Robust estimation of multivariate regression model

This paper studies robust estimation of multivariate regression model using kernel weighted local linear regression. A robust estimation procedure is proposed for estimating the regression function and its partial derivatives. The proposed estimators are jointly asymptotically normal and attain nonparametric optimal convergence rate. One-step approximations to the robust estimators are introduced to reduce computational burden. The one-step local M-estimators are shown to achieve the same efficiency as the fully iterative local M-estimators as long as the initial estimators are good enough. The proposed estimators inherit the excellent edge-effect behavior of the local polynomial methods in the univariate case and at the same time overcome the disadvantages of the local least-squares based smoothers. Simulations are conducted to demonstrate the performance of the proposed estimators. Real data sets are analyzed to illustrate the practical utility of the proposed methodology. This work was supported by the National Natural Science Foundation of China (Grant No. 10471006).     

An Efficient Simulation Method for the Computation of a Class of Conditional Expectations

Suppose that the random vector X and the random variable Y are jointly continuous. Also suppose that an observation x of X can be easily simulated and that the probability density function of Y conditional on X = x is known. The paper presents an efficient simulation-based algorithm for estimating E{ g ( X , Y ) | h ( X , Y ) = r } where g and h are real-valued functions. This algorithm is applicable to time series problems in which X = ( X ₁, . . . , X _n−1) and Y = X_n where { x_t } is a discrete time stochastic process for which ( X₁ , . . . , X_n ) is a continuous random vector. A numerical example from time series analysis illustrates the algorithim, for prediction for an ARCH(1) process.     

Randomly weighted average with Dirichlet component proportions

This article introduces a new average of n independent continuous random variables X₁, …, X_n weighted by Dirichlet random components. A relation between the Cauchy–Stieltjes transforms of the distribution functions of this weighted average and X₁, …, X_n is established. Several examples illustrate usefulness and applicability of the result.     

A weighted spatial median for clustered data

A weighted spatial median is proposed for the multivariate one-sample location problem with clustered data. Its limiting distribution is derived under mild conditions (no moment assumptions) and it is shown to be multivariate normal. Asymptotic as well as finite sample efficiencies and breakdown properties are considered, and the theoretical results are supplied with illustrative examples. It turns out that there is a potential for meaningful gains in estimation efficiency: the weighted spatial median has superior efficiency to the unweighted spatial median particularly when the cluster sizes are widely disparate and in the presence of strong intracluster correlation. The unweighted spatial median for clustered data was considered earlier by Nevalainen et al. (Can J Statist, in press, 2007). The proposed weighted estimators provide companion estimates to the weighted affine invariant sign test proposed recently by Larocque et al. (Biometrika, in press, 2007). An affine equivariant weighted spatial median is discussed in parallel.     

REDUCED DISTANCE MEASURES AND TRANSFORMATIONS IN PROCESSING MULTIVARIATE OUTLIERS

The study of multivariate outliers raises many problems of definition, principle and manipulation. Well-authenticated tests of discordancy exist only for the multivariate normal distribution. Detection of outliers in non-normal distributions involves the adoption of appropriate criteria to represent 'extremeness' of observations in a sample; corresponding tests of discordancy usually require tedious, or even intractable, distributional and computational manipulations. A class of transformations of the data is considered with a view of transferring some of the familiar and desirable features of discordancy tests for normal samples to non-normal situations.     

A class of modified stein estimators with easily computable risk functions

Consider the problem of estimating the mean of a p (≥3)-variate multi-normal distribution with identity variance-covariance matrix and with unweighted sum of squared error loss. A class of minimax, noncomparable (i.e. no estimate in the class dominates any other estimate in the class) estimates is proposed; the class contains rules dominating the simple James-Stein estimates. The estimates are essentially smoothed versions of the scaled, truncated James-Stein estimates studied by Efron and Morris. Explicit and analytically tractable expressions for their risks are obtained and are used to give guidelines for selecting estimates within the class.     

A comparison of univariate and multivariate multilevel models for repeated measures of use of antenatal care in Uttar Pradesh

Summary.  We compare two different multilevel modelling approaches to the analysis of repeated measures data to assess the effect of mother level characteristics on women's use of prenatal care services in Uttar Pradesh, India. We apply univariate multilevel models to our data and find that the model assumptions are severely violated and the parameter estimates are not stable, particularly for the mother level random effect. To overcome this we apply a multivariate multilevel model. The correlation structure shows that, once the decision has been made regarding use of antenatal care by the mother for her first observed birth in the data, she does not tend to change this decision for higher order births.     

Simultaneous modelling of multiple traffic safety performance indicators by using a multivariate generalized linear mixed model

Summary.  Traffic safety in the UK is one of the increasing number of areas where central government sets targets based on 'outcome-focused' performance indicators (PIs). Judgments about such PIs are often based solely on rankings of raw indicators and simple league tables dominate centrally published analyses. There is a considerable statistical literature examining health and education issues which has tended to use the generalized linear mixed model (GLMM) to address variability in the data when drawing inferences about relative performance from headline PIs. This methodology could obviously be applied in contexts such as traffic safety. However, when such models are applied to the fairly crude data sets that are currently available, the interval estimates generated, e.g. in respect of rankings, are often too broad to allow much real differentiation between the traffic safety performance of the units that are being considered. Such results sit uncomfortably with the ethos of 'performance management' and raise the question of whether the inference from such data sets about relative performance can be improved in some way. Motivated by consideration of a set of nine road safety performance indicators measured on English local authorities in the year 2000, the paper considers methods to strengthen the weak inference that is obtained from GLMMs of individual indicators by simultaneous, multivariate modelling of a range of related indicators. The correlation structure between indicators is used to reduce the uncertainty that is associated with rankings of any one of the individual indicators. The results demonstrate that credible intervals can be substantially narrowed by the use of the multivariate GLMM approach and that multivariate modelling of multiple PIs may therefore have considerable potential for introducing more robust and realistic assessments of differential performance in some contexts.     

An analysis of models for the dilution and adulteration of fruit juice

We present models based on the multivariate normal distribution to represent the process of dilution and adulteration of citrius juice. The models specify a common dilution parameter for those components of the juice which are affected by dilution but not adulteration.Statistical testing of the hypothesis of no dilution or adulteration presents theoretical difficulties. These difficulties are resolved by model comparisons based on averaged, rather than maximized, likelihoods.     

On the Moments of Multivariate Discrete Distributions Using Finite Difference Operators

This article discusses a general approach to finding the moments of two classes of multivariate discrete distributions, which include those widely used in applied and theoretical statistics. The two classes of multivariate discrete distributions are the multivariate generalized power series distributions (GPSD) and the unified multivariate hypergeometric (UMH) Distributions. The results of Link (1981) follow as special cases.