On Necessary and Sufficient Conditions for Ordinary Least Squares Estimators to Be Best Linear Unbiased Estimators

Two often-quoted necessary and sufficient conditions for ordinary least squares estimators to be best linear unbiased estimators are described. Another necessary and sufficient condition is described, providing an additional tool for checking to see whether the covariance matrix of a given linear model is such that the ordinary least squares estimator is also the best linear unbiased estimator. The new condition is used to show that one of the two published conditions is only a sufficient condition.     

On a Relation between Principal Components and Regression Analysis

A regression approach to principal component analysis is presented in this note. We provide an alternative interpretation of principal components that illustrates the relation between the extra sum of squares in regression analysis and the eigenvalues associated with the principal components.     

Time series count data regression

The count data model studied in the paper extends the Poisson model by al-lowing for overdispersion and serial correlation. Alternative approaches to esti-mate nuisance parameters, required for the correction of the Poisson maximum likelihood covariance matrix estimator and for a quasi-likelihood estimator, are studied. The estimators are evaluated by finite sample Monte Carlo experi-mentation. It is found that the Poisson maximum likelihood estimator with corrected covariance matrix estimators provide reliable inferences for longer time series. Overdispersion test statistics are wellbehaved, while conventional portmanteau statistics for white noise have too large sizes. Two empirical illustrations are included.     

Assessing robustness of crossover designsto subjects dropping out

In some crossover experiments, particularly in medical applications, subjects may fail to complete their sequences of treatments for reasons unconnected with the treatments received. A method is described of assessing the robustness of a planned crossover design, with more than two periods, to subjects leaving the study prematurely. The method involves computing measures of efficiency for every possible design that can result, and is therefore very computationally intensive. Summaries of these measures are used to choose between competing designs. The computational problem is reduced to a manageable size by a software implementation of Polya theory. The method is applied to comparing designs for crossover studies involving four treatments and four periods. Designs are identified that are more robust to subjects dropping out in the final period than those currently favoured in medical and clinical trials.     

Flexible regression modeling

The author proposes a new method for flexible regression modeling of multi‐dimensional data, where the regression function is approximated by a linear combination of logistic basis functions. The method is adaptive, selecting simple or more complex models as appropriate. The number, location, and (to some extent) shape of the basis functions are automatically determined from the data. The method is also affine invariant, so accuracy of the fit is not affected by rotation or scaling of the covariates. Squared error and absolute error criteria are both available for estimation. The latter provides a robust estimator of the conditional median function. Computation is relatively fast, particularly for large data sets, so the method is well suited for data mining applications.     

Asymptotic Bias in Parameter Estimation of AR-Processes Using Recursive Least Squares with Exponential Forgetting

The recursive least squares technique is often extended with exponential forgetting as a tool for parameter estimation in time-varying systems. The distribution of the resulting parameter estimates is, however, unknown when the forgetting factor is less than one. In this paper an approximative expression for bias of the recursively obtained parameter estimates in a time-invariant AR( n_a ) process with arbitrary noise is given, showing that the bias is non-zero and giving bounds on the approximation errors. Simulations confirm the approximation expressions.     

The additive hazards model with high-dimensional regressors

This paper considers estimation and prediction in the Aalen additive hazards model in the case where the covariate vector is high-dimensional such as gene expression measurements. Some form of dimension reduction of the covariate space is needed to obtain useful statistical analyses. We study the partial least squares regression method. It turns out that it is naturally adapted to this setting via the so-called Krylov sequence. The resulting PLS estimator is shown to be consistent provided that the number of terms included is taken to be equal to the number of relevant components in the regression model. A standard PLS algorithm can also be constructed, but it turns out that the resulting predictor can only be related to the original covariates via time-dependent coefficients. The methods are applied to a breast cancer data set with gene expression recordings and to the well known primary biliary cirrhosis clinical data.     

Root n consistent and optimal density estimators for moving average processes

Abstract.  The marginal density of a first order moving average process can be written as a convolution of two innovation densities. Saavedra & Cao [Can. J. Statist. (2000), 28, 799] propose to estimate the marginal density by plugging in kernel density estimators for the innovation densities, based on estimated innovations. They obtain that for an appropriate choice of bandwidth the variance of their estimator decreases at the rate 1/ n . Their estimator can be interpreted as a specific U -statistic. We suggest a slightly simplified U -statistic as estimator of the marginal density, prove that it is asymptotically normal at the same rate, and describe the asymptotic variance explicitly. We show that the estimator is asymptotically efficient if no structural assumptions are made on the innovation density. For innovation densities known to have mean zero or to be symmetric, we describe improvements of our estimator which are again asymptotically efficient.     

Minimax estimation of a probability of success under LINEX loss

Minimax estimation of a binomial probability under LINEX loss function is considered. It is shown that no equalizer estimator is available in the statistical decision problem under consideration. It is pointed out that the problem can be solved by determining the Bayes estimator with respect to a least favorable distribution having finite support. In this situation, the optimal estimator and the least favorable distribution can be determined only by using numerical methods. Some properties of the minimax estimators and the corresponding least favorable prior distributions are provided depending on the parameters of the loss function. The properties presented are exploited in computing the minimax estimators and the least favorable distributions. The results obtained can be applied to determine minimax estimators of a cumulative distribution function and minimax estimators of a survival function.     

欧氏环中最大公因子与最小公倍子的统一求法

通过对欧氏环上矩阵的讨论 ,给出了欧氏环中两个元素的最大公因子与最小公倍子的统一求法 .该方法对整数环 Z和多项式环 F[x]中的问题的解决有指导意义