In various settings, the observation of a stochastic process at a finite number of locations leads to natural prediction and design questions. General problems of this type are introduced and then related to specific areas of application. A class of processes called G-MAPs is studied with reference to their predictive and other behavior. These processes include many familiar ones and, through being tied to Markov processes, allow a fresh view of prediction. Among other things, G-MAPs stand as reasonably workable possibilities for Bayesian priors in some complex contexts.

For the general linear model with independent errors, we propose and examine the large sample properties of an estimator of the regression parameter. In the location model, the estimator has the same properties as the trimmed mean and the robustness and efficiency properties of the trimmed mean carry over to the general model. The estimator depends on a preliminary estimate of the regression parameter and the residuals based on it. The properties of the adaptive estimator with data-determined trimming proportions are also investigated.

Limit distributions of prediction errors when the parameters are estimated are obtained for a general class of nonstationary processes that includes supercritical branching processes and explosive autoregressive processes as typical examples. The estimated prediction errors are used to construct new tests of fit whose limit distributions are also derived.

A new sieve estimator for the mean function $m(t)$ of a general Gaussian process of known covariance is presented. The estimator $\hat{m}(t)$ is given explicitly from the data and has a simple distribution. It is shown that $\hat{m}(t)$ is asymptotically unbiased and consistent (weakly and in mean square) at each $t$, and that $\hat{m}$ is strongly consistent for $m$ in an appropriate norm. No assumptions are made about the "time" parameter or the covariance.

For the analysis of nonstationary categorical time series, a parsimonious and flexible class of models is proposed. These models are generalizations of regression models for stochastically independent categorical observations. Consistency, asymptotic normality and efficiency of the maximum likelihood estimator are shown under weak and easily verifiable requirements. Some models for binary time series are discussed in detail. To demonstrate asymptotic properties, a theorem is given addressing maximum likelihood estimation for general stochastic processes. Then it is shown that the assumptions of this theorem are consequences of the requirements for categorical time series. For this proof some lemmas are used which may be of interest in similar cases.

This paper addresses the problem of linear regression estimation when the disturbances follow a stationary process with its spectral density known only to be in a neighborhood of some specified spectral density, for instance, that of white noise. Rather than trying to adapt to a small unspecified autocorrelation, we follow here the robustness approach, and establish the extent of the regressors and disturbance spectra interaction which require serial correlation correction. We consider a class of generalized least-squares estimates, and find the estimator in this class which optimally robustifies the least-squares estimator against serial correlation. The estimator, when considered in the frequency domain, is of a form of weighted least squares with the most prominent frequencies of the regression spectrum being downweighted in a way similar to Huber's robust regression estimator.

We consider the estimation problem for the parameter $\vartheta_0$ of a stationary ARMA $(p, q)$ process, with independent and identically, but not necessary normally distributed errors. First we prove local asymptotic normality (LAN) for this model. Then we construct locally asymptotically minimax (LAM) estimators, which asymptotically achieve the smallest possible covariance matrix. Utilizing these, we finally obtain strongly adaptive estimators, by using usual kernel estimators for the score function $\dot{\varphi} = -f'/2 f$, where $f$ denotes the density of the error distribution. These estimates turn out to be asymptotically optimal in the LAM sense for a wide class of symmetric densities $f$.

This paper is concerned with establishing a broad class of estimators (and the limiting behavior, thereof) for parametrizations of higher than second-order structure. This includes parametrizations which reflect, for example, such properties as nonlinearity and/or non-Gaussianity and/or time irreversibility. Asymptotic distributions, almost-sure convergence, and probability-one bounds for such estimators are established. Several applications of such estimators are discussed.

In the setting of nonparametric multivariate density estimation, theorems are established which allow a comparison of the Kullback-Leibler and the least-squares cross-validation methods of smoothing parameter selection. The family of delta sequence estimators (including kernel, orthogonal series, histogram and histospline estimators) is considered. These theorems also show that either type of cross validation can be used to compare different estimators (e.g., kernel versus orthogonal series).

In the setting of kernel density estimation, data-driven bandwidth, i.e., smoothing parameter, selectors are considered. It is seen that there is a well-defined, and surprisingly restrictive, bound on the rate of convergence of any automatic bandwidth selection method to the optimum. The method of least squares cross-validation achieves this bound.

In the model $Y_i = g(t_i) + \varepsilon_i,\quad i = 1,\cdots, n,$ where $Y_i$ are given observations, $\varepsilon_i$ i.i.d. noise variables and $t_i$ nonrandom design points, kernel estimators for the regression function $g(t)$ with variable bandwidth (smoothing parameter) depending on $t$ are proposed. It is shown that in terms of asymptotic integrated mean squared error, kernel estimators with such a local bandwidth choice are superior to the ordinary kernel estimators with global bandwidth choice if optimal bandwidths are used. This superiority is maintained in a certain sense if optimal local bandwidths are estimated in a consistent manner from the data, which is proved by a tightness argument. The finite sample behavior of a specific local bandwidth selection procedure based on the Rice criterion for global bandwidth choice [Rice (1984)] is investigated by simulation.

By the term $k$-statistic or polykay, we mean an unbiased estimate of a cumulant or product of cumulants [Fisher (1929) and Tukey (1950), 1956)]. In this paper, two sets of unbiased estimates are given for the case where the mean response, $E(Y)$, depends linearly on known covariates $x$. The $k$'s are symmetric functions of the least-squares residuals and have previously been discussed by Anscombe (1961; 1981, Appendix 2). The $l$'s are optimal in the sense of having minimum variance under the ideal assumption of normality [Pukelsheim (1980)]. The emphasis here on computability leads to the algebraic inversion of direct product matrices of order $n^3 \times n^3$ and $n^4 \times n^4$, a computation that is rarely feasible numerically, even on the fastest computers. This algebra leads to simple straightforward formulae for all statistics up to degree four. Conditions are given under which the $k$'s are nearly or asymptotically optimal in the sense of being asymptotically equivalent to the corresponding $l$'s. A small-scale simulation study provides a comparison between these statistics for finite $n$. An application to detecting heterogeneity of variance, avoiding the assumption of normality, is given. A new test statistic for detecting systematic dispersion effects is introduced and compared to existing ones. Two examples illustrate the methodology.

It is well-known that the ordinary least squares (OLS) estimator $\hat{\beta}$ of the slope and intercept parameters $\beta$ in a linear regression model with errors of measurement for some of the independent variables (predictors) is inconsistent. However, Gallo (1982) has shown that certain linear combinations of $\beta$. In this paper, it is shown that under reasonable regularity conditions such linear combinations of $\hat{\beta}$ are (jointly) asymptotically normally distributed. Some methodological consequences of our results are given in a companion paper (Carroll, Gallo and Gleser (1985)).

The existence and uniqueness of a limiting form of a Huber-type $M$-estimator of multivariate scatter is established under certain conditions on the observed sample. These conditions hold with probability one when sampling randomly from a continuous multivariate distribution. The existence of the estimator is proven by showing that it is the limiting point of a specific algorithm. Hence, the proof is constructive. For continuous populations, the estimator of multivariate scatter is shown to be strongly consistent and asymptotically normal. An important property of the estimator is that its asymptotic distribution is distribution-free with respect to the class of continuous elliptically distributed populations. This distribution-free property also holds for the finite sample size distribution when the location parameter is known. In addition, the estimator is the "most robust" estimator of the scatter matrix of an elliptical distribution in the sense of minimizing the maximum asymptotic variance.

Consider the problem of estimating simultaneously the means $\theta_i$ of independent normal random variables $x_i$ with unit variance. Under the weighted quadratic loss $L(\theta, a) = \sum_i\lambda_i(\theta_i - a_i)^2$ with positive weights it is well known that: (1) An estimator which is admissible under one set of weights is admissible under all weights. (2) Estimating individual coordinates by proper Bayes estimators results in an admissible estimator. (3) Estimating individual coordinates by admissible estimators may result in an inadmissible estimator, when the number of coordinates is large enough. A dominating estimator must link observations in the sense that at least one $\theta_i$ is estimated using observations other than $x_i$. We consider an infinite model with a countable number of coordinates. In the infinite model admissibility does depend on the weights used and by linking coordinates it is possible to dominate even estimators which are proper Bayes for individual coordinates. Specifically, we show that when $\theta_i$ are square summable, the estimator $\delta_i(x) \equiv 1$ is admissible for $\lambda_i = e^{-ic}, c > \frac{1}{2}$, but inadmissible for $\lambda_i = 1/i^{1+c}, c > 0$. In the latter case, a dominating estimator $\pi = (\pi_1, \pi_2, \cdots)$ is of the form $\pi_i(x) = 1 - \varepsilon_i(x)$, where $\varepsilon_i$ links all the observations $x_1, x_2, \cdots$. Infinite models frequently arise in estimation problems for Gaussian processes. For example, in estimating the drift function $\theta$ of the Wiener process $W$ under the loss $L(\theta, a) = \int\lbrack\theta(t) - a(t)\rbrack^2 dt$, the transformation $x_i = \int \Phi_i dW$ with $\Phi_i$ an appropriate complete orthonormal sequence gives rise to a model which is equivalent to an infinite model with $\lambda_i = 1/i^2$.

In Meeden and Ghosh (1983) a theory was developed for choosing, possibly at random, from a group of experiments the one to be observed. Here we consider the problem when the class of possible designs is restricted to a subclass of all designs. A theorem which identifies some admissible decision procedures and generalizes the early work is proved. Some applications to finite population sampling are discussed.

Let $x_1, x_2, \cdots$ be independent and normally distributed with unknown mean $\theta$ and variance 1. Let $\tau = \inf \{n \geq 1: |s_n| \geq \sqrt{2a(n + c)}\}$. Then a repeated significance test for a normal mean rejects the hypothesis $\theta = 0$ if and only if $\tau \leq N_0$ for some positive integer $N_0$. The problem we consider is estimation of $\theta$ based on the data $x_1,\cdots, x_T, T = \min\{\tau, N_0\}$. We shall solve this problem by obtaining the asymptotic expansion of the distribution of $(s_\tau - \tau\theta)/\sqrt{\tau}$ as $a \rightarrow \infty$, and then constructing the confidence intervals for $\theta$.

Asymptotic moments of normed sample size savings are derived for one-sided curtailed nonparametric tests for symmetry for testing location shift hypotheses under the null hypothesis and contiguous alternatives. The contiguous alternative results are derived using Feller's (1943) central limit theorem result for large deviations and techniques used in Albers, Bickel and van Zwet (1976).

When the $\log$-likelihood statistic is divided by its mean, or an approximation to its mean, the limiting chi-squared distribution is often correct to order $n^{-3/2}$. Similarly, when the signed version of the likelihood ratio statistic is standardized with respect to its mean and variance the normal approximation is correct to order $n^{-3/2}$. Proofs for these statements have been given in great generality in the literature for the case of continuous observations. In this paper we consider cases where the minimal sufficient statistic is partly discrete and partly continuous. In particular, we consider testing problems associated with censored exponential life times.

Estimates of the linear model parameters in a linear transformation model with unknown increasing transformation are obtained by maximizing a partial likelihood. A resampling scheme (likelihood sampler) is used to compute the maximum partial likelihood estimates. It is shown that for a certain "local" parameter set where the "signal to noise ratio" is small, it is asymptotically possible to estimate the linear model parameters using the partial likelihood as well as if the transformation were known. In the case of the power transformation model with symmetric error distribution, this result is shown to also hold when the distribution of the error in the transformed linear model is unknown and is estimated. Monte Carlo results are used to show that for moderate sample size and small to moderate signal to noise ratio, the asymptotic results are approximately in effect and thus the partial likelihood estimates perform very well. Estimates of the transformation are introduced and it is shown that the estimates, when centered at the transformation and multiplied by $\sqrt{n}$, converge weakly to Gaussian processes.

Differential geometric methods are used to construct approximate confidence regions for curved exponential families. The $\alpha$-connection geometries discussed by Amari (1982), and another geometry introduced here, the $c$ geometry, are exploited to construct confidence regions. Survival and case-control studies with general relative risk functions are interpreted in the context of curved exponential families, and an example illustrates the construction of confidence regions for matched case-control studies. Simulations indicate that the geometric procedures have good coverage and power properties.

An asymptotic justification of the Bayesian bootstrap is given. Large-sample Bayesian bootstrap probability intervals for the mean, the variance and bands for the distribution, the smoothed density and smoothed rate function are also provided.

The classes of designs produced by well-known confounding methods are compared for single replicate factorial experiments involving a single blocking factor. It is shown that several of these methods produce equivalent classes of designs, whilst others produce a subset of this common class. It is shown that in certain experimental settings the use of pseudofactors does not enlarge the size of the common class of designs.

For testing the goodness-of-fit of a $\log$-linear model to a multi-way contingency table with cell proportions estimated from survey data, Rao and Scott (1984) derived a first-order correction, $\delta\ldot$, to Pearson chi-square statistic, $X^2$ (or the likelihood ratio statistic, $G^2$) that takes account of the survey design. It was also shown that $\delta\ldot$ requires the knowledge of only the cell design effects (deffs) and the marginal deffs provided the model admits direct solution to likelihood equations under multinomial sampling. Simple upper bounds on $\delta\ldot$ are obtained here for models not admitting direct solutions, also requiring only cell deffs and marginal deffs or some generalized deffs not depending on any hypothesis. Applicability of an $F$-statistic used in GLIM to test a nested hypothesis is also investigated. In the case of a logit model involving a binary response variable, simple upper bounds on $\delta\ldot$ are obtained in terms of deffs of response proportions for each factor combination or some generalized deffs not depending on any hypothesis. Applicability of the GLIM $F$-statistic for nested hypotheses is also studied.

Consider statistics $T_n(\lambda)$ that take the form of limiting chi-square (degenerate) $U$ or $V$ statistics. Here the phrase "limiting chi-square" means they have the same asymptotic distribution as a weighted sum of (possibly infinitely many) independent $\chi^2_1$ random variables. This paper examines the limiting distribution of $T_n(\hat{\lambda})$ and compares it to that of $T_n(\lambda)$, where $\hat{\lambda}$ denotes a consistent estimator of $\lambda$ based on the same data. Whether or not $T_n(\hat{\lambda})$ and $T_n(\lambda)$ have the same limiting distribution is primarily a question of whether or not a certain mean function has a zero derivative. Some statistics that are appropriate for testing hypotheses are used to illustrate the theory.

The Shapiro-Wilk statistic and its modifications are widely applied in tests for normality. We establish the asymptotic equivalence of a class of statistics based on different choices of normal scores. In particular, we conclude that the Shapiro-Francia, Filliben, Weisberg-Bingham and de Wet-Venter versions of the statistic are asymptotically equivalent. Our results also apply to the Type I and Type II censored data cases.

Under a model of random censorship, we consider the test $H_0$: a life distribution is exponential, versus $H_1$: it is new better than used, but not exponential. This paper introduces a class of tests by using the Kaplan-Meier estimator for the sample distribution in the uncensored model. Under some regularity conditions, the asymptotic normality of statistics is derived by an application of von Mises' method, and asymptotically valid tests are obtained by using estimators for the null standard deviations. The efficiency loss in the proportional censoring model is studied and a Monte Carlo study of power is performed.

Hajek's representation theorem states that under certain regularity conditions the limiting distribution of an estimator can be written as the convolution of a certain normal distribution with some other distribution. This result, originally developed for finite dimensional problems, has been extended to a number of infinite dimensional settings where it has been used, for example, to establish the asymptotic efficiency of the Kaplan-Meier estimator. The purpose of this note is to show that the somewhat unintuitive regularity condition on the estimators that is usually used can be replaced by a simple one: It is sufficient for the asymptotic information and the limiting distribution of the estimator to vary continuously with the parameter being estimated.

For jointly distributed random variables $(X, Y)$ having marginal distributions $F$ and $G$ with finite second moments and $F$ continuous, the proportion of $\operatorname{Var}(Y)$ explained by linear regression is $\lbrack\operatorname{Corr}(X, Y)\rbrack^2$ while the proportion explained by $E(Y \mid X)$ can be arbitrarily near 1. However, if the true regression, $E(Y\mid X)$, is monotone, then the proportion of $\operatorname{Var}(Y)$ it explains is at most $\operatorname{Corr}\lbrack Y, G^{-1}(F(X))\rbrack$.

Principles of multiple comparisons with the best treatment are used to construct a sequential procedure for selecting one of $k$ normal populations with equal, known variance. The procedure makes a stronger confidence than traditional Indifference Zone procedures, without (asymptotically) inflating sample sizes. It is shown to be substantially superior in asymptotic sample size behavior to existing procedures making the stronger statement.

Random variables $X_i \sim N(\theta_i, 1), i = 1,2,\cdots, k$, are observed. Suppose $X_S$ is the largest observation. If the inference $\theta_S > \max_{i\neq S}\theta_i$ is made whenever $X_S - \max_{i\neq S}X_i > c$, then the probability of a false inference is maximized when two $\theta_i$ are equal and the rest are $-\infty$. Equivalently, the inference can be made whenever a two-sample two-sided test for difference of means, based on the largest two observations, would reject the hypothesis of no difference. The result also holds in the case of unknown, estimable, common variance, and in fact for location families with monotone likelihood ratio.

We consider the problem of finding $\alpha$-level confidence intervals for the ratio of two normally estimated means. We show that there is no procedure that with probability 1 gives bounded $\alpha$-level confidence intervals for the ratio, and we show that within a large class of sensible procedures the Fieller solution is the only one with exact coverage probability.