It is shown that the covariance operator of a locally stationary process has approximate eigenvectors that are local cosine functions. We model locally stationary processes with pseudo-differential operators that are time-varying convolutions. An adaptive covariance estimation is calculated by searching first for a "best" local cosine basis which approximates the covariance by a band or a diagonal matrix. The estimation is obtained from regularized versions of the diagonal coefficients in the best basis.

We propose a sieve bootstrap procedure for time series with a deterministic trend. The sieve for constructing the bootstrap is based on nonparametric trend estimation and autoregressive approximation for some noise process. The bootstrap scheme itself does i.i.d. resampling of estimated innovations from fitted autoregressive models.

We show the validity and indicate second-order correctness of such sieve bootstrap approximations for the limiting distribution of nonparametric linear smoothers. The resampling can then be used to construct nonparametric confidence intervals for the underlying trend. In particular, we show asymptotic validity for constructing confidence bands which are simultaneously within a neighborhood of size in the order of the smoothing bandwidth.

Our resampling procedure yields satisfactory results in a simulation study for finite sample sizes. We also apply it to the longest series of total ozone measurements from Arosa (Switzerland) and find a significant decreasing trend.

This paper investigates the maximum likelihood estimator (MLE) for unstable autoregressive moving-average (ARMA) time series with the noise sequence satisfying a general autoregressive heteroscedastic (GARCH) process. Under some mild conditions, it is shown that the MLE satisfying the likelihood equation exists and is consistent. The limiting distribution of the MLE is derived in a unified manner for all types of characteristic roots on or outside the unit circle and is expressed as a functional of stochastic integrals in terms of Brownian motions. For various types of unit roots, the limiting distribution of the MLE does not depend on the parameters in the moving-average component and hence, when the GARCH innovations reduce to usual white noises with a constant conditional variance, they are the same as those for the least squares estimators (LSE) for unstable autoregressive models given by Chan and Wei (1988). In the presence of the GARCH innovations, the limiting distribution will involve a sequence of independent bivariate Brownian motions with correlated components. These results are different from those already known in the literature and, in this case, the MLE of unit roots will be much more efficient than the ordinary least squares estimation.

This paper provides a data analysis and some methodological advances which contribute to an ongoing scientific debate about the patterns of aging. One of the problems we address is how to estimate a hazard function when only aggregated information on the lifetimes in the form of a lifetable is available. This problem affects the the estimation of oldest-old mortality which in turn plays an important role in the quantification of biological lifespan and longevity. We illustrate these issues with an analysis of mortality data obtained from cohorts of nematodes. The methods involve data transformation with the aim of bias reduction when estimating the hazard function. We provide rigorous asymptotic results for the smoothing of lifetables and show that the transformation approach is supported by both asymptotic and simulation results. We also demonstrate how the information contained in many samples of lifetables, as typically obtained in aging experiments, can be summarized in a two-dimensional hazard surface.

The Kaplan-Meier estimator of a survival function is well known to be asymptotically efficient when cause of failure is always observed. It has been an open problem, however, to find an efficient estimator when failure indicators are missing at random. Lo showed that nonparametric maximum likelihood estimators are inconsistent, and this has led to several proposals of ad hoc estimators, none of which are efficient. We now introduce a sieved nonparametric maximum likelihood estimator, and show that it is efficient. Our approach is related to the estimation of a bivariate survival function from bivariate right-censored data.

The frailty model is a generalization of Cox's proportional hazard model, where a shared unobserved quantity in the intensity induces a positive correlation among the survival times. Murphy showed consistency and asymptotic normality of the nonparametric maximum likelihood estimator (NPMLE) for the shared gamma-frailty model without covariates. In this paper we extend this result to the correlated gamma-frailty model, and we allow for covariates. We discuss the definition of the nonparametric likelihood function in terms of a classical proof of consistency for the maximum likelihood estimator, which goes back to Wald. Our proof of the consistency for the NPMLE is essentially the same as the classical proof for the maximum likelihood estimator. A new central limit theorem for processes of bounded variation is given. Furthermore, we prove that a consistent estimator for the asymptotic variance of the NPMLE is given by the inverse of a discrete observed information matrix.

A semiparametric hazard model with parametrized time but general covariate dependency is formulated and analyzed inside the framework of counting process theory. A profile likelihood principle is introduced for estimation of the parameters: the resulting estimator is $n^{1/2}$-consistent, asymptotically normal and achieves the semiparametric efficiency bound. An estimation procedure for the nonparametric part is also given and its asymptotic properties are derived. We provide an application to mortality data.

A general theory on rates of convergence of the least-squares projection estimate in multiple regression is developed. The theory is applied to the functional ANOVA model, where the multivariate regression function is modeled as a specified sum of a constant term, main effects (functions of one variable) and selected interaction terms (functions of two or more variables). The least-squares projection is onto an approximating space constructed from arbitrary linear spaces of functions and their tensor products respecting the assumed ANOVA structure of the regression function. The linear spaces that serve as building blocks can be any of the ones commonly used in practice: polynomials, trigonometric polynomials, splines, wavelets and finite elements. The rate of convergence result that is obtained reinforces the intuition that low-order ANOVA modeling can achieve dimension reduction and thus overcome the curse of dimensionality. Moreover, the components of the projection estimate in an appropriately defined ANOVA decomposition provide consistent estimates of the corresponding components of the regression function. When the regression function does not satisfy the assumed ANOVA form, the projection estimate converges to its best approximation of that form.

A data-driven estimate is given that, over a Sobolev space, is simultaneously asymptotically sharp minimax for estimating both the function and its derivatives under integrated squared error loss. It is also shown that linear estimates cannot be simultaneously asymptotically sharp minimax over a given Sobolev space.

An example is provided to show that the natural asymptotic equivalence does not hold between any pairs of three nonparametric experiments: density problem, white noise with drift and nonparametric regression, when the smoothness index of the unknown nonparametric function class is ½

Suppose one observes a process V on the unit interval, where $dV = f_o + dW$ with an unknown parameter $f_o \epsilon L_1[0, 1]$ and standard Brownian motion W. We propose a particular test of one-point hypotheses about $f_o$ which is based on suitably standardized increments of V. This test is shown to have desirable consistency properties if, for instance, $f_o$ is restricted to various Hölder classes of functions. The test is mimicked in the context of nonparametric density estimation, nonparametric regression and interval-censored data. Under shape restrictions on the parameter, such as monotonicity or convexity, we obtain confidence sets for $f_o$ adapting to its unknown smoothness.

When using nonparametric estimates of the mean curve, surface or image underlying noisy observations, the selection of "smoothing parameters" is generally crucial. This paper gives a theoretical comparison of the performances of generalized cross-validation (GCV) and of its fast randomized version (RGCV), as selection criteria. This is mainly done by studying the asymptotic distribution of the excess error for each selector, that is, the difference between the (data-driven) resulting average squared error (ASE) and the best possible ASE. We show here that, by using randomization, this distribution is dilated, as compared to that for CV or GCV, only by a factor always lower than $1 + 1/n_R$, where $n_R$ is the number of primary randomized trace estimates one uses in RGCV. We include in the compared selectors, the partial cross-validation (PCV) approach where only a fraction of all the possible "leave-one-out" validation tests are evaluated; so that PCV is a common practice to reduce the computational cost in many contexts. In this paper, PCV will in fact appear as quite inefficient as compared to RGCV from this computational point of view. Moreover, we show that a precise comparison (and interpretation of the gain of using $n_R \geq 2$) is possible in terms of equivalent (in distribution) excess errors, if PCV uses a certain percentage of the test points greater than 50%. The obtained comparisons will be seen as quite reassuring on what is "sacrificed" in using randomized selectors. We give rigorous results mainly for the kernel regression setting as in the previous detailed study by Härdle, Hall and Marron of standard selectors, except that we do not restrict this one to an equidistant design.

We estimate a probability density function p which is related by a linear operator K to a density function q in sequences of regular exponential families based on a random sample from q. In this paper deconvolution and positron emission tomography are considered. The logarithm of the density function is approximated by basis functions consisting of singular functions of K. While direct maximum likelihood (or minimum Kullback-Leibler) density estimation in exponential families selects the parameters to match the moments of the basis functions to the sample moments, in the inverse problem the moment of each singular function is related to a corresponding moment of the direct problem by a factor given by a singular value $\lambda_{\nu}$ of K. Thus an appropriate analogue of the maximum likelihood estimate is obtained by matching moments with respect to p to $l/\lambda_{\nu}$ times the empirical moments associated with the sample from q. Bounds on the Kullback-Leibler distance between the true density and the estimators are obtained and rates of convergence are established for log-density functions having a measure of smoothness. The density estimator converges to the unknown density in the Kullback-Leibler sense and in the $L_2$-sense at a rate determined not only by the order of smoothness of the log-density and the dimension of data but also by the decay rate of the singular values of the operator. A minimax lower bound for deconvolution is provided under certain conditions. Numerical examples using simulated data are provided to illustrate the finite-sample performance of the proposed method for deconvolution and positron emission tomography.

We construct Markov chain algorithms for sampling from discrete exponential families conditional on a sufficient statistic. Examples include contingency tables, logistic regression, and spectral analysis of permutation data. The algorithms involve computations in polynomial rings using Gröbner bases.

We begin by showing how piecewise linear bounds may be devised, which bound both above and below any concave log-density in general dimensions. We then show how these bounds may be used to gain an upper bound to the volume in the tails outside the convex hull of the sample path in order to assess how well the sampler has explored the target distribution. This method can be used as a stand-alone diagnostic to determine when the sampler output provides a reliable basis for inference on the stationary density, or in conjunction with existing convergence diagnostics to ensure that they are based upon good sampler output. We provide an example and briefly discuss possible extensions to the method and alternative applications of the bounds.

Hall and Welsh established the best attainable rate of convergence for estimates of a positive extreme value index $\gamma$ under a certain second order condition implying that the distribution function of the maximum of n random variables converges at an algebraic rate to the pertaining extreme value distribution. As a first generalization, we obtain a surprisingly sharp bound on the estimation error if $\gamma$ is still assumed to be positive, but the rate of convergence of the maximum may be nonalgebraic. This result allows a more accurate evaluation of the asymptotic performance of an estimator for $\gamma$ than the Hall and Welsh theorem. For example, it is proved that the Hill and the Pickands estimators achieve the optimal rate, but only the Hill estimator attains the sharp bound. Finally, an analogous result is derived for a general, not necessarily positive, extreme value index. In this situation it turns out that location invariant estimators show the best performance.