Hierarchical Generalized Linear Models and Frailty Models with Bayesian Nonparametric Mixing

This paper proposes Bayesian nonparametric mixing for some well-known and popular models. The distribution of the observations is assumed to contain an unknown mixed effects term which includes a fixed effects term, a function of the observed covariates, and an additive or multiplicative random effects term. Typically these random effects are assumed to be independent of the observed covariates and independent and identically distributed from a distribution from some known parametric family. This assumption may be suspect if either there is interaction between observed covariates and unobserved covariates or the fixed effects predictor of observed covariates is misspecified. Another cause for concern might be simply that the covariates affect more than just the location of the mixed effects distribution. As a consequence the distribution of the random effects could be highly irregular in modality and skewness leaving parametric families unable to model the distribution adequately. This paper therefore proposes a Bayesian nonparametric prior for the random effects to capture possible deviances in modality and skewness and to explore the observed covariates' effect on the distribution of the mixed effects.     

D- and V-optimal population designs for the quadratic regression model with a random intercept term

In this paper D- and V-optimal population designs for the quadratic regression model with a random intercept term and with values of the explanatory variable taken from a set of equally spaced, non-repeated time points are considered. D-optimal population designs based on single-point individual designs were readily found but the derivation of explicit expressions for designs based on two-point individual designs was not straightforward and was complicated by the fact that the designs now depend on ratio of the variance components. Further algebraic results pertaining to d-point D-optimal population designs where d≥3 and to V-optimal population designs proved elusive. The requisite designs can be calculated by careful programming and this is illustrated by means of a simple example.     

Aggregation of isotropic autoregressive fields

This note constitutes a corrigendum to the article of Azomahou [2009, Memory properties and aggregation of spatial autoregressive models. J. Statist. Plann. Inference, 139, 2581-2597]. The aggregation of isotropic four nearest neighbors autoregressive models on the lattice Z², with random coefficient, is investigated. The spectral density of the resulting random field is studied in details for a large class of law of the AR coefficient. Depending on this law, the aggregated field may exhibit short memory or isotropic long memory.     

A review on strong convergence of weighted sums of random elements based on Wasserstein metrics

In a previous paper the authors proposed a simple method to extend results about almost sure convergence for weighted sums of real random variables to the case of Banach-valued random elements. The method arises from the extension of Skorohod's Representation Theorem for weakly convergent sequences due to Blackwell and Dubins, applied to the general framework of weakly equivalent tight sequences of probability measures. This provides a scheme which permits us to handle separately a problem that behaves like the Glivenko-Cantelli Theorem and a question on uniform integrability which generally is reduced to the real valued version of the general problem to be solved.

In this paper we prove that Wasserstein's metrics can play the same role as Skorohod's Representation Theorem in the preceding scheme. We also show that our method can be applied to obtain results with respect to various summability methods (Abel, Euler, …) even in the case in which the ‘weights’ are linear operators.     

燃烧诱发的炉墙结构振动

本文根据实测得到的国产300MW锅炉燃烧压力场功率谱密度函数,研究了用计算方法估计由燃烧诱发的结构振动,并与用实测方法获得的振动值作了对比,结果很接近。从而证明计算所用的数据与方法是可靠的。     

Bayesian palaeoclimate reconstruction

Summary.  We consider the problem of reconstructing prehistoric climates by using fossil data that have been extracted from lake sediment cores. Such reconstructions promise to provide one of the few ways to validate modern models of climate change. A hierarchical Bayesian modelling approach is presented and its use, inversely, is demonstrated in a relatively small but statistically challenging exercise: the reconstruction of prehistoric climate at Glendalough in Ireland from fossil pollen. This computationally intensive method extends current approaches by explicitly modelling uncertainty and reconstructing entire climate histories. The statistical issues that are raised relate to the use of compositional data (pollen) with covariates (climate) which are available at many modern sites but are missing for the fossil data. The compositional data arise as mixtures and the missing covariates have a temporal structure. Novel aspects of the analysis include a spatial process model for compositional data, local modelling of lattice data, the use, as a prior, of a random walk with long-tailed increments, a two-stage implementation of the Markov chain Monte Carlo approach and a fast approximate procedure for cross-validation in inverse problems. We present some details, contrasting its reconstructions with those which have been generated by a method in use in the palaeoclimatology literature. We suggest that the method provides a basis for resolving important challenging issues in palaeoclimate research. We draw attention to several challenging statistical issues that need to be overcome.     

An extension of the multivariate Chebyshev's inequality to a random vector with a singular covariance matrix

ABSTRACT

We extend Chebyshev's inequality to a random vector with a singular covariance matrix. Then we consider the case of a multivariate normal distribution for this generalization.     

ASYMPTOTIC AND SMALL SAMPLE STATISTICAL PROPERTIES OF RANDOM FRAILTY VARIANCE ESTIMATES FOR SHARED GAMMA FRAILTY MODELS

This paper concerns maximum likelihood estimation for the semiparametric shared gamma frailty model; that is the Cox proportional hazards model with the hazard function multiplied by a gamma random variable with mean 1 and variance θ. A hybrid ML-EM algorithm is applied to 26 400 simulated samples of 400 to 8000 observations with Weibull hazards. The hybrid algorithm is much faster than the standard EM algorithm, faster than standard direct maximum likelihood (ML, Newton Raphson) for large samples, and gives almost identical results to the penalised likelihood method in S-PLUS 2000. When the true value θ₀ of θ is zero, the estimates of θ are asymptotically distributed as a 50–50 mixture between a point mass at zero and a normal random variable on the positive axis. When θ₀ > 0, the asymptotic distribution is normal. However, for small samples, simulations suggest that the estimates of θ are approximately distributed as an x ? (100 ? x)% mixture, 0 ≤ x ≤ 50, between a point mass at zero and a normal random variable on the positive axis even for θ₀ > 0. In light of this, p-values and confidence intervals need to be adjusted accordingly. We indicate an approximate method for carrying out the adjustment.     

Likelihood Ratio Order of m-Spacings in Multiple-Outlier Models

This article deals with the multiple-outlier exponential model. The likelihood ratio order between m-spacings of the combined sample is developed, some results extend the conclusions on simple spacings in Wen et al. (2007 Wen , S. , Lu , Q. , Hu , T. ( 2007 ). Likelihood ratio order of spacings of heterogeneous exponential random variables . J. Multivariate Anal. 98 : 743 – 756 .[Crossref], [Web of Science ®] , [Google Scholar]).     

Some New Results on Likelihood Ratio Orderings for Spacings of Heterogeneous Exponential Random Variables

Let X ₁, X ₂,…, X _n be independent exponential random variables with X _i having failure rate λ _i for i = 1,…, n. Denote by D _i:n  = X _i:n  ? X _i?1:n the ith spacing of the order statistics X _1:n  ≤ X _2:n  ≤ ··· ≤ X _n:n , i = 1,…, n, where X _0:n ≡ 0. It is shown that if λ _n+1 ≤ [≥] λ _k for k = 1,…, n then D _n:n  ≤ _lr D _n+1:n+1 and D _1:n  ≤ _lr D _2:n+1 [D _2:n+1 ≤ _lr D _2:n ], and that if λ _i  + λ _j  ≥ λ _k for all distinct i,j, and k then D _n?1:n  ≤ _lr D _n:n and D _n:n+1 ≤ _lr D _n:n , where ≤ _lr denotes the likelihood ratio order. We also prove that D _1:n  ≤ _lr D _2:n for n ≥ 2 and D _2:3 ≤ _lr D _3:3 for all λ _i 's.