The paper develops a one-step triangular array Edgeworth expansion for multivariate martingales that are, essentially, asymptotically ergodic. Both discrete and continuous time are covered. The expansion is in a test function topology. We investigate when the expansion has the usual Edgeworth form, looking in particular at likelihood inference, including Cox regression, and at inference for stationary time series. The triangular array nature of the results make them useful for bootstrapping, and a result pointing in that direction is shown for Cox regression.

In this paper, the problem of computing the exact value of the asymptotic efficiency of maximum likelihood estimators of a discontinuous signal in a Gaussian white noise is considered. A method based on constructing difference equations for the appropriate moments is presented and used to show that the exact variance of the Pitman estimator is $16\zeta(3)$, where $\zeta$ is the Riemann zeta function.

This paper analyzes the Gibbs sampler applied to a standard variance component model, and considers the question of how many iterations are required for convergence. It is proved that for $K$ location parameters, with $J$ observations each, the number of iterations required for convergence (for large $K$ and $J$) is a constant times $(1 + \log K/\log J)$. This is one of the first rigorous, a priori results about time to convergence for the Gibbs sampler. A quantitative version of the theory of Harris recurrence (for Markov chains) is developed and applied.

The common unknown probability law $P$ of a random sample $Y_1,\ldots, Y_n$ is assigned a Dirichlet process prior with index $\alpha$. It is shown that the posterior joint density of several moments of $P$ converges, as $\alpha(\mathbb{R})\rightarrow 0$, to a multivariate B-spline, which is, therefore, the Bayesian bootstrap joint density of the moments. The result provides the basis for possible default nonparametric Bayesian inference on unknown moments.

Let $X_{(1)} \leq X_{(2)} \leq \cdots \leq X_{(n)}$ be the order statistics of a random sample of $n$ lifetimes. The total-time-on-test statistic at $X_{(i)}$ is defined by $S_{i,n} = \sum^i_{j = 1}(n - j + 1)(X_{(j)} - X_{(j - 1)}), 1 \leq i \leq n$. A type II censored sample is composed of the $r$ smallest observations and the remaining $n - r$ lifetimes which are known only to be at least as large as $X_{(r)}$. Dufour conjectured that if the vector of proportions $(S_{1,n}/S_{r,n}, \ldots, S_{r - 1,n}/S_{r,n})$ has the distribution of the order statistics of $r - 1$ uniform(0, 1) random variables, then $X_1$ has an exponential distribution. Leslie and van Eeden proved the conjecture provided $n - r$ is no longer than $(1/3)n - 1$. It is shown in this note that the conjecture is true in general for $n \geq r \geq 5$. If the random variable under consideration has either NBU or NWU distribution, then it is true for $n \geq r \geq 2, n \geq 3$. The lower bounds obtained here do not depend on the sample size.

Heitjan and Rubin proposed a concept, "coarsening at random," generalizing Rubin's theory of missing at random. Their analysis was done in discrete sample spaces. We propose a generalization to general sample spaces. Among Heitjan and Rubin's applications was right-censoring in survival analysis. We discuss the application of the generalized theory to various censoring patterns in continuous time and connect to the modern theory of random censoring.

Most hazard regression models in survival analysis specify a given functional form to describe the influence of the covariates on the hazard rate. For instance, Cox's model assumes that the covariates act multiplicatively on the hazard rate, and Aalen's additive risk model stipulates that the covariates have a linear additive effect on the hazard rate. In this paper we study a fully nonparametric model which makes no assumption on the association between the hazard rate and the covariates. We propose a class of estimators for the conditional hazard function, the conditional cumulative hazard function and the conditional survival function, and study their large sample properties. When the size of a data set is relatively large, this fully nonparametric approach may provide more accurate information than that acquired from more restrictive models. It may also be used to test whether a particular submodel gives a good fit to a given data set. Because our results are obtained under the multivariate counting process setting of Aalen, they apply to a number of models arising in survival analysis, including various censoring and random truncation models. Our estimators are related to the conditional Nelson-Aalen estimators proposed by Beran for the random censorship model.

It is shown in infinite-dimensional Gaussian problems that affine estimators minimax the variance among all estimators of a linear functional subject to a constraint on the bias. Likewise, affine estimators also minimax the square of the bias among all estimates of a linear functional subject to a constraint on the variance.

Consider a statistical procedure (Method A) which is based on $n$ observations and a less effective procedure (Method B) which requires a larger number $k_n$ of observations to give equal performance under a certain criterion. To compare two different procedures, Hodges and Lehmann suggested that the difference $k_n - n$, called the deficiency of Method B with respect to Method A, is the most natural quantity to examine. In this article, the performance of two kernel quantile estimators is examined versus the sample quantile estimator under the criterion of equal covering probability for randomly right-censored data. We shall show that the deficiency of the sample quantile estimator with respect to the kernel quantile estimators is convergent in infinity with the maximum rate when the bandwidth is chosen to be optimal. A Monte Carlo study is performed, along with an illustration on a real data set.

By using empirical process theory, the so-called excess mass approach is studied. It can be applied to various statistical problems, especially in higher dimensions, such as testing for multimodality, estimating density contour clusters, estimating nonlinear functionals of a density, density estimation, regression problems and spectral analysis. We mainly consider the problems of testing for multimodality and estimating density contour clusters, but the other problems also are discussed. The excess mass (over $\mathbb{C})$ is defined as a supremum of a certain functional defined on $\mathbb{C}$, where $\mathbb{C}$ is a class of subsets of the $d$-dimensional Euclidean space. Comparing excess masses over different classes $\mathbb{C}$ yields information about the modality of the underlying probability measure $F$. This can be used to construct tests for multimodality. If $F$ has a density $f$, the maximizing sets of the excess mass are level sets or density contour clusters of $f$, provided they lie in $\mathbb{C}$. The excess mass and the density contour clusters can be estimated from the data. Asymptotic properties of these estimators and of the test statistics are studied for general classes $\mathbb{C}$, including the classes of balls, ellipsoids and convex sets.

The traditional kernel density estimator of an unknown density is by construction completely nonparametric in the sense that it has no preferences and will work reasonably well for all shapes. The present paper develops a class of semiparametric methods that are designed to work better than the kernel estimator in a broad nonparametric neighbourhood of a given parametric class of densities, for example, the normal, while not losing much in precision when the true density is far from the parametric class. The idea is to multiply an initial parametric density estimate with a kernel-type estimate of the necessary correction factor. This works well in cases where the correction factor function is less rough than the original density itself. Extensive comparisons with the kernel estimator are carried out, including exact analysis for the class of all normal mixtures. The new method, with a normal start, wins quite often, even in many cases where the true density is far from normal. Procedures for choosing the smoothing parameter of the estimator are also discussed. The new estimator should be particularly useful in higher dimensions, where the usual nonparametric methods have problems. The idea is also spelled out for nonparametric regression.

We provide an asymptotic formula for the mean integrated squared error (MISE) of nonlinear wavelet-based density estimators. We show that, unlike the analogous situation for kernel density estimators, this MISE formula is relatively unaffected by assumptions of continuity. In particular, it is available for densities which are smooth in only a piecewise sense. Another difference is that in the wavelet case the classical MISE formula is valid only for sufficiently small values of the bandwidth. For larger bandwidths MISE assumes a very different form and hardly varies at all with changing bandwidth. This remarkable property guarantees a high level of robustness against oversmoothing, not encountered in the context of kernel methods. We also use the MISE formula to describe an asymptotically optimal empirical bandwidth selection rule.

This paper concerns estimating a mixing density function $g$ and its derivatives based on iid observations from $f(x) = \int f(x \mid \theta)g(\theta)d\theta$, where $f(x \mid \theta)$ is a known exponential family of density functions with respect to the counting measure on the set of nonnegative integers. Fourier methods are used to derive kernel estimators, upper bounds for their rate of convergence and lower bounds for the optimal rate of convergence. If $f(x \mid \theta_0) \geq \varepsilon^{x + 1} \forall x$, for some positive numbers $\theta_0$ and $\varepsilon$, then our estimators achieve the optimal rate of convergence $(\log n)^{-\alpha + m}$ for estimating the $m$th derivative of $g$ under a Lipschitz condition of order $\alpha > m$. The optimal rate of convergence is almost achieved when $(x!)^\beta f(x \mid \theta_0) \geq \varepsilon^{x + 1}$. Estimation of the mixing distribution function is also considered.

We propose a general semiparametric variance function model in a fixed design regression setting. In this model, the regression function is assumed to be smooth and is modelled nonparametrically, whereas the relation between the variance and the mean regression function is assumed to follow a generalized linear model. Almost all variance function models that were considered in the literature emerge as special cases. Least-squares-types estimates for the parameters of this model and the simultaneous estimation of the unknown regression and variance functions by means of nonparametric kernel estimates are combined to infer the parametric and nonparametric components of the proposed model. The asymptotic distribution of the parameter estimates is derived and is shown to follow usual parametric rates in spite of the presence of the nonparametric component in the model. This result is applied to obtain a data-based test for heteroscedasticity under minimal assumptions on the shape of the regression function.

Bounded influence functions are used for robust estimation in semiparametric models. In this paper, we generalize Hampel's variational problem to semiparametric models and define the optimal B-robust influence function as the one solving the variational problem. We identify the lowest bounds for influence functions and establish the existence and uniqueness of the optimal influence functions in general semiparametric models. Explicit optimal influence functions are given for a special case. Examples are provided to illustrate the procedures for calculating the optimal influence functions and for constructing the corresponding optimal estimators.

Dehling and Taqqu established the weak convergence of the empirical process for a long-range dependent stationary sequence under Gaussian subordination. We show that the corresponding density process, based on kernel estimators of the marginal density, converges weakly with the same normalization to the derivative of their limiting process. The phenomenon, which carries on for higher derivatives and for functional laws of the iterated logarithm, is in contrast with independent or weakly dependent situations, where the density process cannot be tight in the usual function spaces with supremum distances.

We consider the fixed-design regression model with long-range dependent normal errors and show that the finite-dimensional distributions of the properly normalized Gasser-Muller and Priestley-Chao estimators of the regression function converge to those of a white noise process. Furthermore, the distributions of the suitably renormalized maximal deviations over an increasingly finer grid converge to the Gumbel distribution. These results contrast with our previous findings for the corresponding problem of estimating the marginal density of long-range dependent stationary sequences.

This paper addresses the problem of testing the hypothesis that an observed series is difference stationary. The alternative hypothesis is that the series is another nonstationary process; in particular, an autoregressive model with a random parameter is used. A locally best invariant test is developed assuming Gaussianity, and a representation of its asymptotic distribution as a mixture of Brownian motions is found. The performance of the test in finite samples is investigated by simulation. An example is given where the difference stationary assumption for a well-known data series is rejected.

In this paper we construct an efficient weighted least squares estimator for the mean and more generally for the regression parameters in certain Gaussian long range dependent regression models, including polynomial regression. The form of the estimator does not depend on the whole dependence structure of the residuals, but only on the local behaviour of the spectral density at zero. By using an estimator of the self-similarity parameter, we give a fully efficient estimator. Furthermore, we construct efficient weighted $M$-estimators.

This paper discusses the estimation of multiple time series models which allow elements of the spectral density matrix to tend to infinity or zero at zero frequency and be unrestricted elsewhere. A form of log-periodogram regression estimate of differencing and scale parameters is proposed, which can provide modest efficiency improvements over a previously proposed method (for which no satisfactory theoretical justification seems previously available) and further improvements in a multivariate context when differencing parameters are a priori equal. Assuming Gaussianity and additional conditions which seem mild, asymptotic normality of the parameter estimates is established.