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.     

Penalized maximum likelihood estimation in the modified extended Weibull distribution

We address the issue of performing inference on the parameters that index the modified extended Weibull (MEW) distribution. We show that numerical maximization of the MEW log-likelihood function can be problematic. It is even possible to encounter maximum likelihood estimates that are not finite, that is, it is possible to encounter monotonic likelihood functions. We consider different penalization schemes to improve maximum likelihood point estimation. A penalization scheme based on the Jeffreys’ invariant prior is shown to be particularly useful. Simulation results on point estimation, interval estimation, and hypothesis testing inference are presented. Two empirical applications are presented and discussed.     

A prior for the variance in hierarchical models

The choice of prior distributions for the variances can be important and quite difficult in Bayesian hierarchical and variance component models. For situations where little prior information is available, a ‘nonin-formative’ type prior is usually chosen. ‘Noninformative’ priors have been discussed by many authors and used in many contexts. However, care must be taken using these prior distributions as many are improper and thus, can lead to improper posterior distributions. Additionally, in small samples, these priors can be ‘informative’. In this paper, we investigate a proper ‘vague’ prior, the uniform shrinkage prior (Strawder-man 1971; Christiansen & Morris 1997). We discuss its properties and show how posterior distributions for common hierarchical models using this prior lead to proper posterior distributions. We also illustrate the attractive frequentist properties of this prior for a normal hierarchical model including testing and estimation. To conclude, we generalize this prior to the multivariate situation of a covariance matrix.     

普通高校赛前短期田径训练方法探索

田径赛前短期训练是普通高校经常采用的训练方法。由于训练时间短、任务重,比赛成绩往往很难把握。浓缩训练阶段的时段,合理安排运动负荷等训练方法,对运动员短期内提高运动成绩,获得最佳训练效果具有积极意义。     

Bayes minimax estimation of a bernoulli p in a restricted parameter space

In many estimation problems the parameter of interest is known,a priori, to belong to a proper subspace of the natural parameter space. Although useful in practice this type of additional information can lead to surprising theoretical difficulties. In this paper the problem of minimax estimation of a Bernoulli pwhen pis restricted to a symmetric subinterval of the natural parameter space is considered. For the sample sizes n = 1,2,3, and 4 least favorable priors with finite support are provided and the corresponding Bayes estimators are shown to be minimax. For n = 5 and 6 the usual constant risk minimax estimator is shown to be the Bayes minimax estimator corresponding to a least favorable prior with finite support, provided the restriction on the parameter space is not too tight.     

Empirical bayes estimation of a distribution function with a gamma process prior

A sequence of empirical Bayes estimators is given for estimating a distribution function. It is shown that ‘i’ this sequence is asymptotically optimum relative to a Gamma process prior, ‘ii’ the overall expected loss approaches the minimum Bayes risk at a rate of n , and ‘iii’ the estimators form a sequence of proper distribution functions. Finally, the numerical example presented by Susarla and Van Ryzin ‘Ann. Statist., 6, 1978’ reworked by Phadia ‘Ann. Statist., 1, 1980, to appear’ has been analyzed and the results are compared to the numerical results by Phadia     

Interval estimation via tail functions

The authors describe a new method for constructing confidence intervals. Their idea consists in specifying the cutoff points in terms of a function of the target parameter rather than as constants. When it is suitably chosen, this so‐called tail function yields shorter confidence intervals in the presence of prior information. It can also be used to improve the coverage properties of approximate confidence intervals. The authors illustrate their technique by application to interval estimation of the mean of Bernoulli and normal populations. They further suggest guidelines for choosing the optimal tail function and discuss the relationship with Bayesian inference.     

Reparameterization strategies for hidden Markov models and Bayesian approaches to maximum likelihood estimation

This paper synthesizes a global approach to both Bayesian and likelihood treatments of the estimation of the parameters of a hidden Markov model in the cases of normal and Poisson distributions. The first step of this global method is to construct a non-informative prior based on a reparameterization of the model; this prior is to be considered as a penalizing and bounding factor from a likelihood point of view. The second step takes advantage of the special structure of the posterior distribution to build up a simple Gibbs algorithm. The maximum likelihood estimator is then obtained by an iterative procedure replicating the original sample until the corresponding Bayes posterior expectation stabilizes on a local maximum of the original likelihood function.     

A generalization and Bayesian interpretation of ridge-type estimators with good prior means

Pliskin (1987) and Trenkler (1988) compared ridge-type estimators with good prior means. From a Bayesian viewpoint, these estimators are special cases of Bayesestimators and the mean square error matrix comparisons can be made in the more general case.     

Analysis of rounded exponential data

The problem of inference based on a rounded random sample from the exponential distribution is treated. The main results are given by an explicit expression for the maximum-likelihood estimator, a confidence interval with a guaranteed level of confidence, and a conjugate class of distributions for Bayesian analysis. These results are exemplified on two concrete examples. The large and increasing body of results on the topic of grouped data has been mostly focused on the effect on the estimators. The methods and results for the derivation of confidence intervals here are hence of some general theoretical value as a model approach for other parametric models. The Bayesian credibility interval recommended in cases with a lack of other prior information follows by letting the prior equal the inverted exponential with a scale equal to one divided by the resolution. It is shown that this corresponds to the standard non-informative prior for the scale in the case of non-rounded data. For cases with the absence of explicit prior information it is argued that the inverted exponential prior with a scale given by the resolution is a reasonable choice for more general digitized scale families also.