Bandwidth selection for procedures such as kernel density estimation and local regression have been widely studied over the past decade. Substantial “evidence” has been collected to establish superior performance of modern plug-in methods in comparison to methods such as cross validation; this has ranged from detailed analysis of rates of convergence, to simulations, to superior performance on real datasets.

In this work we take a detailed look at some of this evidence, looking into the sources of differences. Our findings challenge the claimed superiority of plug-in methods on several fronts. First, plug-in methods are heavily dependent on arbitrary specification of pilot bandwidths and fail when this specification is wrong. Second, the often-quoted variability and undersmoothing of cross validation simply reflects the uncertainty of band-width selection; plug-in methods reflect this uncertainty by oversmoothing and missing important features when given difficult problems. Third, we look at asymptotic theory. Plug-in methods use available curvature information in an inefficient manner, resulting in inefficient estimates. Previous comparisons with classical approaches penalized the classical approaches for this inefficiency. Asymptotically, the plug-in based estimates are beaten by their own pilot estimates.

More and more often, the outcome of a study is not a random variable but a noisy function for each experimental unit, resulting in a sample of curves. Typically, the individual curves vary not only in amplitude or intensity, but also with respect to the time axis: different subjects experience certain events sooner or later. Analyzing such data involves finding out the time changes (or curve registration) among curves. Following our previous work where modified dynamic time warping is applied to align two curves, we formulate a global minimization problem to align all curves in a sample and to compute the aligned average curve. Algorithms for solving the minimization problem are presented and tested with simulated and real data. The test results are promising. The method, which involves kernel smoothing of regression functions, estimates the time changes and the average of the aligned curves from noisy data. Large sample asymptotics is derived.

The problem of extracting a signal $x_{n}$ from a noise-corrupted time series $y_{n} = x_{n}+e_{n}$ is considered. The signal $x_{n}$ is assumed to be generated by a discrete-time, deterministic, chaotic dynamical system $F$, in particular, $x_{n} = F^{n}(x_{0})$, where the initial point $x_{0}$ is assumed to lie in a compact hyperbolic $F$-invariant set. It is shown that (1) if the noise sequence $e_{n}$ is Gaussian then it is impossible to consistently recover the signal $x_{n}$ , but (2) if the noise sequence consists of i.i.d. random vectors uniformly bounded by a constant $\delta > 0$, then it is possible to recover the signal $x_{n}$ provided $\delta < 5\Delta$, where $\Delta$ is a separation threshold for $F$. A filtering algorithm for the latter situation is presented.

We study estimation in the class of stationary variable length Markov chains (VLMC) on a finite space. The processes in this class are still Markovian of high order, but with memory of variable length yielding a much bigger and structurally richer class of models than ordinary high-order Markov chains. From an algorithmic view, the VLMC model class has attracted interest in information theory and machine learning, but statistical properties have not yet been explored. Provided that good estimation is available, the additional structural richness of the model class enhances predictive power by finding a better tradeoff between model bias and variance and allowing better structural description which can be of specific interest. The latter is exemplified with some DNA data.

A version of the tree-structured context algorithm, proposed by Rissanen in an information theoretical setup is shown to have new good asymptotic properties for estimation in the class of VLMCs. This remains true even when the underlying model increases in dimensionality. Fur-thermore, consistent estimation of minimal state spaces and mixing properties of fitted models are given.

We also propose a new bootstrap scheme based on fitted VLMCs. We show its validity for quite general stationary categorical time series and for a broad range of statistical procedures.

State space models is a very general class of time series models capable of modelling dependent observations in a natural and interpretable way. Inference in such models has been studied by Bickel, Ritov and Rydén, who consider hidden Markov models, which are special kinds of state space models, and prove that the maximum likelihood estimator is asymptotically normal under mild regularity conditions. In this paper we generalize the results of Bickel, Ritov and Rydén to state space models, where the latent process is a continuous state Markov chain satisfying regularity conditions, which are fulfilled if the latent process takes values in a compact space.

We give conditions that guarantee that the posterior probability of every Hellinger neighborhood of the true distribution tends to 1 almost surely. The conditions are (1) a requirement that the prior not put high mass near distributions with very rough densities and (2) a requirement that the prior put positive mass in Kullback-Leibler neighborhoods of the true distribution. The results are based on the idea of approximating the set of distributions with a finite-dimensional set of distributions with sufficiently small Hellinger bracketing metric entropy. We apply the results to some examples.

This paper is concerned with nonparametric Bayesian inference of the Aalen’s multiplicative counting process model. For a desired nonparametric prior distribution of the cumulative intensity function, a class of Lévy processes is considered, and it is shown that the class of Lévy processes is conjugate for the multiplicative counting process model, and formulas for obtaining a posterior process are derived. Finally, our results are applied to several practically important models such as one point processes for right-censored data, Poisson processes and Markov processes.

In this paper we present a new characterization and perspective on a neutral to the right prior. This characterization is based on a sequence of predictive laws which provides explicitly the posterior parameters and Bayes estimators for such a prior.

This paper derives an extended version of the Haff or, more appropriately, Stein-Haff identity for an elliptically contoured distribution (ECD) . This identity is then used to show that the minimax estimators of the covariance matrix obtained under normal models remain robust under the ECD model.

A widely held notion of classical conditional theory is that statistical inference in the presence of ancillary statistics should be independent of the distribution of those ancillary statistcs. In this paper, ancillary paradoxes which contradict this notion are presented for two scenarios involving confidence estimation. These results are related to Brown’s ancillary paradox in point estimation. Moreover, the confidence coefficient, the usual constant coverage probability estimator, is shown to be inadmissible for confidence estimation in the multiple regression model with random predictor variables if the dimension of the slope parameters is greater than five. Some estimators better than the confidence coefficient are provided in this paper. These new estimators are constructed based on empirical Bayes estimators.

For a version of the interval censoring model, case 2, in which the observation intervals are allowed to be arbitrarily small, we consider estimation of functionals that are differentiable along Hellinger differentiable paths. The asymptotic information lower bound for such functionals can be represented as the squared $L_{2}$-norm of the canonical gradient in the observation space. This canonical gradient has an implicit expression as a solution of an integral equation that does not belong to one of the standard types. We study an extended version of the integral equation that can also be used for discrete distribution functions like the nonparametric maximum likelihood estimator (NPMLE) , and derive the asymptotic normality and efficiency of the NPMLE from properties of the solutions of the integral equations.

Some sufficient conditions to establish the rate of convergence of certain $M$-estimators in a Gaussian white noise model are presented. They are applied to some concrete problems, including jump point estimation and nonparametric maximum likelihood estimation, for the regression function. The results are shown by means of a maximal inequality for continuous martingales and some techniques developed recently in the context of empirical processes.

Sliced inverse regression (SIR), formally introduced by Li, is a very general procedure for performing dimension reduction in nonparametric regression. This paper considers a version of SIR in which the “slices” are determined by nearest neighbors and the response variable takes value possibly in a multidimensional space. It is shown, under general conditions, that the “effective dimension reduction space” can be estimated with rate $n^{-1/2}$ where $n$ is the sample size.

Let $(X_{1},Y_{1}), (X_{2},Y_{2}),\dots,(X_{n},Y_{n})$ be a random sample from a bivariate distribution function $F$ which is in the domain of attraction of a bivariate extreme value distribution function $G$. A subset $C$ of $\mathbb{R}^{2}$ is given, which contains none of the observations. We shall give an asymptotic confidence interval for $\Pr((X_{i},Y_{i}) \in C)$ under certain conditions.

We suggest two semiparametric methods for accommodating departures from a Pareto model when estimating a tail exponent by fitting the model to extreme-value data. The methods are based on approximate likelihood and on least squares, respectively. The latter is somewhat simpler to use and more robust against departures from classical extreme-value approximations, but produces estimators with approximately 64% greater variance when conventional extreme-value approximations are appropriate. Relative to the conventional assumption that the sampling population has exactly a Pareto distribution beyond a threshold, our methods reduce bias by an order of magnitude without inflating the order of variance. They are motivated by data on extrema of community sizes and are illustrated by an application in that context.