Consider a $p$-times differentiable unknown regression function $\theta$ of a $d$-dimensional measurement variable. Let $T(\theta)$ denote a derivative of $\theta$ of order $m$ and set $r = (p - m)/(2p + d)$. Let $\hat{T}_n$ denote an estimator of $T(\theta)$ based on a training sample of size $n$, and let $\| \hat{T}_n - T(\theta)\|_q$ be the usual $L^q$ norm of the restriction of $\hat{T}_n - T(\theta)$ to a fixed compact set. Under appropriate regularity conditions, it is shown that the optimal rate of convergence for $\| \hat{T}_n - T(\theta)\|_q$ is $n^{-r}$ if $0 < q < \infty$; while $(n^{-1} \log n)^r$ is the optimal rate if $q = \infty$.

The role of functional models and their associated fiducial analysis is explored in an attempt to uncover a general theory of fiducial inference.

The introductory section of Thomas Bayes's famous essay on probability contains two arguments for what we now call the rule of conditioning. The first argument, which leads to Bayes's third proposition, can be made rigorous if we use rooted trees to represent the step-by-step determination of events. The second argument, which leads to Bayes's fifth proposition, does not stand up to scrutiny.

How do Bayesians justify using conjugate priors on grounds other than mathematical convenience? In the 1920's the Cambridge philosopher William Ernest Johnson in effect characterized symmetric Dirichlet priors for multinomial sampling in terms of a natural and easily assessed subjective condition. Johnson's proof can be generalized to include asymmetric Dirichlet priors and those finitely exchangeable sequences with linear posterior expectation of success. Some interesting open problems that Johnson's result raises, and its historical and philosophical background, are also discussed.

The Cox regression model for censored survival data specifies that covariates have a proportional effect on the hazard function of the life-time distribution of an individual. In this paper we discuss how this model can be extended to a model where covariate processes have a proportional effect on the intensity process of a multivariate counting process. This permits a statistical regression analysis of the intensity of a recurrent event allowing for complicated censoring patterns and time dependent covariates. Furthermore, this formulation gives rise to proofs with very simple structure using martingale techniques for the asymptotic properties of the estimators from such a model. Finally an example of a statistical analysis is included.

In this issue Andersen and Gill (hereafter AG) present a stimulating development of asymptotic distribution theory for the Cox regression model with time-dependent covariates. They use a counting process formulation for the failure time data and martingale covergence results. This approach involves such conditions as $\sigma$-algebra right continuity and predictable, locally bounded, covariate processes. In this commentary we consider the implications of such assumptions for likelihood factorization and covariate modeling. In particular, it is noted that the partial likelihood function modeled by AG cannot, in general, involve covariate measurements at the random failure times. Some related work by the authors on a partial likelihood function that may involve covariate values at the random failure times is briefly discussed. Assumptions under which the intensity process modeled by AG has a standard "hazard" function interpretation are described and some generalizations of the AG results are mentioned.

Let $\mathbf{X}$ be an absolutely continuous random variable in $\mathbb{R}^k$ with distribution function $F(\mathbf{x})$ and density $f(\mathbf{x})$. Let $\mathbf{X}_1, \cdots, \mathbf{X}_n$ be independent random variables distributed according to $F$. Mapping the spatial distribution of $\mathbf{X}$ normally entails drawing a map of the isopleths, or level curves, of $f$. In this paper, it is shown how to map the isopleths of $f$ nonparametrically according to the criterion of maximum likelihood. The procedure involves specification of a class $\mathscr{L}$ of sets whose boundaries constitute admissible isopleths and then maximizing the likelihood $\prod^n_{i = 1} g(\mathbf{x}_i)$ over all $g$ whose isopleths are boundaries of $\mathscr{L}$-sets. The only restrictions on $\mathscr{L}$ are that it be a $\sigma$-lattice and an $F$-uniformity class. The computation of the estimate is normally straightforward and easy. Extension is made to the important case where $\mathscr{L}$ may be data-dependent up to locational and/or rotational translations. Strong consistency of the estimator is shown in the most general case.

Admissibility problems involving simultaneous estimation in discrete exponential families are studied by solving difference inequalities. It is shown that if an estimator is admissible under the loss function $L_m(\mathbf{\theta, a)} = \sum^p_{i = 1} \theta^{m_i}_i (\theta_i - a_i)^2$, then in the tail (i.e., for large values of the observations), this estimator has to be less than certain bounds. Specific bounds, called Semi Tail Upper Bounds (STUB), are given here. These STUBs are not only of theoretical interest, but also are sharp enough that they establish many new results. Two of the most interesting ones are: (i) the establishment of Brown's conjecture concerning inadmissibility of some of the estimators proposed by Clevenson and Zidek (1975), and (ii) the establishment of inadmissibility of Hudson's (1978) estimator which improves upon the uniformly minimum variance unbiased estimator in Negative Binomial families.

Necessary and sufficient conditions for the existence of adaptive procedures for identification of one of several probability distributions or for testing a simple hypothesis against a simple alternative are obtained. By definition, adaptive procedures are required to exhibit the same asymptotic behavior for several parametric families as do the optimal (minimax) estimators for each of these families. The proofs are based on a multivariate version of Chernoff's theorem, providing asymptotic formulas for probabilities of large deviations for sums of i.i.d. random vectors. Some examples of adaptive procedures are considered, and the non-existence of such rules is established in certain situations.

Tests of fit that assume i.i.d. observations may be affected by dependence among the observations. This effect is studied for the Pearson chi squared test when the data form a stationary stochastic process. The general form of asymptotic distribution theory under the null hypothesis is outlined and examples are given. For testing fit to a specified normal law, it is shown that when observations come from a quite general class of Gaussian stationary processes, positive correlation among the observations is confounded with lack of normality.

We consider examples where it is possible to transform a contingency table and associated model into a form where it is easier to compute estimates and where closed-form estimates may be recognized. The motivating example comes from the Theory of Social Networks.

Generalizations of the arc sine laws are shown to provide insight into the operating characteristics of certain techniques for selecting models to fit a given data set, when the available models are nested. As a corollary, one sees that a popular technique may be expected to include about one superfluous parameter, even if the sample size is large.

NonGaussian linear processes are considered. It is shown that the phase of the transfer function can be estimated under broad conditions. This is not true of Gaussian linear processes and in this sense Gaussian linear processes are atypical. The asymptotic behavior of a phase estimate is determined. The phase estimates make use of bispectral estimates. These ideas are applied to a problem of deconvolution which is effective even when the transfer function is not minimum phase. A number of computational illustrations are given.

Let $Y_t$ be an autoregressive process satisfying $Y_t = \alpha_1 Y_{t - 1} + \alpha_2 Y_{t - d} + \alpha_3 Y_{t - d - 1} + e_t$, where $\{e_t\}^\infty_{t = 0}$ is a sequence of $\operatorname{iid}(0, \sigma^2)$ random variables and $d \geq 2$. Such processes have been used as parametric models for seasonal time series. Typical values of $d$ are 2, 4, and 12 corresponding to time series observed semi-annually, quarterly, and monthly, respectively. If $\alpha_1 = 1, \alpha_2 = 1, \alpha_3 = - 1$ then $\Delta_1\Delta_d Y_t = e_t$, where $\Delta_r Y_t$ denotes $Y_t - Y_{t - r}$. If $(\alpha_1, \alpha_2, \alpha_3) = (1, 1, - 1)$ the process is nonstationary and the theory for stationary autoregressive processes does not apply. A methodology for testing the hypothesis $(\alpha_1, \alpha_2, \alpha_3) = (1, 1, - 1)$ is presented and percentiles for test statistics are obtained. Extensions are presented for multiplicative processes, for higher order processes, and for processes containing deterministic trend and seasonal components.

We consider kernel estimation of a smooth density $f$ at a point, but depart from the usual approach in admitting an adaptive dependence of the sharpness of the kernels on the underlying density. Proportionally varying the bandwidths like $f^{-1/2}$ at the contributing readings lowers the bias to a vanishing fraction of the usual value, and makes for performance seen in well-known estimators that force moment conditions on the kernel (and so sacrifice positivity of the curve estimate). Issues of equivariance and variance stabilitization are treated.

In a heteroscedastic linear model, it is known that if the variances are a parametric function of the design, then one can construct an estimate of the regression parameter which is asymptotically equivalent to the weighted least squares estimate with known variances. We show that the same is true when the only thing known about the variances is that they are determined by an unknown but smooth function of the design or the mean response.

Consider the hypothesis $H_1: \theta_1 \geq \theta_2 \geq \cdots \geq \theta_k$ regarding a collection, $\theta_1, \theta_2, \cdots, \theta_k$, of unknown parameters. It is clear that this trend is reflected in certain possible parameter sets more than in others. A quantification of this notion of conformity to a trend is studied. Applications of the resulting theory to several order restricted hypothesis tests are presented.

Likelihood ratio statistics for (i) testing the homogeneity of a collection of multinomial parameters against the alternative which accounts for the restriction that those parameters are starshaped (cf. Shaked, Ann. Statist., 1979), and for (ii) testing the null hypothesis that this parameter vector is starshaped, are considered. For both tests the asymptotic distribution of the test statistic under the null hypothesis is a version of the Chi-bar-square distribution. Analogous tests on a collection of Poisson means are also found to have asymptotic Chi-bar-square distributions.

The statement that one region has more probability content than another with respect to all spherically symmetric (rotation invariant) distributions is a partial ordering among such regions. A simple geometric characterization of this ordering is given for star-shaped regions containing the origin. This characterization has several different interpretations. New techniques, inequalities, and examples are produced from this geometrical approach. The results have particular application to simultaneous confidence levels.

Assuming a multinomial sampling model, prior distributions are developed which can accept prior information about symmetry and independence in a two-way contingency table. Bayesian estimates for the cell probabilities are obtained from the posterior distributions which are attractive alternatives to the usual classical estimates when vague prior information about symmetry or independence is available.

The problem of $n$ rankings is considered and the asymptotic distributions of measures of concordance based on rank correlations are derived under the null model of complete randomness. The Bahadur efficiencies of the measures are computed. A matrix analysis then reveals the asymptotic distribution and superior efficiency of average Kendall tau. Some interpretation of the results is also made.

Since Robbins (1956), a large literature has evolved treating the empirical Bayes formulation of a sequence of decision problems. In this paper we look at the formulation and the asymptotic optimality criterion as a classical iid problem and a consistency property. With this point of view and the finite state component problem, we evoke classical results on Bayes procedures to prove a complete class theorem and to establish the asymptotic optimality of Bayes empirical Bayes procedures.

A method is given for constructing a confidence set having smallest expected measure within the class of invariant level $1 - \alpha$ confidence sets. The main assumptions are (i) that the invariance group acts transitively on the parameter space and also acts on the parametric function of interest, and (ii) that the measure satisfies a certain equivariance property. When the invariance group satisfies the conditions of the Hunt-Stein Theorem, the optimal invariant confidence set is shown to minimize the maximum expected measure among all level $1 - \alpha$ confidence sets. The method is applied in several estimation problems, including the GMANOVA problem.

The problem of regression design in the presence of correlated errors is considered. Under the assumption that derivative information is available on the response process, uniqueness results for optimal designs given in Eubank, Smith and Smith (1981) for the BLUE of the regression coefficient are extended to a wider class of processes. In the event that derivatives cannot be sampled, an asymptotic solution is developed and extended to the multiparameter setting.

Let $\alpha$ be a finite nonnull measure on $\mathbb{R}$, and let the random distribution function $F$ be distributed according to $\mathbb{D}_\alpha$, where $\mathbb{D}_\alpha$ is the Dirichlet process prior with parameter $\alpha$; see Ferguson (1973). This note points out that, almost surely, the tails of $F$ are much smaller than the tails of $\alpha$.

The difference between the average of the top $k$ out of $n$ order statistics and the population mean expressed in population standard deviation units is known as the selection differential. This paper obtains some nondegenerate limit laws for this quantity. The results are applied to the construction of tables used in testing for outliers.