Bayesian Prediction Analysis for Growth Curve Model Using Noninformative Priors

We apply a Bayesian approach to the problem of prediction in an unbalanced growth curve model using noninformative priors. Due to the complexity of the model, no analytic forms of the predictive densities are available. We propose both approximations and a prediction-oriented Metropolis-Hastings sampling algorithm for two types of prediction, namely the prediction of future observations for a new subject and the prediction of future values for a partially observed subject. They are illustrated and compared through real data and simulation studies. Two of the approximations compare favorably with the approximation in Fearn (1975, Biometrika, 62, 89–100) and are very comparable to the more accurate Rao-Blackwellization from Metropolis-Hastings sampling algorithm.     

Bayesian linear prediction in finite populations

In this paper, Bayesian linear prediction of the total of a finite population is considered in situations where the observation error variance is parameter dependent. Connections with least squares prediction (Royall (1976, J. Amer. Statist. Assoc., 71, 657–664)) in mixed linear models (Theil (1971, Principles of Econometrics, Wiley, New York)), are established. Extensions to the case of dynamic (state dependent) superpopulation models are also proposed.     

Predicting a cyclic Poisson process

We construct and investigate a (1−α)-upper prediction bound for a future observation of a cyclic Poisson process using past data. A normal based confidence interval for our upper prediction bound is established. A comparison of the new prediction bound with a simpler nonparametric prediction bound is also given.     

Prediction of Fractional Brownian Motion-Type Processes

Abstract

We introduce a class of continuous-time Gaussian processes with stationary increments via moving-average representation with good MA coefficient. The class includes fractional Brownian motion with Hurst index less than 1/2 as a typical example. It also includes processes which have different indices corresponding to the local and long-time properties, repsectively. We derive some basic properties of the processes, and, using the results, we establish a prediction formula for them. The prediction kernel in the formula is given explicitly in terms of MA and AR coefficients.     

Robust Bayesian prediction and estimation under a squared log error loss function

Robust Bayesian analysis is concerned with the problem of making decisions about some future observation or an unknown parameter, when the prior distribution belongs to a class Γ instead of being specified exactly. In this paper, the problem of robust Bayesian prediction and estimation under a squared log error loss function is considered. We find the posterior regret Γ-minimax predictor and estimator in a general class of distributions. Furthermore, we construct the conditional Γ-minimax, most stable and least sensitive prediction and estimation in a gamma model. A prequential analysis is carried out by using a simulation study to compare these predictors.     

Solutions for the Hyperbolic Dirac Equation on R1, m

In this article the hyperbolic unit ball in R ^m will be identified with the manifold of rays in the future null cone in R ^m+1. By means of the induced Clifford algebra structure there, one can introduce a definition of Dirac operators on sections of homogeneous line bundles. An infinite class of solutions for the resulting hyperbolic Dirac-equation will be constructed, in case of an odd dimension. In order to obtain these solutions a geometrical picture will be used, because each ray in the future cone will be identified with a point on a surface Σ in the future cone. The original Dirac-equation can then be rewritten in terms of the coordinates on this surface, and the resulting equation will be solved by means of Frobenius' method.     

Block Coordinate Relaxation Methods for Nonparametric Wavelet Denoising

Abstract

An important class of nonparametric signal processing methods entails forming a set of predictors from an overcomplete set of basis functions associated with a fast transform (e.g., wavelet packets). In these methods, the number of basis functions can far exceed the number of sample values in the signal, leading to an ill-posed prediction problem. The “basis pursuit” denoising method of Chen, Donoho, and Saunders regularizes the prediction problem by adding an l ₁ penalty term on the coefficients for the basis functions. Use of an l ₁ penalty instead of l ₂ has significant benefits, including higher resolution of signals close in time/frequency and a more parsimonious representation. The l ₁ penalty, however, poses a challenging optimization problem that was solved by Chen, Donoho and Saunders using a novel application of interior-point algorithms (IP). This article investigates an alternative optimization approach based on block coordinate relaxation (BCR) for sets of basis functions that are the finite union of sets of orthonormal basis functions (e.g., wavelet packets). We show that the BCR algorithm is globally convergent, and empirically, the BCR algorithm is faster than the IP algorithm for a variety of signal denoising problems.     

Nonparametric prediction intervals for progressive Type-II censored order statistics based on k-records

In this paper, we consider the prediction problem in two-sample case and study the non-parametric predicting future progressively Type-II censored order statistics based on observed $k$ -records from the same distribution. Also, prediction intervals for progressively Type-II censored spacings are obtained based on $k$ -record spacings. It is shown that the coverage probabilities of these intervals are exact and do not depend on the underlying distribution. Moreover, optimal prediction intervals are derived for each case. Finally, for illustrating the proposed procedure, we consider a real data set and numerical computations are given. The results of Ahmadi and Balakrishnan (Statistics 44:417–430, 2010) can be achieved as special cases of our results.     

Simultaneous prediction regions

Summary Two new methods for constructing simultaneous prediction regions are the subject of this article. Both methods simultaneously assert a collection of prediction regions, one prediction region for each future observable of interest. Both methods have the same aims: to control the overall coverage probability of the simultaneous prediction region and to keep equal the coverage probabilities of the individual prediction statements that make up the simultaneous region. The latter property is called balance.The two approaches differ in their choice of critical values. For leading cases, the first method achieves the desired overall coverage probability and the desired balance up to errors of ordern ^–1, wheren is the size of the learning sample. The second method reduces both errors to ordern ^–2. Calculating critical values in the second approach usually relies on a bootstrap algorithm.If overall coverage probability and degree of balance are instead calculatedconditionally given the learning sample, the two methods show the same asymptotic performance. This result reflects intrinsic limits on the extent to which conditional coverage probabilities can be controlled in prediction.This research was supported in part by NSF Grant DMS-87-01426. Part of the work was done while the author was a guest of Sonderforschungsbereich 123 at Universität Heidelberg     

Dynamics of Bianchi Type I Solutions of the Einstein Equations with Anisotropic Matter

We analyze the global dynamics of Bianchi type I solutions of the Einstein equations with anisotropic matter. The matter model is not specified explicitly but only through a set of mild and physically motivated assumptions; thereby our analysis covers matter models as different from each other as, e.g., collisionless matter, elastic matter and magnetic fields. The main result we prove is the existence of an ‘anisotropy classification’ for the asymptotic behaviour of Bianchi type I cosmologies. The type of asymptotic behaviour of generic solutions is determined by one single parameter that describes certain properties of the anisotropic matter model under extreme conditions. The anisotropy classification comprises the following types. The convergent type A₊: Each solution converges to a Kasner solution as the singularity is approached and each Kasner solution is a possible past asymptotic state. The convergent types B₊ and C₊: Each solution converges to a Kasner solution as the singularity is approached; however, the set of Kasner solutions that are possible past asymptotic states is restricted. The oscillatory type D₊: Each solution oscillates between different Kasner solutions as the singularity is approached. Furthermore, we investigate non-generic asymptotic behaviour and the future asymptotic behaviour of solutions. Submitted: October 28, 2008.; Accepted: January 26, 2009.     

Prediction of the Maximum Size in Wicksell's Corpuscle Problem, II

This is a continuing paper of the authors (1998, Ann. Inst. Statist. Math., 50, 361–377). In the Wicksell corpuscle problem, the maximum size of random spheres in a volume part is to be predicted from the sectional circular distribution of spheres cut by a plane. The size of the spheres is assumed to follow the three-parameter generalized gamma distribution. Prediction methods based on the moment estimation are proposed and their performances are evaluated by simulation. For a practically probable case, one of these prediction methods is as good as a method previously proposed by the authors where the two shape parameters are assumed to be known.     

On the future infimum of positive self-similar Markov processes†

We establish integral tests and laws of the iterated logarithm for the upper envelope of the future infimum of positive self-similar Markov processes and for increasing self-similar Markov processes at 0 and +∞. Our proofs are based on the Lamperti representation and time reversal arguments due to Chaumont, L. and Pardo, J.C. (Prépublication (L'université de Paris 6), 2005). These results extend laws of the iterated logarithm for the future infimum of Bessel processes due to Khoshnevisan, D., Lewis, T.M. and Li, W.V. (On the future infima of some transient processes, Probability Theory and Related Fields, 99, 337–360, 1994).     

BSDEs and SDEs with time-advanced and -delayed coefficients

ABSTRACT

This paper introduces a class of backward stochastic differential equations (BSDEs), whose coefficients not only depend on the value of its solutions of the present but also the past and the future. For a sufficiently small time delay or a sufficiently small Lipschitz constant, the existence and uniqueness of such BSDEs is obtained. As an adjoint process, a class of stochastic differential equations (SDEs) is introduced, whose coefficients also depend on the present, the past and the future of its solutions. The existence and uniqueness of such SDEs is proved for a sufficiently small time advance or a sufficiently small Lipschitz constant. A duality between such BSDEs and SDEs is established.     

Prediction intervals based on Gompertz doubly censored data

This article is concerned with the problem of deriving Bayesian prediction bounds for the Gompertz \([Gomp(\alpha ,\,\beta )]\) distribution. Based on doubly Type II censored data, Bayesian prediction bounds for both the future observations will be derived. Two different sampling schemes have been considered. A conjugate prior for the one parameter case, as well as a joint prior for the two parameters case are outlined. Some numerical examples are given to illustrate the results.     

Assessing Mutational Capital: An Imperative for Policy

Abstract

Instructional technology (IT) is transforming collegiate teaching and learning. While IT is wondrously exciting, it is also threatening to teaching traditions that have evolved over centuries. Peter Drucker captured our fears with his recent pronouncement that in 30 years colleges and universities, as we know them today, will be “relics.” At the least, IT commands our urgent attention as we prepare for a future that we know will be dramatically different.     

METHODS OF OBTAINING COLLISION OUTCOMES FOR 3-D LATTICE GASES

Abstract

The isometric collision rules for the FCHC lattice gas, as invented by Hénon [4], are implemented. Some of the necessary background history and theory is summarised, and the implementation is discussed. Due to the findings of the present paper, this method is considered unsuitable for the computer hardware used in these tests. Since this is the only available hardware to Aerotek, recommendations for a future strategy are given, based on other methods discussed in the literature.     

On the role of heterogeneous and imperfect information in a laboratory financial market

In this paper, we analyze the behavior of a group of heterogeneously informed investors in an laboratory asset market. Our experimental setting is inspired by Huber et al. (On the benefit of information in markets with heterogeneously informed traders: an experimental study, 2004). However, instead of their system of cumulative and exogenously given information structure, we introduce an information market where the traders can buy an imperfect prediction of the future value of the dividend with a maximum anticipation of four periods. The accuracy of the prediction decreases with the chosen time horizon, whereas its price remains constant. Our results confirm a non-strictly monotonic increasing value of the information.     

Predictive Inference for the Elliptical Linear Model

This paper derives the prediction distribution of future responses from the linear model with errors having an elliptical distribution with known covariance parameters. For unknown covariance parameters, the marginal likelihood function of the parameters has been obtained and the prediction distribution has been modified by replacing the covariance parameters by their estimates obtained from the marginal likelihood function. It is observed that the prediction distribution with elliptical error has a multivariate Student'st-distribution with appropriate degrees of freedom. The results for some special cases such as the Intra-class correlation model, AR(1), MA(1), and ARMA(1,1) models have been obtained from the general results. As an application, theβ-expectation tolerance region has been constructed. An example has been added.     

Adaptive Estimation of FCG Using Nonlinear State-Space Models

Abstract

In this paper, an efficient adaptive nonlinear algorithm for estimation and identification, the so-called adaptive Lainiotis filter (ALF), is applied to the problem of fatigue crack growth (FCG) estimation, identification, and prediction of the final crack (failure). A suitable nonlinear state-space FCG model is introduced for both ALF and extended Kalman filter (EKF). Both algorithms are tested in order to compare their efficiency. Through extensive analysis and simulation, it is demonstrated that the ALF has superior performance both in FCG estimation, as well as in predicting the remaining lifetime to failure. Furthermore, it is shown that the ALF is faster and easier to implement in a parallel/distributed processing mode, and much more robust than the classic EKF.