This paper surveys recent results involving boundary crossing probabilities and related statistical applications. The first part is concerned with problems of sequential analysis, especially repeated significance tests and their application to sequential clinical trials involving survival data. The second part develops the probability theory motivated by the problems of Part 1. A method for computing first passage distributions of Brownian motion to linear boundaries is introduced and then modified to handle problems problems in discrete time and those involving nonlinear boundaries. The third part is concerned with fixed sample statistical problems, especially change-point problems, which involve boundary crossing probabilities. Examples are given of problems for which the methods of Part 2 appear adequate and of problems which require new methods.

Maximum likelihood estimators are obtained for multivariate components of variance models under the condition that the effect covariance matrix is positive semidefinite with a maximum rank. The rank of the estimator is random. The estimation procedure leads to a likelihood ratio test that the rank of the effect matrix is not greater than a given number against the alternative that the rank is not greater than a larger specified number. Linear structural relationship models and some factor analytic models can be put into this framework.

Under the common principal component model $k$ covariance matrices $\mathbf{\Sigma}_1,\cdots,\mathbf{\Sigma}_k$ are simultaneously diagonalizable, i.e., there exists an orthogonal matrix $\mathbf{\beta}$ such that $\mathbf{\beta'\Sigma_i\beta = \Lambda_i}$ is diagonal for $i = 1,\cdots, k$. In this article we give the asymptotic distribution of the maximum likelihood estimates of $\mathbf{\beta}$ and $\mathbf{\Lambda}_i$. Using these results, we derive tests for (a) equality of eigenvectors with a given set of orthonormal vectors, and (b) redundancy of $p - q$ (out of $p$) principal components. The likelihood-ratio test for simultaneous sphericity of $p - q$ principal components in $k$ populations is derived, and some of the results are illustrated by a biometrical example.

The confidence sets for a $q$-dimensional distribution studied in this paper have several attractive features: affine invariance, correct asymptotic level whatever the actual distribution may be, numerical feasibility, and a local asymptotic minimax optimality property. When dimension $q$ equals one, the confidence sets reduce to the usual Kolmogorov-Smirnov confidence bands, except that critical values are determined by bootstrapping.

Under the assumption of normal errors, confidence spheres for $p(p \geq 3)$ coefficients of a linear model centered at the positive part James-Stein estimators were recently proved, by Hwang and Casella, to dominate the usual confidence set with the same radius. In this paper, the same domination results are established under various spherically symmetric distributions. These distributions include uniform distributions, double exponential distributions, and multivariate $t$ distributions.

For Bayesian analysis, an attractive method of modelling uncertainty in the prior distribution is through use of $\varepsilon$-contamination classes, i.e., classes of distributions which have the form $\pi = (1 - \varepsilon)\pi_0 + \varepsilon q, \pi_0$ being the base elicited prior, $q$ being a "contamination," and $\varepsilon$ reflecting the amount of error in $\pi_0$ that is deemed possible. Classes of contaminations that are considered include (i) all possible contaminations, (ii) all symmetric, unimodal contaminations, and (iii) all contaminations such that $\pi$ is unimodal. Two issues in robust Bayesian analysis are studied. The first is that of determining the range of posterior probabilities of a set as $\pi$ ranges over the $\varepsilon$-contamination class. The second, more extensively studied, issue is that of selecting, in a data dependent fashion, a "good" prior distribution (the Type-II maximum likelihood prior) from the $\varepsilon$-contamination class, and using this prior in the subsequent analysis. Relationships and applications to empirical Bayes analysis are also discussed.

When a panel of experts is asked to provide some advice in the form of a group probability distribution, the question arises as to whether they should synthesize their opinions before or after they learn the outcome of an experiment. If the group posterior distribution is the same whatever the order in which the pooling and the updating are done, the pooling mechanism is said to be externally Bayesian by Madansky (1964). In this paper, we characterize all externally Bayesian pooling formulas and we give conditions under which the opinion of the group will be proportional to the geometric average of the individual densities.

This paper is concerned with model selection in time series analysis. An identification criterion is presented that is asymptotically equivalent to a Bayes decision rule. The discussion is conducted in the context of a general class of parametric time series models and consideration is given to the special case of order determination in autoregressive moving-average representations. Consistency of the criterion is proved.

A strongly dependent Gaussian sequence has a spectral density $f(x, \theta)$ satisfying $f(x, \theta) \sim |x|^{-\alpha(\theta)} L_\theta(x)$ as $x \rightarrow 0$, where $0 < \alpha(\theta) < 1$ and $L_\theta(x)$ varies slowly at 0. Here $\theta$ is a vector of unknown parameters. An estimator for $\theta$ is proposed and shown to be consistent and asymptotically normal under appropriate conditions. These conditions are satisfied by fractional Gaussian noise and fractional ARMA, two examples of strongly dependent sequences.

Let $X_t = \sum^\infty_{j=-\infty} c_jZ_{t-j}$ be a moving average process where the $Z_t$'s are iid and have regularly varying tail probabilities with index $\alpha > 0$. The limit distribution of the sample covariance function is derived in the case that the process has a finite variance but an infinite fourth moment. Furthermore, in the infinite variance case $(0 < \alpha < 2)$, the sample correlation function is shown to converge in distribution to the ratio of two independent stable random variables with indices $\alpha$ and $\alpha/2$, respectively. This result immediately gives the limit distribution for the least squares estimates of the parameters in an autoregressive process.

This paper is concerned with the problem of estimating the variable time delays of a signal arriving at an array of sensors. A procedure to estimate the parameters of a linear time delay model is proposed. The procedure compares the Fourier transforms at different frequencies (thereby taking the Doppler effect into consideration). Under regularity conditions, the estimate obtained is shown to be consistent and asymptotically normal. Simulations were carried out and the results were found to agree well with the theoretical results. The procedure was applied to the records of the Imperial Valley earthquake of October 15, 1979, as recorded by the El Centro differential array.

Estimation by the method of sieves for a semimartingale regression model introduced by Aalen (1980) is studied. It is of interest to estimate functions which describe the influence of the covariates over time. An estimator for these functions is introduced and conditions which ensure consistency of the estimator in $L^2$-norm are given. Applications to diffusion processes and point processes with censored data are also discussed.

Let $(X, Y)$ be a pair of random variables such that $X = (X_1,\cdots, X_J)$ ranges over $C = \lbrack 0, 1\rbrack^J$. The conditional distribution of $Y$ given $X = x$ is assumed to belong to a suitable exponential family having parameter $\eta \in \mathbb{R}$. Let $\eta = f(x)$ denote the dependence of $\eta$ on $x$. Let $f^\ast$ denote the additive approximation to $f$ having the maximum possible expected log-likelihood under the model. Maximum likelihood is used to fit an additive spline estimate of $f^\ast$ based on a random sample of size $n$ from the distribution of $(X, Y)$. Under suitable conditions such an estimate can be constructed which achieves the same (optimal) rate of convergence for general $J$ as for $J = 1$.

In the context of lifetesting, an asymptotically risk-efficient procedure for the estimation of the exponential mean lifetime is considered when the survival times of the units are subject to random censorship. The loss function is the sum of squared error due to estimation, cost of recruitment of the units, and cost of total time on test. Asymptotic properties of the sequential estimator and stopping time are described as the per unit cost of total time on test decreases to zero.

A class of empirical processes associated with $V$-statistics ($V$-empirical process) under random censoring, and a class of nonparametric estimators based on the corresponding quantile process are defined. The $V$-empirical process is the censored data analogue of the $U$-empirical process considered by Silverman (1976, 1983). The class of estimators is the analogue of the class of generalized $L$-statistics introduced by Serfling (1984) and it includes the results of Sander (1975). The weak convergence of the $V$-empirical process and the corresponding quantile process is obtained and, through that, the asymptotic behavior of the estimators is studied. Linear bounds for the Kaplan-Meier estimator near the origin are established. A number of examples are given, including the generalization of the Hodges-Lehmann estimator for estimating the treatment effect in the two-sample problem under random censoring. A measure of spread, a procedure for estimation in the two-way ANOVA model, and a modified version of the two-sample Hodges-Lehmann estimator, all of which are new even in the uncensored case, are proposed.

We prove a Donsker-type invariance principle for a nearest-neighbor-type conditional empirical process. As an application we show asymptotic normality of conditional quantiles and derive large-sample distribution-free tests and confidence bands for a conditional distribution function.

The performance of a sequence of estimators $\{T_n\}$ of $g(\theta)$ can be measured by its inaccuracy rate $-\lim \inf_{n\rightarrow\infty} n^{-1} \log \mathbb{P}_\theta(\|T_n - g(\theta)\| > \varepsilon)$. For fixed $\varepsilon > 0$ optimality of consistent estimators $\operatorname{wrt}$ the inaccuracy rate is investigated. It is shown that for exponential families in standard representation with a convex parameter space the maximum likelihood estimator is optimal. If the parameter space is not convex, which occurs for instance in curved exponential families, in general no optimal estimator exists. For the location problem the inaccuracy rate of $M$-estimators is established. If the underlying density is sufficiently smooth an optimal $M$-estimator is obtained within the class of translation equivariant estimators. Tail-behaviour of location estimators is studied. A connection is made between gross error and inaccuracy rate optimality.

Let $\mathscr{E}^n$ be a statistical experiment based on $n$ i.i.d. observations. We compare $\mathscr{E}^n$ with $\mathscr{E}^{n+r_n}$. The gain of information due to the $r_n$ additional observations is measured by the deficiency distance $\Delta (\mathscr{E}^n, \mathscr{E}^{n+r_n})$, i.e., the maximum diminution of the risk functions. We show that under general dimensionality conditions $\Delta(\mathscr{E}^n, \mathscr{E}^{n+r_n})$ is of order $r_n/n$. Further the behavior of $\Delta$ is studied and compared for asymptotically Gaussian experiments. We show that the information gain increases logarithmically. The Gaussian and the binomial family turn out to be--in some sense--opposite extreme cases, with the increase of information asymptotically minimal in the Gaussian case and maximal in the binomial.

First, an approach to an upper bound for the risk matrix of GLSE's is established when the information on the parameter space of the structural parameter in the covariance matrix of the error can be utilized. Second, this result is applied to regression with (i) serial correlation and (ii) heteroscedastic covariance structure. In the heteroscedastic regression, the problem of estimating the common mean of two normal populations is studied in detail.

Motivated by the apportionment of diversity analysis due to C. R. Rao, a general approach to comparing populations of rankers is proposed. Each permutation metric corresponds to a particular population characteristic that forms the basis of the comparison. Tests of hypotheses concerning equality of characteristics are developed. Throughout, comparison is made with earlier work, most of which is based on the use of only the Spearman metric. Extension to tied rankings is discussed. Examples for two groups are presented which illustrate the computational feasibility as well as the value of the proposed procedures.

The univariate weak convergence theorem of Murota and Takeuchi (1981) is extended for the Mahalanobis transform of the $d$-variate empirical characteristic function, $d \geq 1$. Then a maximal deviation statistic is proposed for testing the composite hypothesis of $d$-variate normality. Fernique's inequality is used in conjunction with a combination of analytic, numerical analytic, and computer techniques to derive exact upper bounds for the asymptotic percentage points of the statistic. The resulting conservative large sample test is shown to be consistent against every alternative with components having a finite variance. (If $d = 1$ it is consistent against every alternative.) Monte Carlo experiments and the performance of the test on some well-known data sets are also discussed.

We exhibit those distributions with minimum Fisher information for location in various Kolmogorov neighbourhoods $\{F|\sup_x|F(x) - G(x)| \leq \varepsilon\}$ of a fixed, symmetric distribution $G$. The associated $M$-estimators are then most robust (in Huber's minimax sense) for location estimation within these neighbourhoods. The previously obtained solution of Huber (1964) for $G = \Phi$ and "small" $\varepsilon$ is shown to apply to all distributions with strongly unimodal densities whose score functions satisfy a further condition. The "large" $\varepsilon$ solution for $G = \Phi$ of Sacks and Ylvisaker (1972) is shown to apply under much weaker conditions. New forms of the solution are given for such distributions as "Student's" $t$, with nonmonotonic score functions. The general form of the solution is discussed.

The problem of deciding the signs of $k$ parameters $(\theta_1, \cdots, \theta_k) \equiv \mathbf{theta}$ based on $(\hat{\theta}_1, \cdots, \hat{\theta}_k) \sim N(\mathbf{\theta,\Sigma})$ such that $p_\mathbf{\theta} \{$\text{any error$\} \leq \alpha \forall \mathbf{\theta}$ is discussed by Bohrer and Schervish (1980). They characterize a desirable class of procedures called locally optimal. For the case $k = 2, \mathbf{\Sigma = I}$, and $\alpha \leq \frac{1}{3}$, they present a particular rule from this class called the double cross. In this paper, we address the problem of selecting a best rule from among all locally optimal rules when $k = 2$ and $\mathbf{\Sigma = I}$. When $\alpha \leq \frac{1}{3}$, the double cross is shown to be an attractive choice. Other rules are obtained for higher values of $\alpha$. We also examine a more general optimization criterion than the one used by Bohrer and Schervish and obtain different optimal rules for several classes of problems. The optimal rule corresponding to one of these classes has no two-decision region. A modification of the formulation is offered under which a well-known rule (with two decision regions) emerges as the unique optimal procedure.

A necessary and sufficient condition is given for a specified factorial effect to be orthogonal to every other factorial effect, after adjustment is made for blocks. The results are extended to the case of regular disconnected designs. The structure of a generalized inverse of the intrablock matrix is investigated when certain pairs of factorial spaces are orthogonal. A useful class of designs exhibiting partial orthogonal factorial structure is identified and examples are given.

If $S(x_1, x_2,\cdots, x_n)$ is any function of $n$ variables and if $X_i, \hat{X}_i, 1 \leq i \leq n$ are $2n$ i.i.d. random variables then $\operatorname{var} S \leq \frac{1}{2} E \sum^n_{i=1} (S - S_i)^2$ where $S = S(X_1, X_2,\cdots, X_n)$ and $S_i$ is given by replacing the $i$th observation with $\hat{X}_i$, so $S_i = S(X_1, X_2,\cdots, \hat{X}_i,\cdots, X_n)$. This is applied to sharpen known variance bounds in the long common subsequence problem.

Two Pearson-type goodness-of-fit test statistics for parametric families are considered for randomly right-censored data. Asymptotic distribution theory for the test statistics is based on the result that the product-limit process with MLE for nuisance parameters converges weakly to a Gaussian process. The Chernoff-Lehmann (1954) result extends to a generalized Pearson statistic. A modified Pearson statistic is shown to have a limiting chi-square null distribution.

The kernel estimator is a widely used tool for the estimation of a density function. In this paper its adaptation to censored data using the Kaplan-Meier estimator is considered. Asymptotic properties of four estimators, arising naturally as a result of considering various types of bandwidths, are investigated. In particular we show that (i) both proposed estimators stemming from the nearest neighbor estimator have censoring-free variances and (ii) one of them is pointwise mean consistent.

A Cramer type large deviation theorem for signed linear rank statistics under the symmetry hypothesis is obtained. The theorem is proved for a wide class of scores covering most of the commonly used ones (including the normal scores). Furthermore, the optimal range $0 < x \leq o(n^{1/4})$ can be obtained for bounded scores, whereas the range $0 < x \leq o(n^\delta), \delta \in (0, \frac{1}{4})$ is obtainable for many unbounded ones. This improves the earlier result under the symmetry hypothesis in Seoh, Ralescu, and Puri (1985).