parameter, we propose a framework for both the vertical density representation (VDR) test and the construction of a confidence region. This approach is explained with the aid of examples. For the equality hypothesis of multiple normal means without the condition of variance homogeneity, we present an exact VDR test, which is shown as an extension of one-way analysis of variance (ANOVA). In the case of two populations, the PDF of the Welch statistics is given using RE. Furthermore, through simulations, we show that the empirical distribution function, the approximated t, and the RE distribution function of Welch statistics are almost equal. The VDR test of the homogeneity of variance is shown to be more efficient than both the Bartlett test and the revised Bartlett test. Finally, we discuss the prospects of RI. [ABSTRACT FROM AUTHOR]

Model averaging is commonly used to allow for model uncertainty in parameter estimation. As well as providing a point estimate that is a natural compromise between the estimates from different models, it also provides confidence intervals with better coverage properties, compared to those based on a single best model. In recent years, the concept of a confidence distribution has been promoted as a frequentist analogue of a Bayesian posterior distribution. The confidence distribution for a parameter is a visual representation of the set of 100 (1 - α) % confidence intervals for all possible α , and was first proposed over 60 years ago. The purpose of this paper is to promote the use of model-averaged confidence distributions. One of the advantages of doing so is the ability to see unusual shapes in the distribution, such as multi-modality. This allows a more comprehensive assessment of the uncertainty about the parameter of interest, in exactly the same way that a model-averaged posterior distribution can be more useful than a model-averaged credible interval. We show that the model-averaged tail-area (MATA) method for calculating a model-averaged confidence interval leads to the corresponding MATA confidence distribution being a mixture of the confidence distributions associated with the individual models, the mixing being determined by the model weights. We consider two ecological examples that illustrate the advantages of a model-averaged confidence distribution over a model-averaged confidence interval. [ABSTRACT FROM AUTHOR]

In this paper, we consider inference about the shape parameters of several inverse Gaussian distributions. At first, an approach is given to test the equality of these parameters based on modified likelihood ratio test. Then, five approaches are presented to construct confidence intervals for the common shape parameter. The performance of these approaches is studied using Monte Carlo simulation, and illustrated using a real data set. [ABSTRACT FROM AUTHOR]

Just as frequentist hypothesis tests have been developed to check model assumptions, prior predictive p-values and other Bayesian p-values check prior distributions as well as other model assumptions. These model checks not only suffer from the usual threshold dependence of p-values, but also from the suppression of model uncertainty in subsequent inference. One solution is to transform Bayesian and frequentist p-values for model assessment into a fiducial distribution across the models. Averaging the Bayesian or frequentist posterior distributions with respect to the fiducial distribution can reproduce results from Bayesian model averaging or classical fiducial inference. [ABSTRACT FROM AUTHOR]

This paper proposes a general framework for prediction in which a prediction is presented in the form of a distribution function, called predictive distribution function . This predictive distribution function is well suited for the notion of confidence subscribed in the frequentist interpretation, and it can provide meaningful answers for questions related to prediction. A general approach under this framework is formulated and illustrated by using the so-called confidence distributions (CDs). This CD-based prediction approach inherits many desirable properties of CD, including its capacity for serving as a common platform for connecting and unifying the existing procedures of predictive inference in Bayesian, fiducial and frequentist paradigms. The theory underlying the CD-based predictive distribution is developed and some related efficiency and optimality issues are addressed. Moreover, a simple yet broadly applicable Monte Carlo algorithm is proposed for the implementation of the proposed approach. This concrete algorithm together with the proposed definition and associated theoretical development produce a comprehensive statistical inference framework for prediction. Finally, the approach is applied to simulation studies, and a real project on predicting the incoming volume of application submissions to a government agency. The latter shows the applicability of the proposed approach to dependence data settings. [ABSTRACT FROM AUTHOR]

Combining estimates for a fixed but unknown parameter to obtain a better estimate is an important problem, but even for independent estimates not straightforward where they involve different experimental characteristics. The problem considered here is the case where two such estimates can each be well represented by a probability density function (PDF) for the ratio of two normally-distributed variables. Two different statistical methods – objective Bayesian and frequentist likelihood-ratio – are employed and compared. Each probabilistic estimate of the parameter value is represented by a fitted three-parameter Bayesian posterior PDF providing a close approximation to the ratio of two normals, that can legitimately be factored into a likelihood function and a noninformative prior distribution. The likelihood functions relating to the parameterised fits to the probabilistic estimates are multiplicatively combined and a prior is derived that is noninformative for inference from the combined evidence. An objective posterior PDF that incorporates the evidence from both sources is produced using a single-step approach, which avoids the order-dependency that would arise if Bayesian updating were used. The frequentist signed root likelihood-ratio method is also applied. The probability matching of credible intervals from the posterior distribution and of approximate confidence intervals from the likelihood-ratio method is tested, showing that both methods provide almost exact confidence distributions. The approach developed is applied in the important case of the Earth’s equilibrium climate sensitivity, by combining an estimate from instrumental records with an estimate representing largely independent paleoclimate proxy evidence, resulting in a median estimate of 2.0 °C and a 5%–95% confidence/credibility interval of (1.1, 4.5) ∘ C. [ABSTRACT FROM AUTHOR]

In its orthodoxy standard frequentist statistics deals only with aleatory probability, suppressing the intuitive epistemic probability representing inferential uncertainty. Confidence distributions, which are posterior distributions not based on any Bayesian priors, are discussed in nontechnical terms, with emphasis on the confidence curve. The correspondence between confidence curves and likelihoods allows independent confidence curves and confidence intervals to be integrated. Confidence and (serious) p -values are interpreted as epistemic probabilities, which do not fully follow ordinary probability calculus. Dimension reduction and other operations might be done on the likelihood related to the confidence curve. Confidence distributions and objective Bayes have much in common. [ABSTRACT FROM AUTHOR]

We propose a way to construct fiducial distributions for a multidimensional parameter using a step-by-step conditional procedure related to the inferential importance of the components of the parameter. For discrete models, in which the non-uniqueness of the fiducial distribution is well known, we propose to use the geometric mean of the “extreme cases” and show its good behavior with respect to the more traditional arithmetic mean. Connections with the generalized fiducial inference approach developed by Hannig and with confidence distributions are also analyzed. The suggested procedure strongly simplifies when the statistical model belongs to a subclass of the natural exponential family, called conditionally reducible, which includes the multinomial and the negative-multinomial models. Furthermore, because fiducial inference and objective Bayesian analysis are both attempts to derive distributions for an unknown parameter without any prior information, it is natural to discuss their relationships. In particular, the reference posteriors, which also depend on the importance ordering of the parameters, are the natural terms of comparison. We show that fiducial and reference posterior distributions coincide in the location-scale models, and we characterize the conditionally reducible natural exponential families for which the same coincidence holds. The discussion of some classical examples closes the paper. [ABSTRACT FROM AUTHOR]

By the modified directed likelihood, higher order accurate confidence limits for a scalar parameter are obtained from the likelihood. They are conveniently described in terms of a confidence distribution, that is a sample dependent distribution function on the parameter space. In this paper we explore a different route to accurate confidence limits via tail-symmetric confidence curves, that is curves that describe equal tailed intervals at any level. Instead of modifying the directed likelihood, we consider inversion of the log-likelihood ratio when evaluated at the median of the maximum likelihood estimator. This is shown to provide equal tailed intervals, and thus an exact confidence distribution, to the third-order of approximation in regular one-dimensional models. Median bias correction also provides an alternative approximation to the modified directed likelihood which holds up to the second order in exponential families. [ABSTRACT FROM AUTHOR]

Empirical Bayes estimates of the local false discovery rate can reflect uncertainty about the estimated prior by supplementing their Bayesian posterior probabilities with confidence levels as posterior probabilities. This use of coherent fiducial inference with hierarchical models generates set estimators that propagate uncertainty to varying degrees. Some of the set estimates approach estimates from plug-in empirical Bayes methods for high numbers of comparisons and can come close to the usual confidence sets given a sufficiently low number of comparisons. [ABSTRACT FROM PUBLISHER]

Most interval estimates are derived from computable conditional distributions conditional on the data. In this article, we call the random variables having such conditional distributions confidence distribution variables and define their finite-sample breakdown values. Based on this, the definition of breakdown value of confidence intervals is introduced, which covers the breakdowns in both the coverage probability and interval length. High-breakdown confidence intervals are constructed by the structural method in location-scale families. Simulation results are presented to compare the traditional confidence intervals and their robust analogues. [ABSTRACT FROM AUTHOR]

Inferences about unobserved random variables, such as future observations, random effects and latent variables, are of interest. In this paper, to make probability statements about unobserved random variables without assuming priors on fixed parameters, we propose the use of the confidence distribution for fixed parameters. We focus on their interval estimators and related probability statements. In random-effect models, intervals can be formed either for future (yet-to-be-realised) random effects or for realised values of random effects. The consistency of intervals for these two cases requires different regularity conditions. Via numerical studies, their finite sampling properties are investigated. [ABSTRACT FROM AUTHOR]

The usefulness of meta-analysis has been recognized in the evaluation of drug safety, as a single trial usually yields few adverse events and offers limited information. For rare events, conventional meta-analysis methods may yield an invalid inference, as they often rely on large sample theories and require empirical corrections for zero events. These problems motivate research in developing exact methods, including Tian et al.'s method of combining confidence intervals (2009, Biostatistics, 10, 275-281) and Liu et al.'s method of combining p-value functions (2014, JASA, 109, 1450-1465). This article shows that these two exact methods can be unified under the framework of combining confidence distributions (CDs). Furthermore, we show that the CD method generalizes Tian et al.'s method in several aspects. Given that the CD framework also subsumes the Mantel-Haenszel and Peto methods, we conclude that the CD method offers a general framework for meta-analysis of rare events. We illustrate the CD framework using two real data sets collected for the safety analysis of diabetes drugs. [ABSTRACT FROM AUTHOR]

We present a scalable optimization method for neutron scattering problems that determines confidence regions of simulation parameters in lattice dynamics models used to fit neutron scattering data for crystalline solids. The method uses physics-based hierarchical dimension reduction in both the computational simulation domain and the parameter space. We demonstrate for silicon that after a few iterations the method converges to parameters values (interatomic force-constants) computed with density functional theory simulations. [ABSTRACT FROM AUTHOR]

In this paper, we construct a new kind of confidence intervals for the among-group variance component through constructing confidence distribution in one-way random effects model with unequal error variances. At first, a combined asymptotic confidence distribution is constructed for the parameter of interest, then the confidence interval is derived naturally by the property of confidence distribution. Simulation results demonstrate that the new confidence intervals perform very well in terms of both empirical coverage probability and average interval length. Finally, a real data example is illustrated. [ABSTRACT FROM AUTHOR]

Frequentist concepts of confidence distribution, reduced likelihood and likelihood synthesis are presented as tools for integrative statistical analysis, leading to distributional inference. Bias correction, often achieved by simulations, is emphasised. The methods are illustrated on data concerning bowhead whales off Alaska, and they are briefly compared with Bayesian methods.

In this paper, we construct a new kind of confidence intervals for the parameters of interest through constructing confidence distributions for them in one-way random effects model. At first, we use the method of Singh et al. (Ann Stat 33:159-183, ) to derive combined asymptotic confidence distribution, then obtain the confidence interval naturally by the property of confidence distribution. Simulation results demonstrate that the new confidence intervals perform very well in terms of empirical coverage probability and average interval length. Although we focus on confidence interval estimation, our method can also be used to carry out hypothesis tests about the parameters of interest. [ABSTRACT FROM AUTHOR]

The proposed minimax procedure blends strict Bayesian methods with p values and confidence intervals or with default-prior methods. Two applications to hypothesis testing bring some implications to light. First, the blended probability that a point null hypothesis is true is equal to the p value or a lower bound of an unknown posterior probability, whichever is greater. As a result, the p value is reported instead of any posterior probability in the case of complete prior ignorance but is ignored in the case of a fully known prior. In the case of partial knowledge about the prior, the possible posterior probability that is closest to the p value is used for inference. The second application provides guidance on the choice of methods used for small numbers of tests as opposed to those appropriate for large numbers. Whereas statisticians tend to prefer a multiple comparison procedure that adjusts each p value for small numbers of tests, large numbers instead lead many to estimate the local false discovery rate (LFDR), a posterior probability of hypothesis truth. Each blended probability reduces to the LFDR estimate if it can be estimated with sufficient accuracy or to the adjusted p value otherwise. [ABSTRACT FROM AUTHOR]