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To study the roles of translational accuracy, translational efficiency, and the Hill-Robertson effect in codon usage bias, we studied the intragenic spatial distribution of synonymous codon usage bias in four prokaryotic (Escherichia coli, Bacillus subtilis, Sulfolobus tohodaii, and Thermotoga maritima) and two eukaryotic (Saccharomyces cerevisiae and Drosophila melanogaster) genomes. We generated supersequences at each codon position across genes in a genome and computed the overall bias at each codon position. By quantitatively evaluating the trend of spatial patterns using isotonic regression, we show that in yeast and prokaryotic genomes, codon usage bias increases along translational direction, which is consistent with purifying selection against nonsense errors. Fruit fly genes show a nearly symmetric M-shaped spatial pattern of codon usage bias, with less bias in the middle and both ends. The low codon usage bias in the middle region is best explained by interference (the Hill-Robertson effect) between selections at different codon positions. In both yeast and fruit fly, spatial patterns of codon usage bias are characteristically different from patterns of GC-content variations. Effect of expression level on the strength of codon usage bias is more conspicuous than its effect on the shape of the spatial distribution. [ABSTRACT FROM AUTHOR]

Estimation of an unstructured covariance matrix is difficult because of its positive-definiteness constraint. This obstacle is removed by regressing each variable on its predecessors, so that estimation of a covariance matrix is shown to be equivalent to that of estimating a sequence of varying-coefficient and varying-order regression models. Our framework is similar to the use of increasing-order autoregressive models in approximating the covariance matrix or the spectrum of a stationary time series. As an illustration, we adopt Fan & Zhang's (2000) two-step estimation of functional linear models and propose nonparametric estimators of covariance matrices which are guaranteed to be positive definite. For parsimony a suitable order for the sequence of (auto)regression models is found using penalised likelihood criteria like AIC and BIC. Some asymptotic results for the local polynomial estimators of components of a covariance matrix are established. Two longitudinal datasets are analysed to illustrate the methodology. A simulation study reveals the advantage of the nonparametric covariance estimator over the sample covariance matrix for large covariance matrices. [ABSTRACT FROM AUTHOR]