This paper reviews Herbert Robbins' research in sequential analysis (excluding stochastic approximation) from 1952 until roughly 1980. Its relation to the research of his contemporaries and its impact on subsequent research are described.

Empirical Bayes was Herbert Robbins' most influential contribution to statistical theory. It is also an idea of great practical potential. That potential is realized in the analysis of microarrays, a new biogenetic technology for the simultaneous measurement of thousands of gene expression levels.

Stochastic approximation, introduced by Robbins and Monro in 1951, has become an important and vibrant subject in optimization, control and signal processing. This paper reviews Robbins' contributions to stochastic approximation and gives an overview of several related developments.

In components of variance models the data are viewed as arising through a sum of two random variables, representing between- and within-group variation, respectively. The former is generally interpreted as a group effect, and the latter as error. It is assumed that these variables are stochastically independent and that the distributions of the group effect and the error do not vary from one instance to another. If each group effect can be replicated a large number of times, then standard methods can be used to estimate the distributions of both the group effect and the error. This cannot be achieved without replication, however. How feasible is distribution estimation if it is not possible to replicate prolifically? Can the distributions of random effects and errors be estimated consistently from a small number of replications of each of a large number of noisy group effects, for example, in a nonparametric setting? Often extensive replication is practically infeasible, in particular, if inherently small numbers of individuals exhibit any given group effect. Yet it is quite unclear how to conduct inference in this case. We show that inference is possible, even if the number of replications is as small as 2. Two methods are proposed, both based on Fourier inversion. One, which is substantially more computer intensive than the other, exhibits better performance in numerical experiments.

We consider a nonregular Cox model for independent and identically distributed right censored survival times, with a change-point according to the unknown threshold of a covariate. The maximum partial likelihood estimators of the parameters and the estimator of the baseline cumulative hazard are studied. We prove that the estimator of the change-point is n-consistent and the estimator of the regression parameters are $n^{1/2}$-consistent, and we establish the asymptotic distributions of the estimators. The estimators of the regression parameters and of the baseline cumulative hazard are adaptive in the sense that they do not depend on the knowledge of the change-point.

We propose new procedures for estimating the component functions in both additive and multiplicative nonparametric marker-dependent hazard models. We work with a full counting process framework that allows for left truncation and right censoring and time-varying covariates. Our procedures are based on kernel hazard estimation as developed by Nielsen and Linton and on the idea of marginal integration. We provide a central limit theorem for the marginal integration estimator. We then define estimators based on finite-step backfitting in both additive and multiplicative cases and prove that these estimators are asymptotically normal and have smaller variance than the marginal integration method.

This paper is concerned with Bayesian analysis of the proportional hazard model with left truncated and right censored data. We use a process neutral to the right as the prior of the baseline survival function and a finite-dimensional prior is placed on the regression coefficient. We then obtain the exact form of the joint posterior distribution of the regression coefficient and the baseline cumulative hazard function. As a by-product, we prove the propriety of the posterior distribution with the constant prior on the regression coefficient.

We study nonparametric estimation with two types of data structures. In the first data structure n i.i.d. copies of $(C,N(C))$ are observed, where N is a finite state counting process jumping at time-variables of interest and C a random monitoring time. In the second data structure n i.i.d. copies of $(C\wedge T,I(T\leq C),N (C\wedge T))$ are observed, where N is a counting process with a final jump at time T (e.g., death). This data structure includes observing right-censored data on T and a marker variable at the censoring time.

In these data structures, easy to compute estimators, namely (weighted)-pool-adjacent-violator estimators for the marginal distributions of the unobservable time variables, and the Kaplan-Meier estimator for the time T till the final observable event, are available. These estimators ignore seemingly important information in the data. In this paper we prove that, at many continuous data generating distributions the ad hoc estimators yield asymptotically efficient estimators of $\sqrt{n}$-estimable parameters.

We consider the problem of estimating the mean of an infinite-break dimensional normal distribution from the Bayesian perspective. Under the assumption that the unknown true mean satisfies a "smoothness condition," we first derive the convergence rate of the posterior distribution for a prior that is the infinite product of certain normal distributions and compare with the minimax rate of convergence for point estimators. Although the posterior distribution can achieve the optimal rate of convergence, the required prior depends on a "smoothness parameter" q. When this parameter q is unknown, besides the estimation of the mean, we encounter the problem of selecting a model. In a Bayesian approach, this uncertainty in the model selection can be handled simply by further putting a prior on the index of the model. We show that if q takes values only in a discrete set, the resulting hierarchical prior leads to the same convergence rate of the posterior as if we had a single model. A slightly weaker result is presented when q is unrestricted. An adaptive point estimator based on the posterior distribution is also constructed.

We consider the problem of determining the distribution of means of random probability measures which are obtained by normalizing increasing additive processes. A solution is found by resorting to a well-known inversion formula for characteristic functions due to Gurland. Moreover, expressions of the posterior distributions of those means, in the presence of exchangeable observations, are given. Finally, a section is devoted to the illustration of two examples of statistical relevance.

We establish the validity of an Edgeworth expansion to the distribution of the maximum likelihood estimator of the parameter of a stationary, Gaussian, strongly dependent process. The result covers ARFIMA-type models, including fractional Gaussian noise. The method of proof consists of three main ingredients: (i) verification of a suitably modified version of Durbin's general conditions for the validity of the Edgeworth expansion to the joint density of the log-likelihood derivatives; (ii) appeal to a simple result of Skovgaard to obtain from this an Edgeworth expansion for the joint distribution of the log-likelihood derivatives; (iii) appeal to and extension of arguments of Bhattacharya and Ghosh to accomplish the passage from the result on the log-likelihood derivatives to the result for the maximum likelihood estimators. We develop and make extensive use of a uniform version of a theorem of Dahlhaus on products of Toeplitz matrices; the extension of Dahlhaus' result is of interest in its own right. A small numerical study of the efficacy of the Edgeworth expansion is presented for the case of fractional Gaussian noise.

Let $\{X_t\}$ be a stationary time series and let $d_T(\lambda)$ denote the discrete Fourier transform (DFT) of $\{X_0,\ldots,X_{T-1}\}$ with a data taper. The main results of this paper provide a characterization of asymptotic independence of the DFTs in terms of the distance between their arguments under both short- and long-range dependence of the process $\{X_t\}$. Further, asymptotic joint distributions of the DFTs $d_T(\lambda_{1T})$ and $d_T(\lambda_{2T})$ are also established for the cases $T(\lambda_{1T}- \lambda_{2T})=O(1)$ as $T\to\infty$ (asymptotically close ordinates) and $|T(\lambda_{1_T}-\lambda_{2_T})|\to\infty$ as $T\to\infty$ (asymptotically distant ordinates). Some implications of the main results on the estimation of the index of dependence are also discussed.

This paper considers adaptive estimation in nonstationary autoregressive moving average models with the noise sequence satisfyinga generalized autoregressive conditional heteroscedastic process. The locally asymptotic quadratic form of the log-likelihood ratio for the model is obtained. It is shown that the limit experiment is neither LAN nor LAMN, but is instead LABF. For the model with symmetric density of the rescaled error, a new efficiency criterion is established for a class of defined $M_{\nu}$-estimators. It is shown that such efficient estimators can be constructed when the density is known. Using the kernel estimator for the score function, adaptive estimators are constructed when the density of the rescaled error is symmetric, and it is shown that the adaptive procedure for the parameters in the conditional mean part uses the full sample without splitting. These estimators are demonstrated to be asymptotically efficient in the class of $M_{\nu}$-estimators. The paper includes the results that the stationary ARMA-GARCH model is LAN, and that the parameters in the model with symmetric density of the rescaled error are adaptively estimable after a reparameterization of the GARCH process. This paper also establishes the locally asymptotic quadratic form of the log-likelihood ratio for nonlinear time series models with ARCH-type errors.

The problem of detecting randomness in the coefficients of an AR(p) model, that is, the problem of testing ordinary AR(p) dependence against the alternative of a random coefficient autoregressive [RCAR(p)] model is considered. A nonstandard LAN property is established for RCAR(p) models in the vicinity of AR(p) ones. Two main problems arise in this context. The first problem is related to the statistical model itself: Gaussian assumptions are highly unrealistic in a nonlinear context, and innovation densities should be treated as nuisance parameters. The resulting semiparametric model however appears to be severely nonadaptive. In contrast with the linear ARMA case, pseudo-Gaussian likelihood methods here are invalid under non-Gaussian densities; even the innovation variance cannot be estimated without a strict loss of efficiency. This problem is solved using a general result by Hallin and Werker, which provides semiparametrically efficient central sequences without going through explicit tangent space calculations. The second problem is related to the fact that the testing problem under study is intrinsically one-sided, while the case of multiparameter one-sided alternatives is not covered by classical asymptotic theory under LAN. A concept of locally asymptotically most stringent somewhere efficient test is proposed in order to cope with this one-sided nature of the problem.