We present an up-to-date, comprehensive summary of the rates for all types of compact binary coalescence sources detectable by the initial and advanced versions of the ground-based gravitational-wave detectors LIGO and Virgo. Astrophysical estimates for compact-binary coalescence rates depend on a number of assumptions and unknown model parameters and are still uncertain. The most confident among these estimates are the rate predictions for coalescing binary neutron stars which are based on extrapolations from observed binary pulsars in our galaxy. These yield a likely coalescence rate of 100 Myr−1 per Milky Way Equivalent Galaxy (MWEG), although the rate could plausibly range from 1 Myr−1 MWEG−1 to 1000 Myr−1 MWEG−1 (Kalogera et al 2004 Astrophys. J. 601 L179; Kalogera et al 2004 Astrophys. J. 614 L137 (erratum)). We convert coalescence rates into detection rates based on data from the LIGO S5 and Virgo VSR2 science runs and projected sensitivities for our advanced detectors. Using the detector sensitivities derived from these data, we find a likely detection rate of 0.02 per year for Initial LIGO–Virgo interferometers, with a plausible range between 2 × 10−4 and 0.2 per year. The likely binary neutron–star detection rate for the Advanced LIGO–Virgo network increases to 40 events per year, with a range between 0.4 and 400 per year.

In functional linear regression, the slope ``parameter'' is a function. Therefore, in a nonparametric context, it is determined by an infinite number of unknowns. Its estimation involves solving an ill-posed problem and has points of contact with a range of methodologies, including statistical smoothing and deconvolution. The standard approach to estimating the slope function is based explicitly on functional principal components analysis and, consequently, on spectral decomposition in terms of eigenvalues and eigenfunctions. We discuss this approach in detail and show that in certain circumstances, optimal convergence rates are achieved by the PCA technique. An alternative approach based on quadratic regularisation is suggested and shown to have advantages from some points of view.Comment: Published at http://dx.doi.org/10.1214/009053606000000957 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

The use of principal component methods to analyze functional data is appropriate in a wide range of different settings. In studies of "functional data analysis," it has often been assumed that a sample of random functions is observed precisely, in the continuum and without noise. While this has been the traditional setting for functional data analysis, in the context of longitudinal data analysis a random function typically represents a patient, or subject, who is observed at only a small number of randomly distributed points, with nonnegligible measurement error. Nevertheless, essentially the same methods can be used in both these cases, as well as in the vast number of settings that lie between them. How is performance affected by the sampling plan? In this paper we answer that question. We show that if there is a sample of n functions, or subjects, then estimation of eigenvalues is a semiparametric problem, with root-n consistent estimators, even if only a few observations are made of each function, and if each observation is encumbered by noise. However, estimation of eigenfunctions becomes a nonparametric problem when observations are sparse. The optimal convergence rates in this case are those which pertain to more familiar function-estimation settings. We also describe the effects of sampling at regularly spaced points, as opposed to random points. In particular, it is shown that there are often advantages in sampling randomly. However, even in the case of noisy data there is a threshold sampling rate (depending on the number of functions treated) above which the rate of sampling (either randomly or regularly) has negligible impact on estimator performance, no matter whether eigenfunctions or eigenvectors are being estimated.

We suggest two nonparametric approaches, based on kernel methods and orthogonal series to estimating regression functions in the presence of instrumental variables. For the first time in this class of problems, we derive optimal convergence rates, and show that they are attained by particular estimators. In the presence of instrumental variables the relation that identifies the regression function also defines an ill-posed inverse problem, the "difficulty" of which depends on eigenvalues of a certain integral operator which is determined by the joint density of endogenous and instrumental variables. We de-lineate the role played by problem difficulty in determining both the optimal convergence rate and the appropriate choice of smoothing parameter.

There has been substantial recent work on methods for estimating the slope function in linear regression for functional data analysis. However, as in the case of more conventional finite-dimensional regression, much of the practical interest in the slope centers on its application for the purpose of prediction, rather than on its significance in its own right. We show that the problems of slope-function estimation, and of prediction from an estimator of the slope function, have very different characteristics. While the former is intrinsically nonparametric, the latter can be either nonparametric or semiparametric. In particular, the optimal mean-square convergence rate of predictors is $n^{-1}$, where $n$ denotes sample size, if the predictand is a sufficiently smooth function. In other cases, convergence occurs at a polynomial rate that is strictly slower than $n^{-1}$. At the boundary between these two regimes, the mean-square convergence rate is less than $n^{-1}$ by only a logarithmic factor. More generally, the rate of convergence of the predicted value of the mean response in the regression model, given a particular value of the explanatory variable, is determined by a subtle interaction among the smoothness of the predictand, of the slope function in the model, and of the autocovariance function for the distribution of explanatory variables.Comment: Published at http://dx.doi.org/10.1214/009053606000000830 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

ABSTRACT. Many practical problems, especially some connected with forecasting, require nonparametric estimation of conditional densities from mixed data. For example, given an explanatory data vector X for a prospective customer, with components that could include the customer's salary, occupation, age, sex, marital status and address, a company might wish to estimate the density of the expenditure, Y , that could be made by that person, basing the inference on observations of (X, Y ) for previous clients. Choosing appropriate smoothing parameters for this problem can be tricky, not least because plug-in rules take a particularly complex form in the case of mixed data. An obvious difficulty is that there exists no general formula for the optimal smoothing parameters. More insidiously, and more seriously, it can be difficult to determine which components of X are relevant to the problem of conditional inference. For example, if the jth component of X is independent of Y then that component is irrelevant to estimating the density of Y given X, and ideally should be dropped before conducting inference. In this paper we show that cross-validation overcomes these difficulties. It automatically determines which components are relevant and which are not, through assigning large smoothing parameters to the latter and consequently shrinking them towards the uniform distribution on the respective marginals. This effectively removes irrelevant components from contention, by suppressing their contribution to estimator variance; they already have very small bias, a consequence of their independence of Y . Crossvalidation also gives us important information about which components are relevant: the relevant components are precisely those which cross-validation has chosen to smooth in a traditional way, by assigning them smoothing parameters of conventional size. Indeed, cross-validation produces asymptotically optimal smoothing for relevant components, while eliminating irrelevant components by oversmoothing. In the problem of nonparametric estimation of a conditional density, cross-validation comes into its own as a method with no obvious peers.

Functional data analysis is intrinsically infinite dimensional; functional principal component analysis reduces dimension to a finite level, and points to the most significant components of the data. However, although this technique is often discussed, its properties are not as well understood as they might be. We show how the properties of functional principal component analysis can be elucidated through stochastic expansions and related results. Our approach quantifies the errors that arise through statistical approximation, in successive terms of orders "n"-super- - 1/2, "n"-super- - 1, "n"-super- - 3/2, …, where "n" denotes sample size. The expansions show how spacings among eigenvalues impact on statistical performance. The term of size "n"-super- - 1/2 illustrates first-order properties and leads directly to limit theory which describes the dominant effect of spacings. Thus, for example, spacings are seen to have an immediate, first-order effect on properties of eigenfunction estimators, but only a second-order effect on eigenvalue estimators. Our results can be used to explore properties of existing methods, and also to suggest new techniques. In particular, we suggest bootstrap methods for constructing simultaneous confidence regions for an infinite number of eigenvalues, and also for individual eigenvalues and eigenvectors. Copyright 2006 Royal Statistical Society.

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