ASSESSING THE RELATIONSHIP BETWEEN SYMPTOMS OF ALLERGIC RHINOCONJUNCTIVITIS AND POLLEN COUNTS

The relationship between daily pollen counts during the peak pollen season and hay fever symptoms in known sufferers of pollen allergy was investigated in a Sydney hospital‐based study. This paper develops statistical models for both the short term (day to day) associations and the longer term relationships between these time series. Possible effects of asthma status are investigated. The analyses illustrate how different relationships between time series may be explored in a simple way by working on different time scales with suitably transformed variables.     

Maximum Likelihood Estimation in Gaussian Chain Graph Models under the Alternative Markov Property

Abstract.  The Andersson–Madigan–Perlman (AMP) Markov property is a recently proposed alternative Markov property (AMP) for chain graphs. In the case of continuous variables with a joint multivariate Gaussian distribution, it is the AMP rather than the earlier introduced Lauritzen–Wermuth–Frydenberg Markov property that is coherent with data-generation by natural block-recursive regressions. In this paper, we show that maximum likelihood estimates in Gaussian AMP chain graph models can be obtained by combining generalized least squares and iterative proportional fitting to an iterative algorithm. In an appendix, we give useful convergence results for iterative partial maximization algorithms that apply in particular to the described algorithm.     

A BAYESIAN INTERPRETATION OF MULTIPLE POINT ESTIMATES

Consider a large number of econometric investigations using different estimation techniques and/or different subsets of all available data to estimate a fixed set of parameters. The resulting empirical distribution of point estimates can be shown - under suitable conditions - to coincide with a Bayesian posterior measure on the parameter space induced by a minimum information procedure. This Bayesian interpretation makes it easier to combine the results of various empirical exercises for statistical decision making. The collection of estimators may be generated by one investigator to ensure the satisfaction of our conditions, or they may be collected from published works, where behavioral assumptions need to be made regarding the dependence structure of econometric studies.     

DENSITY ESTIMATION FOR CLUSTERED DATA

The commonly used survey technique of clustering introduces dependence into sample data. Such data is frequently used in economic analysis, though the dependence induced by the sample structure of the data is often ignored. In this paper, the effect of clustering on the non-parametric, kernel estimate of the density, f(x), is examined. The window width commonly used for density estimation for the case of i.i.d. data is shown to no longer be optimal. A new optimal bandwidth using a higher-order kernel is proposed and is shown to give a smaller integrated mean squared error than two window widths which are widely used for the case of i.i.d. data. Several illustrations from simulation are provided.     

Smoothing for discrete-valued time series

We deal with smoothed estimators for conditional probability functions of discrete-valued time series { Y_t } under two different settings. When the conditional distribution of Y_t given its lagged values falls in a parametric family and depends on exogenous random variables, a smoothed maximum (partial) likelihood estimator for the unknown parameter is proposed. While there is no prior information on the distribution, various nonparametric estimation methods have been compared and the adjusted Nadaraya–Watson estimator stands out as it shares the advantages of both Nadaraya–Watson and local linear regression estimators. The asymptotic normality of the estimators proposed has been established in the manner of sparse asymptotics, which shows that the smoothed methods proposed outperform their conventional, unsmoothed, parametric counterparts under very mild conditions. Simulation results lend further support to this assertion. Finally, the new method is illustrated via a real data set concerning the relationship between the number of daily hospital admissions and the levels of pollutants in Hong Kong in 1994–1995. An ad hoc model selection procedure based on a local Akaike information criterion is proposed to select the significant pollutant indices.     

Improving coverage accuracy of nonparametric prediction intervals

Methods are suggested for improving the coverage accuracy of intervals for predicting future values of a random variable drawn from a sampled distribution. It is shown that properties of solutions to such problems may be quite unexpected. For example, the bootstrap and the jackknife perform very poorly when used to calibrate coverage, although the jackknife estimator of the true coverage is virtually unbiased. A version of the smoothed bootstrap can be employed for successful calibration, however. Interpolation among adjacent order statistics can also be an effective way of calibrating, although even there the results are unexpected. In particular, whereas the coverage error can be reduced from O ( n ^-1) to orders O ( n ^-2) and O ( n ^-3) (where n denotes the sample size) by interpolating among two and three order statistics respectively, the next two orders of reduction require interpolation among five and eight order statistics respectively.     

Error rates in classification consisting of discrete and continuous variables in the presence of covariates

We consider classifying an object based on mixed continuous and discrete variables between two populations. Mixed discrete and continuous covariates with identical means in both populations are amongst the variables. Under the location model with homogeneous location specific conditional dispersion matrices for both populations, the Bayes rule is given. Classification is implemented by a plug-in version of the Bayes rule with full covariate adjustment. An asymptotic expansion of the overall expected error of the procedure is derived. Our findings generalize several classical results.     

Estimation of relaxation time distributions in magnetic resonance imaging

Magnetic resonance imaging techniques can be used to measure some biophysical properties of tissue. In this context, the T₂ relaxation time is an important parameter for soft‐tissue contrast. The authors develop a new technique to estimate the integral of the distribution of T₂ relaxation time without imposing any constraint other than the monotonicity of the underlying cumulative relaxation time distribution. They explore the properties of the estimation and its applications for the analysis of breast tissue data. As they show, an extension of linear discriminant analysis is found to distinguish well between two classes of breast tissue.     

Nonparametric empirical Bayes for the Dirichlet process mixture model

The Dirichlet process prior allows flexible nonparametric mixture modeling. The number of mixture components is not specified in advance and can grow as new data arrive. However, analyses based on the Dirichlet process prior are sensitive to the choice of the parameters, including an infinite-dimensional distributional parameter G ₀. Most previous applications have either fixed G ₀ as a member of a parametric family or treated G ₀ in a Bayesian fashion, using parametric prior specifications. In contrast, we have developed an adaptive nonparametric method for constructing smooth estimates of G ₀. We combine this method with a technique for estimating α, the other Dirichlet process parameter, that is inspired by an existing characterization of its maximum-likelihood estimator. Together, these estimation procedures yield a flexible empirical Bayes treatment of Dirichlet process mixtures. Such a treatment is useful in situations where smooth point estimates of G ₀ are of intrinsic interest, or where the structure of G ₀ cannot be conveniently modeled with the usual parametric prior families. Analysis of simulated and real-world datasets illustrates the robustness of this approach.     

Adaptive weights smoothing with applications to image restoration

We propose a new method of nonparametric estimation which is based on locally constant smoothing with an adaptive choice of weights for every pair of data points. Some theoretical properties of the procedure are investigated. Then we demonstrate the performance of the method on some simulated univariate and bivariate examples and compare it with other nonparametric methods. Finally we discuss applications of this procedure to magnetic resonance and satellite imaging.