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.     

Generalized fiducial inference for generalized exponential distribution

This article mainly considers interval estimation of the scale and shape parameters of the generalized exponential (GE) distribution. We adopt the generalized fiducial method to construct a kind of new confidence intervals for the parameters of interest and compare them with the frequentist and Bayesian methods. In addition, we give the comparison of the point estimation based on the frequentist, generalized fiducial and Bayesian methods. Simulation results show that a new procedure based on generalized fiducial inference is more applicable than the non-fiducial methods for the point and interval estimation of the GE distribution. Finally, two lifetime data sets are used to illustrate the application of our new procedure.     

Statistical Accuracy of iPad Applications: An Initial Examination

A general explanation of the fiducial confidence interval and its construction for a class of parameters in which the distributions are stochastically increasing or decreasing is provided. Major differences between the fiducial interval and Bayesian and frequentist intervals are summarized. Applications of fiducial inference in evaluating pre-data frequentist intervals and general post-data intervals are discussed.     

Bayesian revision of a prior given prior-data conflict,expert opinion,or a similar insight: a large-deviation approach

Learning from model diagnostics that a prior distribution must be replaced by one that conflicts less with the data raises the question of which prior should instead be used for inference and decision. The same problem arises when a decision maker learns that one or more reliable experts express unexpected beliefs. In both cases, coherence of the solution would be guaranteed by applying Bayes's theorem to a distribution of prior distributions that effectively assigns the initial prior distribution a probability arbitrarily close to 1. The new distribution for inference would then be the distribution of priors conditional on the insight that the prior distribution lies in a closed convex set that does not contain the initial prior. A readily available distribution of priors needed for such conditioning is the law of the empirical distribution of sufficiently large number of independent parameter values drawn from the initial prior. According to the Gibbs conditioning principle from the theory of large deviations, the resulting new prior distribution minimizes the entropy relative to the initial prior. While minimizing relative entropy accommodates the necessity of going beyond the initial prior without departing from it any more than the insight demands, the large-deviation derivation also ensures the advantages of Bayesian coherence. This approach is generalized to uncertain insights by allowing the closed convex set of priors to be random.     

Alternatives to post‐processing posterior predictive p values

The posterior predictive p value (ppp) was invented as a Bayesian counterpart to classical p values. The methodology can be applied to discrepancy measures involving both data and parameters and can, hence, be targeted to check for various modeling assumptions. The interpretation can, however, be difficult since the distribution of the ppp value under modeling assumptions varies substantially between cases. A calibration procedure has been suggested, treating the ppp value as a test statistic in a prior predictive test. In this paper, we suggest that a prior predictive test may instead be based on the expected posterior discrepancy, which is somewhat simpler, both conceptually and computationally. Since both these methods require the simulation of a large posterior parameter sample for each of an equally large prior predictive data sample, we furthermore suggest to look for ways to match the given discrepancy by a computation‐saving conflict measure. This approach is also based on simulations but only requires sampling from two different distributions representing two contrasting information sources about a model parameter. The conflict measure methodology is also more flexible in that it handles non‐informative priors without difficulty. We compare the different approaches theoretically in some simple models and in a more complex applied example.     

Bayesian Inference in Generalized Error and Generalized Student-t Regression Models

This study takes up inference in linear models with generalized error and generalized t distributions. For the generalized error distribution, two computational algorithms are proposed. The first is based on indirect Bayesian inference using an approximating finite scale mixture of normal distributions. The second is based on Gibbs sampling. The Gibbs sampler involves only drawing random numbers from standard distributions. This is important because previously the impression has been that an exact analysis of the generalized error regression model using Gibbs sampling is not possible. Next, we describe computational Bayesian inference for linear models with generalized t disturbances based on Gibbs sampling, and exploiting the fact that the model is a mixture of generalized error distributions with inverse generalized gamma distributions for the scale parameter. The linear model with this specification has also been thought not to be amenable to exact Bayesian analysis. All computational methods are applied to actual data involving the exchange rates of the British pound, the French franc, and the German mark relative to the U.S. dollar.     

Fiducial inference under nonparametric situations

The object of the paper is to provide recipes for various fiducial inferences on a parameter under nonparametric situations. First, the fiducial empirical distribution of a random variable was introduced under nonparametric situations. And its almost sure behavior was established. Then based on it, fiducial model and hence fiducial distribution of a parameter are obtained. Further fiducial intervals of parameters as functionals of the population were constructed. Some of their frequentist properties were investigated under some mild conditions. Besides, p-values of some test hypotheses and their asymptotical properties were also given. Three applications of above results and further results were provided. For the mean, simulations on its interval estimator and hypothesis testing were conducted and their results suggest that the fiducial method performs better than others considered here.     

Inference and Decision Making for 21st-Century Drug Development and Approval

ABSTRACT

The cost and time of pharmaceutical drug development continue to grow at rates that many say are unsustainable. These trends have enormous impact on what treatments get to patients, when they get them and how they are used. The statistical framework for supporting decisions in regulated clinical development of new medicines has followed a traditional path of frequentist methodology. Trials using hypothesis tests of “no treatment effect” are done routinely, and the p-value < 0.05 is often the determinant of what constitutes a “successful” trial. Many drugs fail in clinical development, adding to the cost of new medicines, and some evidence points blame at the deficiencies of the frequentist paradigm. An unknown number effective medicines may have been abandoned because trials were declared “unsuccessful” due to a p-value exceeding 0.05. Recently, the Bayesian paradigm has shown utility in the clinical drug development process for its probability-based inference. We argue for a Bayesian approach that employs data from other trials as a “prior” for Phase 3 trials so that synthesized evidence across trials can be utilized to compute probability statements that are valuable for understanding the magnitude of treatment effect. Such a Bayesian paradigm provides a promising framework for improving statistical inference and regulatory decision making.     

Confidence distributions applied to propagating uncertainty to inference based on estimating the local false discovery rate: A fiducial continuum from confidence sets to empirical Bayes set estimates as the number of comparisons increases

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.     

Criticism of the predictive distribution

The predictive distribution is a mixture of the original distribution model and is used for predicting a future observation. Therein, the mixing distribution is the posterior distribution of the distribution parameters in the Bayesian inference. The mixture can also be computed for the frequentist inference because the Bayesian posterior distribution has the same meaning as a frequentist confidence interval. I present arguments against the concept of predictive distribution. Examples illustrate these. The most important argument is that the predictive distribution can depend on the parameterization. An improvement of the theory of the predictive distribution is recommended.     

Dempster's combination rule and sufficient statistics

In his articles (1966-1968) concerning statistical inference based on lower and upper probabilities, Dempster refers to the connection between Fisher's fiducial argument and his own ideas of statistical inference. Dempster's main concern however focuses on the “Bayesian” aspects of his theory and not on an elaboration of the relation between Fisher's and his ideas. This article attempts to work out the connection between those two approaches and focuses primarily on the question, whether Dempster's combination rule, his upper and lower probabilty based on sufficient statistics and inference based on sufficient statistics in Fisher's sense are consistent. To be adequate to Fisher's reasoning, we deal with absolutely continuous, one parametric families of distributions.This is certainly not the usual assumption in context with Dempster's theory and implies a normative but straightforward definition concerning the underlying conditional distribution; this definition however is done in Dempster's spirit as can be seen from his articles, (1966, 1968,a,b). Under those assumptions it can be shown that - similar to Lindley's results concerning consistency in fiducial reasoning (1958) - the combination rule, Dempster's procedure based on sufficient statistics and fiducial inference by sufficient statistics agree iff the parametric family under consideration can be transformed to location parameter form.     

A generalization of Jeffreys' rule for non regular models

ABSTRACT

We propose a generalization of the one-dimensional Jeffreys' rule in order to obtain non informative prior distributions for non regular models, taking into account the comments made by Jeffreys in his article of 1946. These non informatives are parameterization invariant and the Bayesian intervals have good behavior in frequentist inference. In some important cases, we can generate non informative distributions for multi-parameter models with non regular parameters. In non regular models, the Bayesian method offers a satisfactory solution to the inference problem and also avoids the problem that the maximum likelihood estimator has with these models. Finally, we obtain non informative distributions in job-search and deterministic frontier production homogenous models.     

Hybrid Bayesian inference on HIV viral dynamic models

Modelling of HIV dynamics in AIDS research has greatly improved our understanding of the pathogenesis of HIV-1 infection and guided for the treatment of AIDS patients and evaluation of antiretroviral therapies. Some of the model parameters may have practical meanings with prior knowledge available, but others might not have prior knowledge. Incorporating priors can improve the statistical inference. Although there have been extensive Bayesian and frequentist estimation methods for the viral dynamic models, little work has been done on making simultaneous inference about the Bayesian and frequentist parameters. In this article, we propose a hybrid Bayesian inference approach for viral dynamic nonlinear mixed-effects models using the Bayesian frequentist hybrid theory developed in Yuan [Bayesian frequentist hybrid inference, Ann. Statist. 37 (2009), pp. 2458–2501]. Compared with frequentist inference in a real example and two simulation examples, the hybrid Bayesian approach is able to improve the inference accuracy without compromising the computational load.     

Confidence intervals,significance values,maximum likelihood estimates,etc. sharpened into Occam’s razors

Abstract

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.     

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.     

Approximate Bayesianity of Frequentist Confidence Intervals for a Binomial Proportion

The well-known Wilson and Agresti–Coull confidence intervals for a binomial proportion p are centered around a Bayesian estimator. Using this as a starting point, similarities between frequentist confidence intervals for proportions and Bayesian credible intervals based on low-informative priors are studied using asymptotic expansions. A Bayesian motivation for a large class of frequentist confidence intervals is provided. It is shown that the likelihood ratio interval for p approximates a Bayesian credible interval based on Kerman’s neutral noninformative conjugate prior up to O(n^{? 1}) in the confidence bounds. For the significance level α ? 0.317, the Bayesian interval based on the Jeffreys’ prior is then shown to be a compromise between the likelihood ratio and Wilson intervals. Supplementary materials for this article are available online.     

A fiducial-based approach to the one-way ANOVA in the presence of nonnormality and heterogeneous error variances

In this study, we propose a new test for testing the equality of the treatment means in one-way ANOVA when the usual normality and the homogeneity of variances assumptions are not met. In developing the proposed test, we benefit from the Fisher's fiducial inference [1–3]. Distribution of the error terms is assumed to be long-tailed symmetric (LTS) which includes the normal distribution as a limiting case. Modified maximum likelihood (MML) estimators are used in the test statistics rather than the traditional least squares (LS) estimators, since LS estimators have very low efficiencies under nonnormal distributions, see Tiku [4] for the details of MML methodology. An extensive Monte Carlo simulation study is done to compare the efficiency of the proposed test with the corresponding test based on normal theory, see Li et al. [5]. Finally, we give a real life example to show the applicability of the proposed methodology.     

Composite transformation models: A fiducial perspective

This paper discusses inferences for the parameters of a transformation model in the presence of a scalar nuisance parameter that describes the shape of the error distribution. The development is from the point of view of conditional inference and thus is an attempt to extend the classical fiducial (or structural inference) argument. For known shape parameter it is straightforward to derive a fiducial distribution of the transformation parameters from which confidence points can be obtained. For unknown shape parameter, the paper discusses a certain average of these fiducial distributions. The weights used in this averaging process are naturally induced by the action of the underlying group of transformations and correspond to a noninformative prior for the nuisance parameter. This results in a confidence distribution for the transformation parameters which in some cases has good frequentist properties. The method is illustrated by some examples.     

Predictive Inference from a Two-Parameter Rayleigh Life Model Given a Doubly Censored Sample

This article is concerned with making predictive inference on the basis of a doubly censored sample from a two-parameter Rayleigh life model. We derive the predictive distributions for a single future response, the ith future response, and several future responses. We use the Bayesian approach in conjunction with an improper flat prior for the location parameter and an independent proper conjugate prior for the scale parameter to derive the predictive distributions. We conclude with a numerical example in which the effect of the hyperparameters on the mean and standard deviation of the predictive density is assessed.     

Bayesian and frequentist prediction limits for the Poisson distribution

Prediction limits for Poisson distribution are useful in real life when predicting the occurrences of some phenomena, for example, the number of infections from a disease per year among school children, or the number of hospitalizations per year among patients with cardiovascular disease. In order to allocate the right resources and to estimate the associated cost, one would want to know the worst (i.e., an upper limit) and the best (i.e., the lower limit) scenarios. Under the Poisson distribution, we construct the optimal frequentist and Bayesian prediction limits, and assess frequentist properties of the Bayesian prediction limits. We show that Bayesian upper prediction limit derived from uniform prior distribution and Bayesian lower prediction limit derived from modified Jeffreys non informative prior coincide with their respective frequentist limits. This is not the case for the Bayesian lower prediction limit derived from a uniform prior and the Bayesian upper prediction limit derived from a modified Jeffreys prior distribution. Furthermore, it is shown that not all Bayesian prediction limits derived from a proper prior can be interpreted in a frequentist context. Using a counterexample, we state a sufficient condition and show that Bayesian prediction limits derived from proper priors satisfying our condition cannot be interpreted in a frequentist context. Analysis of simulated data and data on Atlantic tropical storm occurrences are presented.