Fixed-smoothing asymptotics for time series

In this paper, we derive higher order Edgeworth expansions for the finite sample distributions of the subsampling-based t-statistic and the Wald statistic in the Gaussian location model under the so-called fixed-smoothing paradigm. In particular, we show that the error of asymptotic approximation is at the order of the reciprocal of the sample size and obtain explicit forms for the leading error terms in the expansions. The results are used to justify the second-order correctness of a new bootstrap method, the Gaussian dependent bootstrap, in the context of Gaussian location model.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1113 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Resampling methods for longitudinal data analysis

Ph.DDOCTOR OF PHILOSOPH

Higher order asymptotic theory for nonparametric time series analysis and related contributions.

We investigate higher order asymptotic theory in nonparametric time series analysis. The aim of these techniques is to approximate the finite sample distribution of estimates and test statistics. This is specially relevant for smoothed nonparametric estimates in the presence of autocorrelation, which have slow rates of convergence so that inference rules based on first-order asymptotic approximations may not be very precise. First we review the literature on autocorrelation-robust inference and higher order asymptotics in time series. We evaluate the effect of the nonparametric estimation of the variance in the studentization of least squares estimates in linear regression models by means of asymptotic expansions. Then, we obtain an Edgeworth expansion for the distribution of nonparametric estimates of the spectral density and studentized sample mean. Only local smoothness conditions on the spectrum of the time series are assumed, so long range dependence behaviour in the series is allowed at remote frequencies, not necessary only at zero frequency but at possible cyclical and seasonal ones. The nonparametric methods described rely on a bandwidth or smoothing number. We propose a cross-validation algorithm for the choice of the optimal bandwidth, in a mean square sense, at a single point without restrictions on the spectral density at other frequencies. Then, we focus on the performance of the spectral density estimates around a singularity due to long range dependence and we obtain their asymptotic distribution in the Gaussian case. Semiparametric inference procedures about the long memory parameter based on these nonparametric estimates are justified under mild conditions on the distribution of the observed time series. Using a fixed average of periodogram ordinates, we also prove the consistency of the log-periodogram regression estimate of the memory parameter for linear but non-Gaussian time series

Convergence of Gaussian quasi-likelihood random fields for ergodic L\'{e}vy driven SDE observed at high frequency

This paper investigates the Gaussian quasi-likelihood estimation of an exponentially ergodic multidimensional Markov process, which is expressed as a solution to a L\'{e}vy driven stochastic differential equation whose coefficients are known except for the finite-dimensional parameters to be estimated, where the diffusion coefficient may be degenerate or even null. We suppose that the process is discretely observed under the rapidly increasing experimental design with step size $h_n$. By means of the polynomial-type large deviation inequality, convergence of the corresponding statistical random fields is derived in a mighty mode, which especially leads to the asymptotic normality at rate $\sqrt{nh_n}$ for all the target parameters, and also to the convergence of their moments. As our Gaussian quasi-likelihood solely looks at the local-mean and local-covariance structures, efficiency loss would be large in some instances. Nevertheless, it has the practically important advantages: first, the computation of estimates does not require any fine tuning, and hence it is straightforward; second, the estimation procedure can be adopted without full specification of the L\'{e}vy measure.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1121 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Computer-intensive statistical methods:saddlepoint approximations with applications in bootstrap and robust inference

The saddlepoint approximation was introduced into statistics in 1954 by Henry E. Daniels. This basic result on approximating the density function of the sample mean has been generalized to many situations. The accuracy of this approximation is very good, particularly in the tails of the distribution and for small sample sizes, compared with normal or Edgeworth approximation methods. Before applying saddlepoint approximations to the bootstrap, this thesis will focus on saddlepoint approximations for the distribution of quadratic forms in normal variables and for the distribution of the waiting time in the coupon collector's problem. Both developments illustrate the modern art of statistics relying on the computer and embodying both numeric and analytic approximations. Saddlepoint approximations are extremely accurate in both cases. This is underlined in the first development by means of an extensive study and several applications to nonparametric regression, and in the second by several examples, including the exhaustive bootstrap seen from a collector's point of view. The remaining part of this thesis is devoted to the use of saddlepoint approximations in order to replace the computer-intensive bootstrap. The recent massive increases in computer power have led to an upsurge in interest in computer-intensive statistical methods. The bootstrap is the first computer-intensive method to become widely known. It found an immediate place in statistical theory and, more slowly, in practice. The bootstrap seems to be gaining ground as the method of choice in a number of applied fields, where classical approaches are known to be unreliable, and there is sustained interest from theoreticians in its development. But it is known that, for accurate approximations in the tails, the nonparametric bootstrap requires a large number of replicates of the statistic. As this is time-intensive other methods should be considered. Saddlepoint methods can provide extremely accurate approximations to resampling distributions. As a first step I develop fast saddlepoint approximations to bootstrap distributions that work in the presence of an outlier, using a saddlepoint mixture approximation. Then I look at robust M-estimates of location like Huber's M-estimate of location and its initially MAD scaled version. One peculiarity of the current literature is that saddlepoint methods are often used to approximate the density or distribution functions of bootstrap estimators, rather than related pivots, whereas it is the latter which are more relevant for inference. Hence the aim of the final part of this thesis is to apply saddlepoint approximations to the construction of studentized confidence intervals based on robust M-estimates. As examples I consider the studentized versions of Huber's M-estimate of location, of its initially MAD scaled version and of Huber's proposal 2. In order to make robust inference about a location parameter there are three types of robustness one would like to achieve: robustness of performance for the estimator of location, robustness of validity and robustness of efficiency for the resulting confidence interval method. Hence in the context of studentized bootstrap confidence intervals I investigate these in more detail in order to give recommendations for practical use, underlined by an extensive simulation study