Pursuing the classification initiated by Morris (1982), we describe all the natural exponential families on the real line such that the variance is a polynomial function of the mean with degree less than or equal to 3. We get twelve different types; the first six appear in the fundamental paper by Morris (1982); most of the other six appear as distributions of first passage times in the literature, the inverse Gaussian type being the most famous example. An explanation of this occurrence of stopping times is provided by the introduction of the notion of reciprocity between two measures or between two natural exponential families, and by classical fluctuation theory.

If $\{g_\theta(t)\}$ is a regular family of probability densities on the real line, with corresponding hazard rates $\{h_\theta(t)\}$, then the Fisher information for $\theta$ can be expressed in terms of the hazard rate as follows: $\mathscr{I}_\theta \equiv \int \big(\frac{\dot{g}_\theta}{g_\theta}\big)^2 g_\theta = \int \big(\frac{\dot{h}_\theta}{h_\theta}\big)^2 g_\theta, \theta \in \mathbb{R},$ where the dot denotes $\partial/\partial\theta$. This identity shows that the hazard rate transform of a probability density has an unexpected length-preserving property. We explore this property in continuous and discrete settings, some geometric consequences and curvature formulas, its connection with martingale theory and its relation to statistical issues in the theory of life-time distributions and censored data.

The comparison of nonparametric regression curves is considered. It is assumed that there are parametric (possibly nonlinear) transformations of the axes which map one curve into the other. Estimation and testing of the parameters in the transformations are studied. The rate of convergence is $n^{-1/2}$ although the nonparametric components of the model typically have a rate slower than that. A statistic is provided for testing the validity of a given completely parametric model.

An empirical likelihood ratio function is defined and used to obtain confidence regions for vector valued statistical functionals. The result is a nonparametric version of Wilks' theorem and a multivariate generalization of work by Owen. Cornish-Fisher expansions show that the empirical likelihood intervals for a one dimensional mean are less adversely affected by skewness than are those based on Student's $t$ statistic. An effective method is presented for computing empirical profile likelihoods for the mean of a vector random variable. The method is a reduction by convex duality to an unconstrained minimization of a convex function on a low dimensional domain. Algorithms exist for finding the unique global minimum at a superlinear rate of convergence. A byproduct is a noncombinatorial algorithm for determining whether a given point lies within the convex hull of a finite set of points. The multivariate empirical likelihood regions are justified for functions of several means, such as variances, correlations and regression parameters and for statistics with linear estimating equations. An algorithm is given for computing profile empirical likelihoods for these statistics.

It is proved that, except for a location term, empirical likelihood does draw contours which are second-order correct for those of a pseudo-likelihood. However, except in the case of one dimension, this pseudo-likelihood is not that which would commonly be employed when constructing a likelihood-based confidence region. It is shown that empirical likelihood regions may be adjusted for location so as to render them second-order correct. Furthermore, it is proved that location-adjusted empirical likelihood regions are Bartlett-correctable, in the sense that a simple empirical scale correction applied to location-adjusted empirical likelihood reduces coverage error by an order of magnitude. However, the location adjustment alters the form of the Bartlett correction. It is also shown that empirical likelihood regions and bootstrap likelihood regions differ to second order, although both are based on statistics whose centered distributions agree to second order.

We analyze the following model: One person, called "helper" observes an outcome $x^n = (x_1, \cdots, x_n) \in \mathscr{X}^n$ of the sequence $X^n = (X_1, \cdots, X_n)$ of i.i.d. RV's and the statistician gets a sample $y^n = (y_1, \cdots, y_n)$ of the sequence $Y^n(\theta, x^n)$ of RV's with a density $\prod^n_{t = 1} f(y_t \mid \theta, x_t)$. The helper can give some (side) information about $x^n$ to the statistician via an encoding function $s_n: \mathscr{X}^n \rightarrow \mathbb{N}$ with rate($s_n)^{def}{=}(1/n)\log {\tt\#}$ range($s_n) \leq R$. Based on the knowledge of $s_n(x^n)$ and $y^n$ the statistician tries to estimate $\theta$ by an estimator $\hat{\theta}_n$. For the maximal mean square error $e_n(R) =^{def} \inf_{\hat\theta_n} \inf_{s_n: \text{rate}}(s_n) \leq R \sup_{\theta \in \Theta} E_\theta|\hat{\theta}_n - \theta|^2$ we establish a Cramer-Rao type bound and, in case of a finite $\mathscr{X}$, prove asymptotic achievability of this bound under certain conditions. The proof involves a nonobvious combination of results (some of which are novel) for both coding and estimation.

Step-down testing procedures for the Hotelling $T^2$ problem are considered. It is shown that the classical step-down procedure proposed by S. N. Roy, R. E. Bargmann and J. Roy is inadmissible among invariant procedures (for a suitable invariance group) in many situations, including all those for which more than two steps are contemplated. It is also noted that in most cases, the power of the step-down procedure decreases in at least one of the noncentrality parameters over part of the parameter space. Finally, several alternative admissible step-down procedures are proposed.

We establish a new functional central limit theorem for empirical processes indexed by classes of functions. In a neighborhood of a fixed parameter point, an $n^{-1/3}$ rescaling of the parameter is compensated for by an $n^{2/3}$ rescaling of the empirical measure, resulting in a limiting Gaussian process. By means of a modified continuous mapping theorem for the location of the maximizing value, we deduce limit theorems for several statistics defined by maximization or constrained minimization of a process derived from the empirical measure. These statistics include the short, Rousseeuw's least median of squares estimator, Manski's maximum score estimator, and the maximum likelihood estimator for a monotone density. The limit theory depends on a simple new sufficient condition for a Gaussian process to achieve its maximum almost surely at a unique point.

Asymptotic properties of multivariate time series with characteristic roots on the unit circle are considered. For a vector autoregressive moving average (ARMA) process, we derive the limiting distributions of certain statistics which are useful in understanding nonstationary processes. These distributions are derived in a unified manner for all types of characteristic roots and are expressed in terms of stochastic integrals of Brownian motions. For applications, we use the limiting distributions to establish the consistency properties of the ordinary least squares (LS) estimates of various autoregressions of a vector process, e.g., the ordinary, forward and shifted autoregressions. For a purely nonstationary vector ARMA($p, q$) process, the LS estimates are consistent if the order of the fitted autoregression is $p$; for a general ARMA model, the limits of the LS estimates exist, but these estimates can only provide consistent estimates of the nonstationary characteristic roots.

Several algorithms for image reconstruction in positron emission tomography (PET) have been described in the medical and statistical literature. We study a continuous idealization of the PET reconstruction problem, considered as an example of bivariate density estimation based on indirect observations. Given a large sample of indirect observations, we consider the size of the equivalent sample of observations, whose original exact positions would allow equally accurate estimation of the image of interest. Both for indirect and for direct observations, we establish exact minimax rates of convergence of estimation, for all possible estimators, over suitable smoothness classes of functions. A key technical device is a modulus of continuity appropriate to global function estimation. For indirect data and (in practice unobservable) direct data, the rates for mean integrated square error are $n^{-p/(p + 2)}$ and $(n/\log n)^{-p/(p + 1)}$, respectively, for densities in a class corresponding to bounded square-integrable $p$th derivatives. We obtain numerical values for equivalent sample sizes for minimax linear estimators using a slightly modified error criterion. Modifications of the model to incorporate attenuation and the third dimension effect do not affect the minimax rates. The approach of the paper is applicable to a wide class of linear inverse problems.

Box and Leon, Shoemaker and Kackar have discussed the problem of closeness to target in quality engineering. If the mean response $f(x, z)$ depends on $(x, z)$, the variance function is a PERMIA if it is $g(z)$, i.e., depends only on $z$. The goal is to find $(x_0, z_0)$ which minimizes variance while achieving a target mean value. We pose and answer the question: For given smoothness assumptions about $f$ and $g$, how accurately can we estimate $x_0$ and $z_0$? As part of the investigation, we also find optimal rates of convergence for estimating $f, g$ and their derivatives.

We consider the semiparametric linear regression model with censored data and with unknown error distribution. We describe estimation equations of the Buckley-James type that admit $\sqrt n$-consistent and asymptotically normal solutions. The derived estimator is efficient at a particular error distribution. We show the equivalence between this type of estimator and an estimator based on a linear rank test suggested by Tsiatis. This equivalence is an extension of a basic equivalence between Doob type martingales and counting process martingales shown by Ritov and Wellner. An extension to an estimator that is efficient everywhere is discussed.

Techniques are developed for nonparametric analysis of data under a Cox-regression-like model permitting time-dependent covariate effects determined by a regression function $\beta_0(t)$. Estimators resulting from maximization of an appropriate penalized partial likelihood are shown to exist and a computational approach is outlined. Weak uniform consistency (with a rate of convergence) and pointwise asymptotic normality of the estimators are established under regularity conditions. A consistent estimator of a common baseline hazard function is presented and used to construct a consistent estimator of the asymptotic variance of the estimator of the regression function. Extensions to multiple covariates, general relative risk functions and time-dependent covariates are discussed.

A class of estimates for regression parameters in a linear model with right censored data is proposed. These estimates are derived by using linear rank tests for right censored data as estimating equations. They are shown to be consistent and asymptotically normal with covariance matrix for which estimates are proposed. Efficient estimates within this class are derived together with conditions when they are fully efficient.

Dose-response experiments are widely used in scientific research and nonmonotone dose-response curves are commonly observed. A class of nonparametric tests for dose-response curves subject to downturns at high doses is examined via its asymptotic properties. This class includes the well-known Jonckheere-Terpstra test as a limiting case. The analysis indicates that the Jonckheere-Terpstra test lacks robustness to a downturn in the dose-response, and that other members of the class provide superior overall performance. A result concerning $U$ statistics in locally asymptotically normal families of distributions facilitates the derivation of the asymptotic power function for the class of tests under consideration. This result may also be used to obtain asymptotic power functions for other tests based on $U$ statistics, for instance, the Mann-Whitney-Wilcoxon test. The accuracy of the asymptotic approximation is examined via Monte Carlo simulation and is found to be quite good for moderate sample sizes, suggesting that the approximation might reasonably be used for sample-size determinations.

Double censoring often occurs in collecting lifetime data and accordingly self-consistent estimators are widely employed for estimating the survival functions. In this paper we prove the weak convergence of self-consistent estimators using classical results on the Fredholm integral equation. Also, a method of calculating the asymptotic variance is presented.

For a distribution $F$ on $\mathbb{R}^p$ and a point $x$ in $\mathbb{R}^p$, the simplical depth $D(x)$ is introduced, which is the probability that the point $x$ is contained inside a random simplex whose vertices are $p + 1$ independent observations from $F$. Mathematically and heuristically it is argued that $D(x)$ indeed can be viewed as a measure of depth of the point $x$ with respect to $F$. An empirical version of $D(\cdot)$ gives rise to a natural ordering of the data points from the center outward. The ordering thus obtained leads to the introduction of multivariate generalizations of the univariate sample median and $L$-statistics. This generalized sample median and $L$-statistics are affine equivariant.

A smoothing spline estimator can be interpreted in two ways: either as the solution to a variational problem or as the posterior mean when a particular Gaussian prior is placed on the unknown regression function. In order to explain the remarkable performance of her Bayesian "confidence intervals" in a simulation study, Wahba conjectured that the average posterior variance of a spline estimate evaluated at the observation points will be close to the expected average squared error. The estimate of the average posterior variance proposed by Wahba is shown to converge in probability to a quantity proportional to the expected average squared error. This result is established by relating this statistic to a consistent risk estimate based on generalized cross-validation.

Let $\chi$ be a random probability measure chosen by a Dirichlet process on $(\mathbb{R}, \mathscr{B})$ with parameter $\alpha$ and such that $\int x\chi(dx)$ turns out to be a (finite) random variable. The main concern of this paper is the statement of a suitable expression for the distribution function of that random variable. Such an expression is deduced through an extension of a procedure based on the use of generalized Stieltjes transforms, originally proposed by the present authors in 1978.

We consider Bayes procedures for a location parameter $\theta$ that are robust with respect to the shape of the distribution $F$ of the data. The case where $F$ is fixed (nonrandom) and the case where $F$ has a Dirichlet distribution are both treated. The procedures are based on the posterior distributions of the location parameter given the partial information contained in a robust estimate of location. We show consistency and asymptotic normality of the procedures and give instances where the Bayes procedure based on the full sample diverges while the Bayes procedures based on partial information converges and is asymptotically normal. Finally, we show that robust confidence procedures can be given a Bayesian interpretation.

We show that the mathematical structure of belief functions makes them suitable for generating classes of prior distributions to be used in robust Bayesian inference. In particular, the upper and lower bounds of the posterior probability content of a measurable subset of the parameter space may be calculated directly in terms of upper and lower expectations (Theorem 4.1). We also extend an integral representation given by Dempster to infinite sets (Theorem 2.1).

In this paper we investigate the power function of the Studentised range test for comparing the means of normal populations in the one-way fixed effects analysis of variance model. The main results provide rigorous proofs of certain least favourable configurations of population means. These results are important in the calculation of the sample sizes required to guarantee power levels under certain restrictions on the ranges of the population means.