Estimation of structural equation models with exact and stochastic prior information

Recently, structural equation models are widely used in assessing data in economical and behavioral researches. To give more freedom in defining the structures of the model and obtain more precise and meaningful interpretations to the data, prior informations about the unknown parameters are usually incorporated in the analysis. In this article, basic estimation theory of structural equation models with both exact and stochastic prior informations is developed via the generalized least squares approach. Asymptotic properties of the estimator are derived and an iterative algorithm is implemented to obtain the estimates. An illustrative example is reported.     

Machine covering with combined partial information

Machine scheduling and covering problems may occur in many applications such as load balancing in network communication channel assignment, parallel processing in large-size computing, task arrangement in flexible manufacturing systems, etc. In this paper we study machine covering problems with combined partial information on m   parallel identical machines. We consider sequences where the processing time of all jobs are at most 1/k$1/k$ times of the optimal value (for an integer k). For the case where the optimal value is not known in advance, we show that LS algorithm is optimal. For the case where the optimal value is known in advance, we give lower bounds and present semi-online algorithms.     

Inferences for hierarchical models with partial prior information

Suppose items can be purchased from one of k-suppliers and it is required to purchase from the one with the smaller failure rate or equivalently from the one with the larger mean-time-to-failure. It is assumed that data $\mathit{d}$, in the form of the times-to-failure for $n_1,\dots ,n_k$ items from suppliers $1,\dots ,k$, respectively is available. There are two suggested selection criteria studied in this paper and when comparing only two suppliers they reduce to$P({\theta }_1<b{\theta }_2|d)\text{and}P(Y_1>{\mathit{cY}}_2|d)\text{,}$where b and c are prespecified practical constants; ${\theta }_1$ and ${\theta }_2$ are the respective mean failure rates; $Y_1$ and $Y_2$ are the predicted times to failure for individual items purchased from each supplier.In addition partial prior information about the $k$-suppliers collectively is assumed to have been elicited. This situation is modelled using the hierarchical Bayesian approach, which easily facilitates interpreting the elicited partial prior information as constraints on the hyperpriors, i.e. hyperpriors that are known only to be contained in families with specified properties. In this paper these properties are assumed to be in the form of specifying certain quantiles arising from the elicited information. Minimum and maximum values of the above selection criteria are obtained and are used to indicate whether or not the elicited prior information is useful. Specific examples are given for comparing two suppliers but generalisation to k-suppliers follows easily.     

Objective Bayesian analysis based on upper record values from two-parameter Rayleigh distribution with partial information

In the life test, predicting higher failure times than the largest failure time of the observed is an important issue. Although the Rayleigh distribution is a suitable model for analyzing the lifetime of components that age rapidly over time because its failure rate function is an increasing linear function of time, the inference for a two-parameter Rayleigh distribution based on upper record values has not been addressed from the Bayesian perspective. This paper provides Bayesian analysis methods by proposing a noninformative prior distribution to analyze survival data, using a two-parameter Rayleigh distribution based on record values. In addition, we provide a pivotal quantity and an algorithm based on the pivotal quantity to predict the behavior of future survival records. We show that the proposed method is superior to the frequentist counterpart in terms of the mean-squared error and bias through Monte carlo simulations. For illustrative purposes, survival data on lung cancer patients are analyzed, and it is proved that the proposed model can be a good alternative when prior information is not given.     

Numerical solution to stochastic partial differential equation with variable repair rate

ABSTRACT

In this paper, a numerical solution technique to stochastic partial differential equations in reliability engineering is presented. The method is based upon finite difference discretization of the governing equations for the Markovian reliability model. In realistic situations, the repair rates and failure rates of engineering system are variable. Such variable repair and failure rates are difficult to account in reliability modeling. The novelty in this work is to present a numerical method to easily take into consideration such variables and give an accurate prediction of reliability measures of engineering systems.     

Analysis of structural equation models with exact and stochastic constraints using a bayesian approach

In this paper, multisample analyses of exactand stochastic constraints with identified structural equation models are investigated using a Bayesian approach. Asymptotic properties of the estimates are developed and a multiplier method is employed to obtain the solution. A numerical example is also included as an illustration.     

Limited information bayesian analysis of a structural coefficient in a simultaneous equations system

An analytical expression is obtained for the marginal posterior density for a structural coefficient in a simultaneous equations system based on a limited information Bayesian analysis. A con- ditional posterior density is obtained given reduced form para- meters. This conditional posterior density is in univariate student t form. Numerical examples suggest that the conditional density hasa tighter distribution around the posterior mean than the unconditional density when the correlation between the endo- genous variables and the structural error term is high.     

On the information content of partial observations

The modified zero order approach to estimating coefficients in the face of missing observations treats them as parameters to be estimated simultaneously with the missing observations. The paper then investigates (in the context of Han's generalized regression model)(i) when parameter estimators don't vary between using the partial data points and using only the complete ones (the informationless result), and (ii) large sample properties of the modified zero order estimator. It's found the sequential cut property is crucial to the informationless result for coefficient estimators; consistency of the modified zero order estimator depends on the percentage of observations with missing elements for large sample sizes or the sequential cut property.     

Single-parameter inference based on partial prior information

Partial specification of a prior distribution can be appealing to an analyst, but there is no conventional way to update a partial prior. In this paper, we show how a framework for Bayesian updating with data can be based on the Dirichlet(a) process. Within this framework, partial information predictors generalize standard minimax predictors and have interesting multiple-point shrinkage properties. Approximations to partial-information estimators for squared error loss are defined straightforwardly, and an estimate of the mean shrinks the sample mean. The proposed updating of the partial prior is a consequence of four natural requirements when the Dirichlet parameter a is continuous. Namely, the updated partial posterior should be calculable from knowledge of only the data and partial prior, it should be faithful to the full posterior distribution, it should assign positive probability to every observed event {X,}, and it should not assign probability to unobserved events not included in the partial prior specification.     

Bayesian analysis of stochastic volatility models with flexible tails

An alternative distributional assumption is proposed for the stochastic volatility model. This results in extremely flexible tail behaviour of the sampling distribution for the observables, as well as in the availability of a simple Markov Chain Monte Carlo strategy for posterior analysis. By allowing the tail behaviour to be determined by a separate parameter, we reserve the parameters of the volatility process to dictate the degree of volatility clustering. Treatment of a mean function is formally integrated in the analysis.

Some empirical examples on both stock prices and exchange rates clearly indicate the presence of fat tails, in combination with high levels of volatility clustering. In addition, predictive distributions indicate a good fit with these typical financial data sets.     

Bayesian analysis of stochastic volatility models with flexible tails

An alternative distributional assumption is proposed for the stochastic volatility model. This results in extremely flexible tail behaviour of the sampling distribution for the observables, as well as in the availability of a simple Markov Chain Monte Carlo strategy for posterior analysis. By allowing the tail behaviour to be determined by a separate parameter, we reserve the parameters of the volatility process to dictate the degree of volatility clustering. Treatment of a mean function is formally integrated in the analysis.

Some empirical examples on both stock prices and exchange rates clearly indicate the presence of fat tails, in combination with high levels of volatility clustering. In addition, predictive distributions indicate a good fit with these typical financial data sets.     

Behaviour of the posterior error rate with partial prior information in auditing

Most of the Bayesian literature on statistical techniques in auditing has focused on assessing appropriate prior density using parameters such as interest, error rate and the mean of the error amount. Frequently, prior beliefs and mathematical tractable reasons are jointly used to assess prior distributions. As a robust Bayesian approach, we propose to replace the prior distribution with a set of prior distributions compatible with auditor's beliefs. We show how an auditor may draw the behaviour of the posterior error rate, using only partial prior information (quartiles of the prior distribution for the error rate O and, very often, the prior distribution is assumed to be unimodal). An example is pursued in depth.     

Empirical bayes tests in a positive exponential family with partial information on priors

We investigate an empirical Bayes testing problem in a positive exponential family having pdf f{x/θ)=c(θ)u(x) exp(?x/θ), x>0, θ>0. It is assumed that θ is in some known compact interval [C₁, C₂]. The value C₁ is used in the construction of the proposed empirical Bayes test δ^* _n. The asymptotic optimality and rate of convergence of its associated Bayes risk is studied. It is shown that under the assumption that θ is in [C₁, C₂] δ^* _n is asymptotically optimal at a rate of convergence of order O(n^?1/n n). Also, δ^* _n is robust in the sense that δ^* _n still possesses the asymptotic optimality even the assumption that "C₁≦θ≦C₂ may not hold.     

Bayesian analysis of multivariate stochastic volatility with skew return distribution

Multivariate stochastic volatility models with skew distributions are proposed. Exploiting Cholesky stochastic volatility modeling, univariate stochastic volatility processes with leverage effect and generalized hyperbolic skew t-distributions are embedded to multivariate analysis with time-varying correlations. Bayesian modeling allows this approach to provide parsimonious skew structure and to easily scale up for high-dimensional problem. Analyses of daily stock returns are illustrated. Empirical results show that the time-varying correlations and the sparse skew structure contribute to improved prediction performance and Value-at-Risk forecasts.     

Global sensitivity analysis of stochastic computer models with joint metamodels

The global sensitivity analysis method used to quantify the influence of uncertain input variables on the variability in numerical model responses has already been applied to deterministic computer codes; deterministic means here that the same set of input variables always gives the same output value. This paper proposes a global sensitivity analysis methodology for stochastic computer codes, for which the result of each code run is itself random. The framework of the joint modeling of the mean and dispersion of heteroscedastic data is used. To deal with the complexity of computer experiment outputs, nonparametric joint models are discussed and a new Gaussian process-based joint model is proposed. The relevance of these models is analyzed based upon two case studies. Results show that the joint modeling approach yields accurate sensitivity index estimators even when heteroscedasticity is strong.     

Fisher information and maximum-likelihood estimation of covariance parameters in Gaussian stochastic processes

The inverse of the Fisher information matrix is commonly used as an approximation for the covariance matrix of maximum-likelihood estimators. We show via three examples that for the covariance parameters of Gaussian stochastic processes under infill asymptotics, the covariance matrix of the limiting distribution of their maximum-likelihood estimators equals the limit of the inverse information matrix. This is either proven analytically or justified by simulation. Furthermore, the limiting behaviour of the trace of the inverse information matrix indicates equivalence or orthogonality of the underlying Gaussian measures. Even in the case of singularity, the estimator of the process variance is seen to be unbiased, and also its variability is approximated accurately from the information matrix.     

A note on estimating parameters of partial prior information

In an empirical Bayes model, examples for estimators of parameters of partial prior information are given. A typical application is the estimation of a probability ?(p≤a),p the “fraction defective”, which is used as prior information in Quality Control.     

Model comparison of linear and nonlinear Bayesian structural equation models with dichotomous data

In this article, dichotomous variables are used to compare between linear and nonlinear Bayesian structural equation models. Gibbs sampling method is applied for estimation and model comparison. Statistical inferences that involve estimation of parameters and their standard deviations and residuals analysis for testing the selected model are discussed. Hidden continuous normal distribution (censored normal distribution) is used to solve the problem of dichotomous variables. The proposed procedure is illustrated by a simulation data obtained from R program. Analyses are done by using R2WinBUGS package in R-program.