Regression with the error measured by the supremum norm is considered. Analytic functions on [0, 1] and functions with a bounded $r$th derivative are considered as functions to be estimated. It is assumed that the experimenter chooses the points at which the observations are taken. Polynomial and piecewise polynomial estimates are considered. Asymptotic and nonasymptotic bounds for the error are obtained.

An estimation procedure for $(b, g)$ is developed for the transformation model $g(Y) = bz + \text{error, where} g$ is an unspecified strictly increasing function. The estimator for $b$ can be viewed as a hybrid between an $M$-estimator and an $R$-estimator. It differs from an $M$-estimator in that the dependent variable is replaced by a score based on ranks and from an $R$-estimator in that the ranks of dependent variable itself are used, not the ranks of the residuals. This provides robustness against the scale on which the variables are thought to be linearly related, as opposed to robustness against misspecification of the error distribution. Existence, uniqueness, consistency and asymptotic normality are studied.

This paper discusses the possibility of truly nonparametric inference about functionals of an unknown density. Examples considered include: discrete functionals, such as the number of modes of a density and the number of terms in the true model; and continuous functionals, such as the optimal bandwidth for kernel density estimates or the widths of confidence intervals for adaptive location estimators. For such functionals it is not generally possible to make two-sided nonparametric confidence statements. However, one-sided nonparametric confidence statements are possible: e.g., "I say with 95% confidence that the underlying distribution has at least three modes." Roughly, this is because the functionals of interest are semicontinuous with respect to the topology induced by a distribution-free metric. Then a neighborhood procedure can be used. The procedure is to find the minimum value of the functional over a neighborhood of the empirical distribution in function space. If this neighborhood is a nonparametric $1 - \alpha$ confidence region for the true distribution, the resulting minimum value lowerbounds the true value with a probability of at least $1 - \alpha$. This lower bound has good asymptotic properties in the high-confidence setting $\alpha$ close to 0.

A kernel density estimator is defined to be admissible if no other kernel estimator has (among all densities and sample sizes) uniformly smaller mean integrated squared error. Admissible kernel density estimators are precisely those using kernels with nonnegative Fourier transforms bounded by 1. Several examples are given.

Strong uniform consistency rates are established for kernel type estimators of functionals of the conditional distribution function, under general conditions. The present treatment unifies a number of specific problems previously studied separately in the literature. Some of these applications we treat in detail, including regression curve estimation, density estimation, estimation of conditional df's, $L$-smoothing and $M$-smoothing. Various previous results in the literature are extended and/or sharpened.

We study semiparametric models where for a fixed value of the finite-dimensional parameter there exists a sufficient statistic for the nuisance parameter. An asymptotically normal sequence of estimators for the parametric component is constructed, which is efficient under the assumption that projecting on the set of nuisance scores is equivalent to taking conditional expectations given the sufficient statistic. The latter property is checked for a number of examples, in particular for mixture models. We discuss the relation of our approach to conditional maximum likelihood estimation.

Estimation of the bivariate survival function from censored data is considered. The product integral representation of univariate survival functions is generalized to the bivariate case and used to determine identifiability of the survival function of the partially observed data. A bivariate analogue of the Kaplan-Meier estimate is introduced and its almost sure consistency is studied. Extensions to the general multivariate case are sketched.

An interval censoring model is carefully defined and the range of its applicability is illustrated. A class of rank tests for the two- and $k$-sample problems is proposed. The statistic is based on the exact ranks of the observed responses and resembles the rank statistic in the usual uncensored case. This statistic is shown to be asymptotically equivalent to the efficient scores test statistic under an assumed parametric model. An alternative rank statistic, based on the estimated ranks of the unobserved variables of interest and applicable under a more general interval censoring model, is also proposed.

We present a solution to the goodness-of-fit problem for multivariate observations, using the innovation process for the (sequential) empirical distribution function with respect to a conveniently chosen linear ordering or scanning system in $R^m$.

We study the asymptotic behavior of the statistical estimators that maximize a not necessarily differentiable criterion function, possibly subject to side constraints (equalities and inequalities). The consistency results generalize those of Wald and Huber. Conditions are also given under which one is still able to obtain asymptotic normality. The analysis brings to the fore the relationship between the problem of finding statistical estimators and that of finding the optimal solutions of stochastic optimization problems with partial information. The last section is devoted to the properties of the saddle points of the associated Lagrangians.

Imputation and multiple-imputation procedures have been used in practice to handle the problem of ignorable nonresponse in sample surveys. We examine the large-sample properties of these procedures where covariates are available for the case when the complete-data analysis is based on least squares. The results provide a formal justification for the inference procedures discussed by Rubin and Schenker for the location problem and suggest new procedures for the regression problem.

Discrete and multinomial analogs are defined for classical (continuous) invariant nonparametric problems of estimating the sample cumulative distribution function (sample c.d.f.) and the sample median. Admissibility of classical estimators and their analogs is investigated. In discrete (including multinomial) settings the sample c.d.f. is shown to be an admissible estimator of the population c.d.f. under the invariant weighted Cramer-von Mises loss function $L_1(F, \hat{F}) = \int \big\lbrack (F(t) - \hat{F}(t))^2/(F(t)(1 - F(t))) \big\rbrack dF(t).$ Ordinary Cramer-von Mises loss--$L_2(F, \hat{F}) = \int \lbrack (F(t) - \hat{F}(t))^2 \rbrack dF(t)$--is also studied. Admissibility of the best invariant estimator is investigated. (It is well known in the classical problem that the sample c.d.f. is not the best invariant estimator, and hence is not admissible.) In most discrete settings this estimator must be modified in an obvious fashion to take into account the end points of the known domain of definition for the sample c.d.f. When this is done the resulting estimator is shown to be admissible in some of the discrete settings. However, in the classical continuous setting and in other discrete settings, the best invariant estimator, or its modification, is shown to be inadmissible. Kolmogorov-Smirnov loss for estimating the population c.d.f. is also investigated, but definitive admissibility results are obtained only for discrete problems with sample size 1. In discrete settings the sample median is an admissible estimator of the population median under invariant loss. In the continuous setting this is not true for even sample sizes.

Controlling Bayes and/or minimax risks under possibly different loss functions is formulated as a problem faced by two or more statisticians who must compromise and agree on the use of a single decision procedure. The theory characterizing solutions to Bayes compromise problems and minimax-Bayes compromise problems is presented. In a Bayes compromise problem, Bayes risks under different prior distributions and/or loss functions are minimized simultaneously. In a minimax-Bayes compromise problem, a Bayes risk under some loss function for a given prior distribution and a maximum risk under a possibly different loss function are minimized simultaneously.

A procedure for estimating the mixing distribution for a normal mean is presented. The estimator is shown to be consistent regardless of the mixing distribution, and it is suggested that the estimation techniques may be applied in a similar manner to estimate mixing distributions for a wide class of location parameter families. Implied marginal density and derivative estimates for the normal case are shown to be consistent, converging at near-optimal rates. In addition, empirical Bayes rules for the quadratic loss mean estimation problem which are derived from the estimated mixing distributions are shown to be asymptotically optimal in average compound risk and Bayes risk (if it is defined), provided the means are assumed to lie within a fixed range. A brief discussion of computational issues related to this estimation procedure is also included, along with some small-sample simulation results for the compound decision problem.

Identifiability problems have previously precluded a general approach to testing the hypothesis of a "pure" distribution against the alternative of a mixture of distributions. Three types of identifiability are defined, and it is shown that $B$-identifiability allows a Bayesian solution to the testing problem. First, an equivalence relation is defined over parametrizations of probability functions. Then the projection onto the quotient space is shown to give a $B$-identifiable parametrization. Bayesian inference proceeds using the Bayes factor as a "test" criterion.

The purpose of this paper is to investigate the convergence rates of a sequence of empirical Bayes decision rules for the two-action decision problems where the distributions of the observations belong to a discrete exponential family. It is found that the sequence of the empirical Bayes decision rules under study is asymptotically optimal, and the order of associated convergence rates is $O(\exp(-cn))$, for some positive constant $c$, where $n$ is the number of accumulated past experience (observations) at hand. Two examples are provided to illustrate the performance of the proposed empirical Bayes decision rules. A comparison is also made between the proposed empirical Bayes rules and some earlier existing empirical Bayes rules.

The problem of approximating the power and significance levels of repeated significance tests (RST) and modified repeated significance tests (MRST) is considered. The method due to Siegmund in the special case of normal observations with known variance is generalized. The main advantages that are claimed for this method are two-fold. First, it can be used to approximate the power of RSTs. Second, it can also be used to approximate the power and significance levels of MRSTs. Numerical and Monte Carlo results are also given for the repeated $t$-test.

A class of procedures is proposed for testing the stationarity of a Gaussian process or the homogeneity of independent processes. Requiring very limited prior knowledge of model structure, the methods can detect changes or differences in mean, in variance, in covariances and even in law. Although the theory of the stationarity test is worked out only for processes whose realizations are stationary over "epochs" separated by known change points, Monte Carlo evidence indicates that it can be useful also in detecting more general forms of nonstationarity. The test statistic is a quadratic form in differences among epoch means of certain "sensing" functions, the choice of which governs sensitivity to specific forms of nonstationarity or inhomogeneity. The applicability of the general asymptotic theory of the test is verified for two specific forms of sensing function, and small-sample properties of tests of each form are studied by means of simulation.

A Bayesian bootstrap for a finite population is introduced; its small-sample distributional properties are discussed and compared with those of the frequentist bootstrap for a finite population. It is also shown that the two are first-order asymptotically equivalent.

It is shown in this article that the classical i.i.d. bootstrap remains a valid procedure for estimating the sampling distributions of certain symmetric estimators of location, as long as the random observations are independently drawn from distributions with (essentially) a common location. This may be viewed as a robust property of the classical i.i.d. bootstrap. Also included is a study of the second order properties of a different bootstrap procedure proposed by Wu in the context of heteroscedasticity in regression.

We prove that the distribution of least-squares estimates in autoregressions can be bootstrapped with accuracy $o(n^{-1/2})$ a.s., thereby improving the normal approximation error of $O(n^{-1/2})$.

The class of single-generator generalized cyclic designs is shown to be degrees of freedom equivalent to a subclass of the class of prime-level pseudofactor designs.

It is an open problem in the literature to derive optimum tests for the equality of treatment effects in an unbalanced two-way classification model. For such models without interaction, optimum tests are derived in the following cases: (i) the locally best invariant unbiased test for the random effects model corresponding to an equiblock and equireplicate design, (ii) the locally best invariant unbiased test for the mixed effects model with mixed treatment effects corresponding to a balanced incomplete block design and (iii) the uniformly most powerful invariant test or the locally best invariant test for the mixed effects model with random treatment effects. Robustness of the optimum invariant tests against suitable deviations from normality is also indicated.