The results obtained in five years of forecasting with Bayesian vector autoregressions (BVAR's) demonstrate that this inexpensive, reproducible statistical technique is as accurate, on average, as those used by the best known commercial forecasting services. This article considers the problem of economic forecasting, the justification for the Bayesian approach, its implementation, and the performance of one small BVAR model over the past five years.
A recently developed statistical model, called Bayesian vector autoregression, has proven to be a useful tool for economic forecasting. Such a model today forecasts a strong resurgence of growth in the second half of 1985 and in 1986.
The kernel method of estimating the cell probabilities of a multivariate categorical distribution, due to Aitchison & Aitken (1976), depends crucially on an unknown smoothing parameter λ. A method of estimating A is introduced which is explicitly connected to multivariate discrimination. The method, based on maximization of the leaving-one-out estimator of the nonerror rate, is shown to be Bayes risk strongly consistent. An example is given to illustrate the application.
A predictive likelihood is given which approximates both Bayes and maximum likelihood predictive inference by expansion of a posterior likelihood. This synthesizes and extends previous results and is widely applicable. The approximation usually differs from exact Bayes posterior predictive density by Op(n–2), and from exact predictive likelihood by Op(n–2) but does not depend on the availability of prior information and is applicable when exact predictive likelihood cannot…
DENNIS D. BOOS · JOHN F. MONAHAN·刊期 1986 · 卷 73（1）·Biometrika已编入正刊
利用先验信息的自助法AI 解读
Bayesian analysis is subject to the same kinds of misspeciflcation problems which motivate the robustness and nonparametric literature. We present a method of incorporating prior information which performs well without direct knowledge of the error distribution. This is accomplished by replacing the likelihood in Bayes's formula by a bootstrap estimate of the sampling density of a robust estimator. The flexibility of the method is illustrated by examples, and its performance…
Assume a priori that the log density is a sample function from a Gaussian process subject to the condition that the density integrates to one. The posterior distribution given a number of observations is then still a Gaussian process with the same condition and the same covariance function. The mean value function is changed according to a simple formula. This prior may thus be regarded as a conjugate prior for an unknown density. The mode of the posterior distribution is…
Sequential selections are to be made from two stochastic processes, or "arms." At each stage the arm selected for observation depends on past observations. The objective is to maximize the expected sum of the first $n$ observations. For arm 1 the observations are identically distributed with probability measure $P$, and for arm 2 the observations have probability measure $Q; P$ is a Dirichlet process and $Q$ is known. An equivalent problem is deciding sequentially when to…
For data $\theta + \varepsilon_i, i = 1, \ldots, n$ where $\varepsilon_i$ are i.i.d. $\sim F$ with the median of $F$ equal to $0$ but $F$ otherwise unknown, it is desired to estimate $\theta$. In Doss (1985) priors are put on the pair $(F, \theta)$, the marginal posterior distribution of $\theta$ is computed, and the mean of the posterior is taken as the estimate of $\theta$. In the present paper a frequentist point of view is adopted. The consistency properties of the Bayes…
Let $X_i, i = 1, \ldots, n$ be i.i.d. $\sim F_\theta$, where $F_\theta(x) = F(x - \theta)$ for some $F$ that has median equal to 0. $F$ is assumed unknown or only partially known, and the problem is to estimate $\theta$. Priors are put on the pair $(F, \theta)$. The priors on $F$ concentrate all their mass on c.d.f.s with median equal to 0. These priors include "Dirichlet-type" priors. The marginal posterior distribution of $\theta$ given $X_1, \ldots, X_n$ is computed. The…
Assuming a model appropriate for many multistage sample surveys, Bayesian predictive inference for a general linear function, ω, of the finite population elements is described. In a broad class of linear estimators of ω, the posterior mean, E″(ω), of ω is shown to have the optimal frequentist property of minimal bounded mean squared error. For the special case of three-stage sampling, E″(ω) is described in detail. Also presented are the results of an investigation of the…
SUMMARY Non-parametric regression using cubic splines is an attractive, flexible and widely-applicable approach to curve estimation. Although the basic idea was formulated many years ago, the method is not as widely known or adopted as perhaps it should be. The topics and examples discussed in this paper are intended to promote the understanding and extend the practicability of the spline smoothing methodology. Particular subjects covered include the basic principles of the…
This paper examines how a Bayesian decision maker would update his/her probability $p$ for the occurrence of an event $A$ in the light of a number of expert opinions expressed as probabilities $q_1, \cdots, q_n$ of $A$. It is seen, among other things, that the linear opinion pool, $\lambda_0p + \sum^n_{i = 1} \lambda_iq_i$, corresponds to an application of Bayes' Theorem when the decision maker has specified only the mean of the marginal distribution for $(q_1, \cdots, q_n)$…
We may take observations sequentially from a population with unknown mean $\theta$. After this sampling stage, we are to decide whether $\theta$ is greater or less than a known constant $\nu$. The net worth upon stopping is either $\theta$ or $\nu$, respectively, minus sampling costs. The objective is to maximize the expected net worth when the probability measure of the observations is a Dirichlet process with parameter $\alpha$. The stopping problem is shown to be…
The problem is to estimate a quantile of a symmetric distribution. The cases of known and unknown center are studied for small and large samples. The estimators for known center are the sample quantile, the symmetrized sample quantile, the sample quantile from flipped over data, the Rao-Blackwellized sample quantile, and a Bayes estimator using a Dirichlet prior. For center unknown, we study the analogues of the first four estimators listed above. For small samples and…
Empirical partially Bayes methods are considered as a means of improving efficiency in a class of problems in which the number of nuisance parameters increases to infinity. In the method used, the parameter of interest is estimated in an asymptotically unbiased way while James-Stein shrinkage is applied to the nuisance parameter estimates. When the shrinkage estimators are carefully chosen, this yields estimators generally more efficient than maximum likelihood. In the…
A new goodness-of-fit statistic for sparse multinomials is proposed. It is assumed that the null distribution exhibits smoothness. The test statistic is based on the maximum posterior estimator probability estimates of Simonoff (1983). Computer simulations are used to estimate the null distribution, significance levels, and the power function of the test. It is shown that power of the test is a great improvement over that of the standard tests if the alternative distribution…
Estimation of an individual's regression coefficients is considered in a multivariate general linear model, where it is assumed that the individual's coefficients β k are subject to both fixed effects and random effects over different individuals. The mean squared error matrix of a natural estimator of β k is derived for any individual k, in the general situation where all parameters of the model must be estimated, and is shown to be smaller than the mean squared error…
A hierarchical logistic regression model is proposed for studying data with group structure and a binary response variable. The group structure is defined by the presence of micro observations embedded within contexts (macro observations), and the specification is at both of these levels. At the first (micro) level, the usual logistic regression model is defined for each context. The same regressors are used in each context, but the micro regression coefficients are free to…
Noel Cressie · Allan Seheult·刊期 1985-08 · 卷 72（2）·Biometrika已编入正刊
抽样检验中的经验贝叶斯估计AI 解读
This paper considers an empirical Bayes approach to estimating the number of defectives in a hypergeometric distribution. It involves application of an empirical Bayes identity which has proved useful in providing simple Bayes estimates for various other models. Data on the reliability of coding the answers to questions from a household survey illustrate how an empirical Bayes estimate can be computed without knowledge of the prior distribution of the number of defectives.
SUMMARY Precise statements of three basic conditions for the asymptotic normality of limiting density functions are given. In the Bayesian context, these conditions as a whole provide a greater generality and more operational flexibility for asymptotic posterior normality than those assumed by previous investigators.
SUMMARY This note describes a Bayesian method for estimation of finite population parameters in general population surveys where acceptable regression-type models are typically unavailable. A categorical data model is adopted as in Ericson (1969, Section 4). However, specifications of smoothness are incorporated into the prior distribution. These smoothness conditions are expressed as unimodal or, possibly, multi-modal order relations among the category probabilities.…
Sequential selections are to be made from two stochastic processes, or "arms." At each stage the arm selected for observation depends on past observations. The objective is to maximize the expected sum of the first n observations. For arm 1 the observations are identically distributed with probability measure P, and for arm 2 the observations have probability measure Q;P is a Dirichlet process and Q is known. An equivalent problem is deciding sequentially when to…
For data θ+εi,i=1,…,n where εi are i.i.d. ∼F with the median of F equal to 0 but F otherwise unknown, it is desired to estimate θ. In Doss (1985) priors are put on the pair (F,θ), the marginal posterior distribution of θ is computed, and the mean of the posterior is taken as the estimate of θ. In the present paper a frequentist point of view is adopted. The consistency properties of the Bayes…
Let Xi,i=1,…,n be i.i.d. ∼Fθ, where Fθ(x)=F(x−θ) for some F that has median equal to 0. F is assumed unknown or only partially known, and the problem is to estimate θ. Priors are put on the pair (F,θ). The priors on F concentrate all their mass on c.d.f.s with median equal to 0. These priors include "Dirichlet-type" priors. The marginal posterior distribution of θ given X1,…,Xn is computed. The…
This paper examines how a Bayesian decision maker would update his/her probability p for the occurrence of an event A in the light of a number of expert opinions expressed as probabilities q1,⋯,qn of A. It is seen, among other things, that the linear opinion pool, λ0p+∑i=1nλiqi, corresponds to an application of Bayes' Theorem when the decision maker has specified only the mean of the marginal distribution for (q1,⋯,qn)…
We may take observations sequentially from a population with unknown mean θ. After this sampling stage, we are to decide whether θ is greater or less than a known constant ν. The net worth upon stopping is either θ or ν, respectively, minus sampling costs. The objective is to maximize the expected net worth when the probability measure of the observations is a Dirichlet process with parameter α. The stopping problem is shown to be…