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Bacterial communities provide important services. They break down pollutants, municipal waste and ingested food, and they are the primary means by which organic matter is recycled to plants and other autotrophs. However, the processes that determine the rate at which these services are supplied are only starting to be identified. Biodiversity influences the way in which ecosystems function, but the form of the relationship between bacterial biodiversity and functioning remains poorly understood. Here we describe a manipulative experiment that measured how biodiversity affects the functioning of communities containing up to 72 bacterial species constructed from a collection of naturally occurring culturable bacteria. The experimental design allowed us to manipulate large numbers of bacterial species selected at random from those that were culturable. We demonstrate that there is a decelerating relationship between community respiration and increasing bacterial diversity. We also show that both synergistic interactions among bacterial species and the composition of the bacterial community are important in determining the level of ecosystem functioning.

Summary A technique for using kernel density estimates to investigate the number of modes in a population is described and discussed. The amount of smoothing is chosen automatically in a natural way.

An empirical Bayes approach to the estimation of possibly sparse sequences observed in Gaussian white noise is set out and investigated. The prior considered is a mixture of an atom of probability at zero and a heavy-tailed density γ, with the mixing weight chosen by marginal maximum likelihood, in the hope of adapting between sparse and dense sequences. If estimation is then carried out using the posterior median, this is a random thresholding procedure. Other thresholding rules employing the same threshold can also be used. Probability bounds on the threshold chosen by the marginal maximum likelihood approach lead to overall risk bounds over classes of signal sequences of length n, allowing for sparsity of various kinds and degrees. The signal classes considered are "nearly black" sequences where only a proportion η is allowed to be nonzero, and sequences with normalized ℓp norm bounded by η, for η > 0 and 0 < p ≤ 2. Estimation error is measured by mean qth power loss, for 0 < q ≤ 2. For all the classes considered, and for all q in (0, 2], the method achieves the optimal estimation rate as n → ∞ and η → 0 at various rates, and in this sense adapts automatically to the sparseness or otherwise of the underlying signal. In addition the risk is uniformly bounded over all signals. If the posterior mean is used as the estimator, the results still hold for q > 1. Simulations show excellent performance. For appropriately chosen functions γ, the method is computationally tractable and software is available. The extension to a modified thresholding method relevant to the estimation of very sparse sequences is also considered.

Wavelets are of wide potential use in statistical contexts. The basics of the discrete wavelet transform are reviewed using a filter notation that is useful subsequently in the paper. A 'stationary wavelet transform', where the coefficient sequences are not decimated at each stage, is described. Two different approaches to the construction of an inverse of the stationary wavelet transfonn are set out. The application of the stationary wavelet transform as an exploratory statistical method is discussed, together with its potential use in nonparametric regression. A method of local spectral density estimation is developed. This involves extensions to the wavelet context of standard time series ideas such as the periodogram and spectrum. The technique is illustrated by its application to data sets from astronomy and veterinary anatomy.

Wavelet threshold estimators for data with stationary correlated noise are constructed by applying a level-dependent soft threshold to the coef®cients in the wavelet transform. A variety of threshold choices is proposed, including one based on an unbiased estimate of mean-squared error. The practical performance of the method is demonstrated on examples, including data from a neurophysiological context. The theoretical properties of the estimators are investigated by comparing them with an ideal but unattainable`benchmark', that can be considered in the wavelet context as the risk obtained by ideal spatial adaptivity, and more generally is obtained by the use of an`oracle' that provides information that is not actually available in the data. It is shown that the level-dependent threshold estimator performs well relative to the bench-mark risk, and that its minimax behaviour cannot be improved on in order of magnitude by any other estimator. The wavelet domain structure of both short-and long-range dependent noise is considered, and in both cases it is shown that the estimators have near optimal behaviour simultaneously in a wide range of function classes, adapting automatically to the regularity properties of the underlying model. The proofs of the main results are obtained by considering a more general multivariate normal decision theoretic problem.

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