Some standard numerical problems become intractable in high dimensions. Yet successes are often achieved in practice. This may be explained in terms of the underlying target function being somehow simpler than assumed in the intractability arguments. A prototypical form of simplicity is approximate linearity. In moderate dimensions, linearity can be investigated by linear regression. In very high dimensions, this becomes computationally inefficient and eventually infeasible, as the cost of regression for $n$ observations in $d$ dimensions grows as $nd^2$. This paper presents a quasi-regression method for determining the degree of linearity in a function, where the cost grows only as $nd$. A bias-corrected version of quasi-regression is able to estimate the degree of linearity with a sample size of order $d^{2 /3}$. An example is given of a function on $[0,1]^ {1,000,000}$, for which the amount of linear variability is accurately estimated from only 100,000 observations.

A new approach to testing for monotonicity of a regression mean, not requiring computation of a curve estimator or a bandwidth, is suggested. It is based on the notion of “running gradients ” over short intervals, although from some viewpoints it may be regarded as an analogue for monotonicity testing of the dip/excess mass approach for testing modality hypotheses about densities. Like the latter methods, the new technique does not suffer difficulties caused by almost-flat parts of the target function. In fact, it is calibrated so as to work well for flat response curves, and as a result it has relatively good power properties in boundary cases where the curve exhibits shoulders. In this respect, as well as in its construction, the “running gradients” approach differs from alternative techniques based on the notion of a critical bandwidth.

Richardson and Green present a method of performing a Bayesian analysis of data from a finite mixture distribution with an unknown number of components. Their method is a Markov Chain Monte Carlo (MCMC) approach, which makes use of the “reversible jump” methodology described by Green. We describe an alternative MCMC method which views the parameters of the model as a (marked) point process, extending methods suggested by Ripley to create a Markov birth-death process with an appropriate stationary distribution. Our method is easy to implement, even in the case of data in more than one dimension, and we illustrate it on both univariate and bivariate data. There appears to be considerable potential for applying these ideas to other contexts, as an alternative to more general reversible jump methods, and we conclude with a brief discussion of how this might be achieved.

General results on adaptive density estimation are obtained with respect to any countable collection of estimation strategies under Kullback-Leibler and squared $L_2$ losses. It is shown that without knowing which strategy works best for the underlying density, a single strategy can be constructed by mixing the proposed ones to be adaptive in terms of statistical risks. A consequence is that under some mild conditions, an asymptotically minimax-rate adaptive estimator exists for a given countable collection of density classes; that is, a single estimator can be constructed to be simultaneously minimax-rate optimal for all the function classes being considered. A demonstration is given for high-dimensional density estimation on $[0,1]^d$ where the constructed estimator adapts to smoothness and interaction-order over some piecewise Besov classes and is consistent for all the densities with finite entropy.

Nonparametric estimators of solutions of renewal-type equations are proposed in terms of the empirical renewal function. Using the approach of Grübel and Pitts [Ann .Stat .21 1431 –1451 (1993)], who studied asymptotic properties of the empirical renewal function, a number of properties of our estimators are established, including strong consistency, asymptotic normality and efficiency, and asymptotic validity of bootstrap confidence bounds. The results are illustrated by some particular examples.

We identify the asymptotic behavior of the estimators proposed by Rojo and Samaniego and Mukerjee of a distribution $F$ assumed to be uniformly stochastically smaller than a known baseline distribution $G$.We show that these estimators are $\sqrt{n}$-convergent to a limit distribution with mean squared error smaller than or equal to the mean squared error of the empirical survival function. By examining the mean squared error of the limit distribution, we are able to identify the optimal estimator within Mukerjee’s class under a variety of different assumptions on $F$ and $G$. Similar theoretical results are developed for the two-sample problem where$F$ and $G$ are both unknown. The asymptotic distribution theory is applied to obtain confidence bands for the survival function $\bar{F}$ based on published data from an accelerated life testing experiment.

Vardi [Ann.Statist.13 178-203 (1985)] introduced an $s$-sample biased sampling model with known selection weight functions, gave a condition under which the common underlying probability distribution $G$ is uniquely estimable and developed simple procedure for computing the nonparametric maximum likelihood estimator (NPMLE) $\mathbb{G}_n$ of $G$. Gill, Vardi and Wellner thoroughly described the large sample properties of Vardi’s NPMLE, giving results on uniform consistency, convergence of $\sqrt{n}(\mathbb{G}-G)$ to a Gaussian process and asymptotic efficiency of $\mathbb{G}_n$. Gilbert, Lele and Vardi considered the class of semiparametric $s$-sample biased sampling models formed by allowing the weight functions to depend on an unknown finite-dimensional parameter $\theta$ .They extended Vardi’s estimation approach by developing a simple two-step estimation procedure in which $\hat{\theta}_n$ is obtained by maximizing a profile partial likelihood and $\mathbb{G}_n \equiv \mathbb{G}_n(\hat{\theta}_n)$ is obtained by evaluating Vardi’s NPMLE at $\hat{\theta}_n$. Here we examine the large sample behavior of the resulting joint MLE $(\hat{\theta}_n,\mathbb{G}_n)$, characterizing conditions on the selection weight functions and data in order that $(\hat{\theta}_n, \mathbb{G}_n)$ is uniformly consistent, asymptotically Gaussian and efficient.

Examples illustrated here include clinical trials (especially HIV vaccine efficacy trials), choice-based sampling in econometrics and case-control studies in biostatistics.

A pretest procedure consists of a preliminary test on a nuisance parameter, investigating whether it equals a given value or not, followed by the main testing problem on the parameter of interest. In case of acceptance of the preliminary test, the main test is applied in the restricted family with the given value of the nuisance parameter, while otherwise the test is performed in the complete family, including the nuisance parameter. For an appropriate class of tests, containing all standard first-order optimal tests, an attractive expression for the difference in size and power between the pretest procedure and the test in the complete family is derived using second-order asymptotics. For a very great part the result is the same for all members of the class. From this expression considerable insight can be obtained in a qualitative and quantitative sense. The results can be applied easily as is illustrated by a number of practical examples, where also the accuracy of the approximations is seen from comparison with numerical results.

The shortcoming of a test is the difference between the maximal attainable power and the power of the test under consideration. Vanishing shortcoming, when the number of observations tends to infinity, is therefore an optimality property of a test. Other familiar optimality criteria are based on the asymptotic relative efficiency of the test. The relations between these optimality criteria are investigated. It turns out that vanishing shortcoming is seemingly slightly stronger than first-order efficiency, but in regular cases there is equivalence. The results are in particular applied to tests for goodness-of-it.

It is shown that the global power function of any nonparametric test is flat on balls of alternatives except for alternatives coming from a finite dimensional subspace. The present benchmark is here the upper one-sided (or two-sided) envelope power function. Every choice of a test fixes a priori a finite dimensional region with high power. It turns out that also the level points are far away from the corresponding Neyman–Pearson test level points except for a finite number of orthogonal directions of alternatives. For certain submodels the result is independent of the underlying sample size. In the last section the statistical consequences and special goodness of fit tests are discussed.

An abundance of high quality data sets requiring heavy tailed models necessitates reliable methods of estimating the shape parameter governing the degree of tail heaviness. The Hill estimator is a popular method for doing this but its practical use is encumbered by several difficulties. We show that an alternative method of plotting Hill estimator values is more revealing than the standard method unless the underlying data comes from a Pareto distribution.

When a unique $M$-estimate exists, its density is obtained as a corollary to a more general theorem which asserts that under mild conditions the intensity function of the point process of solutions of the estimating equations exists and is given by the density of the estimating function standardized by multiplying it by the inverse of its derivative. We apply the results to give a result for Huber’s proposal 2 applied to regression and scale estimates. We also give a saddlepoint approximation for the density and use this to give approximations for tail areas for smooth functions of the $M$-estimates.

We present a procedure associated with nonlinear wavelet methods that provides adaptive confidence intervals around $f (x_0)$, in either a white noise model or a regression setting. A suitable modification in the truncation rule for wavelets allows construction of confidence intervals that achieve optimal coverage accuracy up to a logarithmic factor. The procedure does not require knowledge of the regularity of the unknown function $f$; it is also efficient for functions with a low degree of regularity.