Bayesian estimation of the product of two proportions

Suppose that for i = 1,2, a Bernoulli random variable with success probability θ_i is observable from population i. The problem is to estimate θ = θ₁θ₂ using a Bayesian approach with squared error estimation loss in θ. For estimating θ, the best nonrandom sampling scheme, the two-stage sampling scheme, and the optimal sampling scheme are discussed. It is shown that the two-stage sampling scheme is typically asymptotically optimal, and can improve the Bayes risk (over the best nonrandom allocation) up to fifty percent     

A Bayesian approach to the estimation of maps between Riemannian manifolds

Let Θ be a smooth compact oriented manifold without boundary, imbedded in a Euclidean space E ^s, and let γ be a smooth map of Θ into a Riemannian manifold Λ. An unknown state θ ∈ Θ is observed via X = θ + εξ, where ε > 0 is a small parameter and ξ is a white Gaussian noise. For a given smooth prior λ on Θ and smooth estimators g(X) of the map γ we derive a second-order asymptotic expansion for the related Bayesian risk. The calculation involves the geometry of the underlying spaces Θ and Λ, in particular, the integration-by-parts formula. Using this result, a second-order minimax estimator of γ is found based on the modern theory of harmonic maps and hypo-elliptic differential operators.      

Bayes empirical Bayes estimation of a poisson mean

In the empirical Bayes (EB) decision problem consisting of squared error estimation of a Poisson mean, a prior distribution λ is placed on the gamma family of prior distributions to produce Bayes EB estimators which are admissible. A subclass of such estimators is shown to be asymptotically optimal (a.o.). The results of a Monte Carlo study are presented to demonstrate the favorable a.o. property of the Bayes EB estimators in comparison with other competitors.     

A sequential test for the failure rate of an exponential distribution with censored data

This paper provides an optimal sequential decision procedure for deciding between two composite hypotheses about the unknown failure rate of an exponential distribution, using censored data. The procedure has two components, a stopping time and a decision function. The optimal stopping time minimizes the expected total loss due to a wrong decision plus cost of observing the process. The optimal decision function is easily characterized once a stopping time has been specified. The main result determines the continuation region for the optimal decision procedure     

正态分布在不同先验下的Bayes估计及风险比较

本文讨论在均值未知,方差已知的正态分布情况下通过在共轭先验以及Jeffreys先验二种先验下的Bayes估计问题,在平方损失函数下和线性损失函数下Bayes风险的比较.数据计算可以看出,在Jeffreys先验下的Bayes风险要比在共轭先验下的Bayes风险要大,但是当样本量增大时,两者的后验风险越来越靠近.     

BAYESIAN ANALYSIS OF DATA WITH ONLY ONE FAILURE

The hearings of a certain type have their lives following a Weibull distribution. In a life test with 20 sets of bearings, only one set failed within the specified time, and none of the remainder failed even after the time of to estimate the reliabilWith a set of testing data like that in Table 1, it is required to estimate the reliability at the mission time, In this paper, we first use hierarchical Bayesian method of determine the prior distribution and the Bayesian estimates of various probabilities of failures, pi‘s, then use the method of least squares to estimate the parameters of the Weibull distribution and the reliability. Actual computation shows that the estimates so obtained are rather robust. And the results have been adopted for practical use.     

失效率的E-Bayes估计和多层Bayes估计

提出了一种可靠性参数的估计方法—E-Bayes估计法,对寿命服从指数分布的产品,在无失效数据情形,给出了失效率的E-Bayes估计的定义、E-Bayes估计和多层Bayes估计,并在此基础上给出了E-Bayes估计的性质.最后,结合发动机的实际问题进行了计算,结果表明E-Bayes估计法可行且便于应用.     

E-Bayesian estimation and hierarchical Bayesian estimation of failure rate

This paper introduces a new method, E-Bayesian estimation method, to estimate failure rate. The method is suitable for the censored or truncated data with small sample sizes and high reliability. The definition and properties of E-Bayesian estimation are given. A real data set is discussed, which shows that the method is both efficiency and easy to operate.     

Bayesian analysis of hybrid life tests with exponential failure times

Summary A hybrid life test procedure is discussed from the Bayesian viewpoint. A total ofn items is placed on test, failed items are either not replaced or are replaced, and the test is terminated either when a pre-chosen number,K, of items have failed, or when a pre-determined time on test has been reached. Posterior and predictive distributions are obtained under the assumption of an exponential failure distribution, and point and interval estimates are given for the mean life and the life of an untested item. The results are applied to a numerical example.     

Lower estimation of the difference between quasi-arithmetic means

In the 1960s Cargo and Shisha introduced a metric in a family of quasi-arithmetic means defined on a common interval as the maximal possible difference between these means taken over all admissible vectors with corresponding weights. During the years 2013–2016 we proved that, having two quasi-arithmetic means, we can majorize the distance between them in terms of the Arrow–Pratt index. In this paper we are going to prove that this operator can also be used to establish certain lower bounds of this distance.     

Shrinkage priors for Bayesian estimation of the mean matrix in an elliptically contoured distribution

This paper deals with the problem of estimating the mean matrix in an elliptically contoured distribution with unknown scale matrix. The Laplace and inverse Laplace transforms of the density allow us not only to evaluate the risk function with respect to a quadratic loss but also to simplify expressions of Bayes estimators. Consequently, it is shown that generalized Bayes estimators against shrinkage priors dominate the unbiased estimator.     

Krylov iterative methods for the geometric mean of two matrices times a vector

In this work, we are presenting an efficient way to compute the geometric mean of two positive definite matrices times a vector. For this purpose, we are inspecting the application of methods based on Krylov spaces to compute the square root of a matrix. These methods, using only matrix-vector products, are capable of producing a good approximation of the result with a small computational cost.     

The detection and estimation of the change point in a disccrete-time stochastic system

A piecewise-constant process containing a single jump is observed under noise in the context of discrete time. The conditional density and maximum a posteriori (MAP) estimator of the jump time as well as the Bayes detector of the jump itself are determined using the powerful measure transformation approach. The Bayes detector provides a convenient sequential detection rule for practical on-line implementation. An asymptotic result for the distribution of the MAP estimator's estimation error and the corresponding convergence rate are derived. This result provides a reference measure of optimal performance for jump-time estimators in discrete-time stochastic systems that does not depend on the jump time's prior distribution     

A semiparametric Bayesian approach to the analysis of financial time series with applications to value at risk estimation

GARCH models are commonly used for describing, estimating and predicting the dynamics of financial returns. Here, we relax the usual parametric distributional assumptions of GARCH models and develop a Bayesian semiparametric approach based on modeling the innovations using the class of scale mixtures of Gaussian distributions with a Dirichlet process prior on the mixing distribution. The proposed specification allows for greater flexibility in capturing the usual patterns observed in financial returns. It is also shown how to undertake Bayesian prediction of the Value at Risk (VaR). The performance of the proposed semiparametric method is illustrated using simulated and real data from the Hang Seng Index (HSI) and Bombay Stock Exchange index (BSE30).     

Generalized Bayes minimax estimation of the normal mean matrix with unknown covariance matrix

This paper addresses the problem of estimating the normal mean matrix in the case of unknown covariance matrix. This problem is solved by considering generalized Bayesian hierarchical models. The resulting generalized Bayes estimators with respect to an invariant quadratic loss function are shown to be matricial shrinkage equivariant estimators and the conditions for their minimaxity are given.     

On the admissibility of an estimator of a normal mean vector under a linex loss function

For ap-variate normal mean with known variances, the model proposed by Zellner (1986,J. Amer. Statist. Assoc.,81, 446–451) is discussed in a slightly different framework. A generalized Bayes estimate is derived from a three-stage Bayes point of view under the asymmetric loss function, and the admissibility of such estimators is proved.     

Combining the data from two normal populations to estimate the mean of one when their means difference is bounded

In this paper we address the problem of estimating θ₁ when , are observed and |θ₁−θ₂|?c for a known constant c. Clearly Y₂ contains information about θ₁. We show how the so-called weighted likelihood function may be used to generate a class of estimators that exploit that information. We discuss how the weights in the weighted likelihood may be selected to successfully trade bias for precision and thus use the information effectively. In particular, we consider adaptively weighted likelihood estimators where the weights are selected using the data. One approach selects such weights in accord with Akaike's entropy maximization criterion. We describe several estimators obtained in this way. However, the maximum likelihood estimator is investigated as a competitor to these estimators along with a Bayes estimator, a class of robust Bayes estimators and (when c is sufficiently small), a minimax estimator. Moreover we will assess their properties both numerically and theoretically. Finally, we will see how all of these estimators may be viewed as adaptively weighted likelihood estimators. In fact, an over-riding theme of the paper is that the adaptively weighted likelihood method provides a powerful extension of its classical counterpart.     

A Bayesian approach to the estimation of maps between Riemannian manifolds. II: Examples

Let gJ be a smooth compact oriented manifold without boundary, imbedded in a Euclidean space E ^s , and let γ be a smooth map of gJ into a Riemannian manifold Λ. An unknown state θ ∈ gJ is observed via X = θ + ɛξ, where ɛ > 0 is a small parameter and ξ is a white Gaussian noise. For a given smooth prior λ on gJ and smooth estimators g(X) of the map γ we have derived a second-order asymptotic expansion for the related Bayesian risk [3]. In this paper, we apply this technique to a variety of examples. The second part examines the first-order conditions for equality-constrained regression problems. The geometric tools that are utilized in [3] are naturally applicable to these regression problems.