Posterior analysis of restricted seemingly unrelated regression equation models: a recursive analytical approach

The scope of exact analytical results in Bayesian econometrics is known to be quite limited. It is, however, shown here to be broader than the simple natural-conjugare framework. Restricting the coefficients of a SURE model in a recursive linear way can not be accommodated in a natural-conjugate analysis,but still allows for analytical ingerence, exploiting the recursive characteristics over equations. These finding are used to obtain analytical posterior results in a two-equation model for money and interest rate in the UK. Subsequent research shows that such methods can substantially increase both reliability and efficiency in the analysis of more complicated models than the ine under scrutiny here.     

Estimation in regression models with stationary,dependent errors

Often the unknown covariance structure of a stationary, dependent, Gaussian error sequence can be simply parametrised. The error sequence can either be directly observed or observed only through a random sequence containing a deterministic regression model. The method of scoring is used here, in conjunction with recursive estimation techniques, to effect the maximum likelihood estimation of the covariance parameters. Sequences of recursive residuals, useful in model diagnostics and data analysis, are obtained in the estimation procedure.     

Some Analytical and Numerical Bounds on the Renewal Function

Renewal-type equations are frequently encountered in the study of reliability, warranty analysis, replacement and maintenance policies, and inventory control. Renewal equations usually do not have analytical solutions, and hence, bounds or approximations are very useful. In this article, analytical bounds are studied based on a simple iterative procedure which provides some analytical results and nice convergence properties when the number of iteration increases. Bounds and approximations are also investigated for a recursive algorithm for numerical computation. In addition, some interesting monotonicity properties are introduced and discussed. The approximation error, which is important for determining the stopping rule of the iterative procedure and the numerical algorithm, is also studied.     

On the Estimation of the Density of a Directional Data Stream

Many directional data such as wind directions can be collected extremely easily so that experiments typically yield a huge number of data points that are sequentially collected. To deal with such big data, the traditional nonparametric techniques rapidly require a lot of time to be computed and therefore become useless in practice if real time or online forecasts are expected. In this paper, we propose a recursive kernel density estimator for directional data which (i) can be updated extremely easily when a new set of observations is available and (ii) keeps asymptotically the nice features of the traditional kernel density estimator. Our methodology is based on Robbins–Monro stochastic approximations ideas. We show that our estimator outperforms the traditional techniques in terms of computational time while being extremely competitive in terms of efficiency with respect to its competitors in the sequential context considered here. We obtain expressions for its asymptotic bias and variance together with an almost sure convergence rate and an asymptotic normality result. Our technique is illustrated on a wind dataset collected in Spain. A Monte‐Carlo study confirms the nice properties of our recursive estimator with respect to its non‐recursive counterpart.     

Correlation analysis-based error compensation recursive least-square identification method for the Hammerstein model

In this paper, the correlation analysis based error compensation recursive least-square (RLS) identification method is proposed for the Hammerstein model. Firstly, the covariance matrix between input and output data points of the Hammerstein model is derived by using separable signal to realize that the unmeasurable internal variable is replaced by the covariance matrix of input. Thus, the correlation analysis method can be accordingly used to estimate parameters of the linear part, which results in the identification problem of the nonlinear part separated from the linear part. In addition, a correction term is added to least-square estimation to compensate error caused by output noise, further the error compensation-based RLS method is obtained for the observed data from the Hammerstein model. As a result, the least-square identification method, which generates error in the presence of noise distribution, can be compensated. Finally, simulation experiments are conducted to illustrate the performance of the proposed identification method.     

Choosing a Model Selection Strategy

An important problem in statistical practice is the selection of a suitable statistical model. Several model selection strategies are available in the literature, having different asymptotic and small sample properties, depending on the characteristics of the data generating mechanism. These characteristics are difficult to check in practice and there is a need for a data-driven adaptive procedure to identify an appropriate model selection strategy for the data at hand. We call such an identification a model metaselection, and we base it on the analysis of recursive prediction residuals obtained from each strategy with increasing sample sizes. Graphical tools are proposed in order to study these recursive residuals. Their use is illustrated on real and simulated data sets. When necessary, an automatic metaselection can be performed by simply accumulating predictive losses. Asymptotic and small sample results are presented.     

Smoothed estimates for nonlinear time series models with irregular data

Time series data observed at unequal time intervals (irregular data) occur quite often and this usually poses problems in its analysis. A recursive form of the exponentially smoothed estimated is here proposed for a nonlinear model with irregularly observed data and its asymptotic properties are discussed An alternative smoother to that of Wright (1985) is also derived. Numerical comparison is made between the resulting estimates and other smoothed estimates.     

基于B样条曲线的异常数据剔除与修复

文章针对大量复杂的靶场观测数据,通过构造初始拟合数据,利用B样条曲线的方法构造递推模型,使用基于样条平滑方法估计的判断门限对双向检验的结果数据是否异常进行判定,并且对满足修复条件的数据进行拟合修复,当双向检验的结果不同时,通过构造内推模型来进一步检验。实例分析表明:文章提出的方法相对其他方法能更有效地剔除异常数据,通过数据分段处理能更有效地检验那些可能产生阶段性跳跃的数据,使得模型具有更好的稳定性、更广的适用性和更高的异常数据剔除率。     

Subgroup identification by recursive segmentation

A new modeling approach called ‘recursive segmentation’ is proposed to support the supervised exploration and identification of subgroups or clusters. It is based on the frameworks of recursive partitioning and the Patient Rule Induction Method (PRIM). Through combining these methods, recursive segmentation aims to exploit their respective strengths while reducing their weaknesses. Consequently, recursive segmentation can be applied in a very general way, that is in any (multivariate) regression, classification or survival (time-to-event) problem, using conditional inference, evolutionary learning or the CART algorithm, with predictor variables of any scale and with missing values. Furthermore, results of a synthetic example and a benchmark application study that comprises 26 data sets suggest that recursive segmentation achieves a competitive prediction accuracy and provides more accurate definitions of subgroups by models of less complexity as compared to recursive partitioning and PRIM. An application to the German Breast Cancer Study Group data demonstrates the improved interpretability and reliability of results produced by the new approach. The method is made publicly available through the R-package rseg (http://rseg.r-forge.r-project.org/).     

Variational Bayes for estimating the parameters of a hidden Potts model

Hidden Markov random field models provide an appealing representation of images and other spatial problems. The drawback is that inference is not straightforward for these models as the normalisation constant for the likelihood is generally intractable except for very small observation sets. Variational methods are an emerging tool for Bayesian inference and they have already been successfully applied in other contexts. Focusing on the particular case of a hidden Potts model with Gaussian noise, we show how variational Bayesian methods can be applied to hidden Markov random field inference. To tackle the obstacle of the intractable normalising constant for the likelihood, we explore alternative estimation approaches for incorporation into the variational Bayes algorithm. We consider a pseudo-likelihood approach as well as the more recent reduced dependence approximation of the normalisation constant. To illustrate the effectiveness of these approaches we present empirical results from the analysis of simulated datasets. We also analyse a real dataset and compare results with those of previous analyses as well as those obtained from the recently developed auxiliary variable MCMC method and the recursive MCMC method. Our results show that the variational Bayesian analyses can be carried out much faster than the MCMC analyses and produce good estimates of model parameters. We also found that the reduced dependence approximation of the normalisation constant outperformed the pseudo-likelihood approximation in our analysis of real and synthetic datasets.     

Reml estimation for repeated measures analysis

Models for repeated measures or growth curves consist of a mean response plus error and the errors are usually correlated. Both maximum likelihood and residual maximum likelihood (REML) estimators of a regression model with dependent errors are derived for cases in which the variance matrix of the error model admits a convenient Cholesky factorisation. This factorisation may be linked to methods for producing recursive estimates of the regression parameters and recursive residuals to provide a convenient computational method. The method is used to develop a general approach to repeated measures analysis.     

Testing for random individual effects using recursive residuals

This paper derives a simple ANOVA-F-statistic which tests for random individual effects in a one-way error component model, using recursive residuals. Power comparisons are performed for this F-test when it is computed using true disturbances and recursive residuals from a panel data regression. Under the null, both statistics have an exact F distribution. The standardized version of the Breusch and Pagan (1980) Lagrange Multiplier test (SLM) as well as a fixed effects F-statistic (FE) recommended by Moulton and Randolph (1989), are also included in this comparison. The exact power function can be computed in all cases using Imhof's (1961) procedure. Our results suggest that the F-test based on recursive residuals is inferior to the popular SLM and FE tests based on computational simplicity, power comparisons and its sensitivity to the K observations starting the recursion.     

Automatic breakpoint detection for retrospective cumulative sum charts

Cumulative sum (cusum) methods can be used for monitoring processes and for retrospective (historical) data analysis. Most software only provides the former. The comment by Williamson that retrospective cusum analysis is a neglected area is still true. Though not in vogue, retrospective cusum analysis is useful for investigations such as benchmarking of processes, identifying causes of process decay, selecting reference data sets for typicality studies, and reporting of historical data. Even those texts which cover retrospective analyses, usually ignore the question of identifying multiple points of change (breakpoints), and present essentially manual methods for assessing single breakpoints. Most users of statistical methods want software solutions that are easy to use and require little user intervention or interpretation. Direct implementation of manual method does not give a user robust solution. Problems are illustrated. Attempts to use monitoring CuSums in retrospective analysis can also lead to errors. A practical recursive method is presented for breakpoint identification and significance assessment, which can be automated. Copyright © 2002 John Wiley & Sons, Ltd.     

Comment

The article by Conway and Reinganum demonstrates that cross-validation is a necessary part of any goodness-of-fit evaluation of multifactor asset-pricing models. The use of this procedure guards against overfitting, which is a common occurrence in large samples. It is illustrated here that if cross-validation cannot be used, two alternative goodness-of-fit measures that include a penalty based on parameters fitted can be used to assist in model selection. The two alternative measures include a penalty function based on the number of parameters fitted. The values of the two alternative measures for the Conway and Reinganum data are presented here for comparison purposes. The more conservative of the two measures obtains results comparable to the cross-validation procedure.     

Bias Reduction through First-order Mean Correction, Bootstrapping and Recursive Mean Adjustment

Standard methods of estimation for autoregressive models are known to be biased in finite samples, which has implications for estimation, hypothesis testing, confidence interval construction and forecasting. Three methods of bias reduction are considered here: first-order bias correction, FOBC, where the total bias is approximated by the O(T^-1) bias; bootstrapping; and recursive mean adjustment, RMA. In addition, we show how first-order bias correction is related to linear bias correction. The practically important case where the AR model includes an unknown linear trend is considered in detail. The fidelity of nominal to actual coverage of confidence intervals is also assessed. A simulation study covers the AR(1) model and a number of extensions based on the empirical AR(p) models fitted by Nelson & Plosser (1982). Overall, which method dominates depends on the criterion adopted: bootstrapping tends to be the best at reducing bias, recursive mean adjustment is best at reducing mean squared error, whilst FOBC does particularly well in maintaining the fidelity of confidence intervals.     

NEAREST NEIGHBOUR ANALYSIS OF UNEQUALLY REPLICATED TRIALS

There has been much work on the use of neighbouring plots to control environmental variation in the analysis of agricultural field experiments. In particular, the Residual Maximum Likelihood Neighbour (REMLN) analysis of Gleeson&Cullis (1987) appears very promising. The application of the REMLN analysis to an unequally replicated field trial augmented with an additional variety planted every six plots in a grid system is here compared with a covariance (COV) analysis using the neighbouring grid or check plot values as the covariate. The results indicate that the REMLN analysis gives more accurate estimates of treatment contrasts than the COV analyses, but that the estimate of treatment means can be biased. The bias depends on the mean of the check plot. This bias can be removed by adjusting the estimates of the treatment means such that their average equals the average of the raw means rather than that of the raw data.     

Uncorrelated residuals and an exact test for two variance components in experimental design

The error contrasts from an experimental design can be constructed from uncorrelated residuals normally associated with the linear model. In this paper uncorrelated residuals are defined for the linear model that has a design matrix which is less than full rank, typical of many experimental design representations. It transpires in this setting, that for certain choices of uncorrelated residuals, corresponding to recursive type residuals, there is a natural partition of information when two variance components are known to be present. Under an assumtion of normality of errors this leads to construction of appropriate F-tests for testing heteroscedasticity. The test, which can be optimal, is applied to two well known data sets to illustrate its usefullness.     

Analytical closed-form solution for binary logit regression by categorical predictors

In contrast to the common belief that the logit model has no analytical presentation, it is possible to find such a solution in the case of categorical predictors. This paper shows that a binary logistic regression by categorical explanatory variables can be constructed in a closed-form solution. No special software and no iterative procedures of nonlinear estimation are needed to obtain a model with all its parameters and characteristics, including coefficients of regression, their standard errors and t-statistics, as well as the residual and null deviances. The derivation is performed for logistic models with one binary or categorical predictor, and several binary or categorical predictors. The analytical formulae can be used for arithmetical calculation of all the parameters of the logit regression. The explicit expressions for the characteristics of logit regression are convenient for the analysis and interpretation of the results of logistic modeling.     

STABILITY AND PERFORMANCE OF STACK ALGORITHMS FOR RANDOM ACCESS COMMUNICATION MODELED AS A TREE STRUCTURED QBD MARKOV CHAIN

In this paper, we introduce an analytical model to study the stability and the main performance measures of a binary stack algorithm for random multiple access communication. The input traffic is a discrete time Batch Markovian Arrival Process (D-BMAP). The analytical model is nearly exact (one minor approximation is required) and the analysis is based on recent results obtained from tree structured Quasi-Birth-Death (QBD) Markov chains. Apart from studying the stability of the protocol, we are also able to calculate the mean delay and other important performance measures. The method deployed in this paper can also be extended to evaluate other medium access control (MAC) protocols with an underlying stack structure.     

MINIMUM QUADRATIC DISTANCE ESTIMATION FOR A PARAMETRIC FAMILY OF DISCRETE DISTRIBUTIONS DEFINED RECURSIVELY

The distribution function of a random sum can easily be computed iteratively when the distribution of the number of independent identically distributed elements in the sum is itself defined recursively. Classical estimation procedures for such recursive parametric families often require specific distributional assumptions (e.g. Poisson, Negative Binomial). The minimum distance estimator proposed here is an estimator within a larger parametric family. The estimator is consistent, efficient when the parametric family is truncated, and can be made either robust or asymptotically efficient when the parametric family has infinite range. Its asymptotic distribution is derived. A brief illustration with Automobile Insurance data is included.