The first lecture in this series is devoted to a survey of contributions during the last five years to estimation of parameters by linear functions of observations in the Gauss-Markoff model. Some new results are also given. The classes of BLE (Bayes linear estimators) and ALE (admissible linear estimators) are characterized when the loss function is quadratic. It is shown that ALE's are either BLE's or limits of BLE's. Biased estimators like ridge and shrunken estimators are shown to be special cases of BLE's. Minimum variance unbiased estimation of parameters in a linear model is discussed with the help of a projection operator under very general conditions.

For each $t$ in some subinterval $T$ of the real line let $F_t$ be a distribution function with mean $m(t)$. Suppose $m(t)$ is concave. Let $t_1, t_2, \cdots$ be a sequence of points in $T$ and let $Y_1, Y_2, \cdots$ be an independent sequence of random variables such that the distribution function of $Y_k$ is $F_{t_k}$. We consider estimators $m_n(t) = m_n(t; Y_1, \cdots, Y_n)$ which are concave in $t$ and which minimize $\sum^n_{i=1} \lbrack m_n(t_i; Y_1, \cdots, Y_n) - Y_i\rbrack^2$ over the class of concave functions. We investigate their consistency and the convergence of $\{m_n'(t)\}$ to $m'(t)$.

De Finetti has defined coherent previsions and coherent probabilities, and others have described concepts of coherent actions or coherent decisions. Here we consider a related concept of coherent preferences. Willingness to accept one side of a bet is an example of a preference. A set of preferences is called incoherent if reversal of some subset yields a uniform increase in utility, as with a sure win for a collection of bets. In both probability and statistical models (where preferences are conditional on data) separating hyperplane theorems show that coherence implies existence of a probability measure from which the preferences could have been inferred. Relationships to confidence intervals and to decision theory are indicated. No single definition of coherence is given which covers all cases of interest. The various cases distinguish between probability and statistical models and between finite and infinite spaces. No satisfactory theory is given for continuous statistical models.

Let $X = (X_1, X_2, X_3)$ be a random vector with density $f(x - \theta)$, where $\theta = (\theta_1, \theta_2, \theta_3)$ is unknown. It is desired to estimate $(\theta_1, \theta_2)$ using an estimator $(\delta_1(X), \delta_2(X))$, and under a loss function $L(\delta_1 - \theta_1, \delta_2 - \theta_2)$. (Note that $\theta_3$ is a nuisance parameter.) Under certain conditions on $f$ and $L$, it is shown that the best invariant estimator of $(\theta_1, \theta_2)$ is inadmissible.

Given a procedure $\hat{\theta}^\ast$ for estimating the largest mean $\theta^\ast$ of $k$ normal populations with common known variance, it is desired to choose the common sample size $n$ so that the mean squared error (M.S.E.) of $\hat{\theta}^\ast$ does not exceed a given bound, $r$, regardless of the configuration of values of the $k$ means. Let $\Delta_1 \geqq \cdots \geqq \Delta_k = 0$ be the ordered values of $(\theta^\ast - \theta_i)$, where $\theta_1, \cdots, \theta_k$ are the unknown means. The M.S.E. depends on the $\Delta$'s and the conservative approach chooses a sample size $n^\ast$ to hold M.S.E. $\leqq r$ for all $\Delta$'s. Sequential and multisample procedures are considered which use sample information about the $\Delta$'s to reduce sample sizes. Asymptotic properties of the sample size and M.S.E. of the resulting estimates are developed. Improvements over using $n^\ast$ are possible, but with limitations. The sample size behavior of any $\hat{\theta}^\ast$ depends on the limiting variance of the estimator as all of $\Delta_1, \cdots, \Delta_{k-1}$ become infinite.

A well-known characteristic function representation of the family of symmetric stable distributions $\mathscr{F}$ indexes them with a location, scale, and type parameter. A sample of size $n$ is taken from an unknown member of $\mathscr{F}$. In this paper, an estimator of the location parameter is constructed which is maximum probability. This means that the estimator conventionally normalized converges in distribution to a normal distribution with zero mean and variance the inverse of the Fisher Information.

Gilliland and Hannan (1974, Section 3) consider a general finite state compact risk component and reduce the problem of treating the asymptotic excess compound risk of Bayes compound rules to the question of $L_1$ consistency of certain induced estimators. This present paper considers the two state case and for several classes of diffuse symmetric priors on proportions establishes the $L_1$ consistency with rate. The rate $O(n^{-\frac{1}{2}})$ uniform in state sequences is shown for the uniform prior giving strong affirmation to the asymptotic form of a conjecture by Robbins (1951). The same or logarithmically weakened rate is shown for symmetric priors which are $\Lambda$-mixtures for several classes of $\Lambda$. A corollary shows a nonnull consistency, without regularity conditions, of a maximum likelihood estimator.

Polynomial regression of degree $n$ on an interval is considered. Optimal designs $\xi_n$ are discussed for various optimality criteria. The behavior of $\xi_n$ for large $n$ is investigated and comparisons of $\xi_n$ with the limiting design $\xi_0$ are made.

For an optimality criterion function $\Phi$ and a design $\xi_n$, approximate $\Phi(\mathbf{M}(\xi))$ near $\xi_n$ by a quadratic Taylor expansion, let $\xi_n + \eta$ minimize this approximation, and let $\xi_{n+1} = \xi_n + \alpha\eta$, with $\alpha$ minimizing $\Phi(\mathbf{M}(\xi_{n+1}))$. If $\Phi$ satisfies regularity conditions, including strict convexity, possession of three continuous derivatives, and finiteness only for nonsingular $\mathbf{M}$, then $\mathbf{M}(\xi_n)$ converges to the optimal value for both the Federov steepest descent sequence and the above quadratic sequence, with the quadratic sequence having a faster asymptotic convergence rate. Methods are discussed for collapsing clusters of design points during the iterative process. In a simple example with $D$-optimality, the two methods are comparable. In a more complicated example the quadratic method is far superior.

Measures of dispersion are defined as functionals satisfying certain equivariance and order conditions. In the main part of the paper attention is restricted to symmetric distributions. Different measures are compared in terms of asymptotic relative efficiency, i.e., the inverse ratio of their standardized variances. The efficiency of a trimmed to the untrimmed standard deviation turns out not to have a positive lower bound even over the family of Tukey models. Positive lower bounds for the efficiency (over the family of all symmetric distributions for which the measures are defined) exist if the trimmed standard deviations are replaced by $p$th power deviations. However, these latter measures are no longer robust, although for $p < 2$ they are more robust than the standard deviation. The results of the paper suggest that a positive bound to the efficiency may be incompatible with robustness but that trimmed standard deviations and $p$th power deviations for $p = 1$ or 1.5 are quite satisfactory in practice.

Bayes factors against various hypotheses of independence are proposed for contingency tables and for multidimensional contingency tables. The priors assumed for the nonnull hypothesis are linear combinations of symmetric Dirichlet distributions as in some work of 1965 and later. The results can be used also for probability estimation. The evidence concerning independence, provided by the marginal totals alone, is evaluated, and preliminary numerical calculations suggest it is small. The possibility of applying the Bayes/non-Bayes synthesis is proposed because it was found useful for an analogous problem for multinomial distributions. As a spinoff, approximate formulae are suggested for enumerating "arrays" in two and more dimensions.

A variant of the general multivariate inclusion-exclusion formula of Meyer (1969) is derived for the case where $K$ classes of events are considered and specific subsets of the events, one from each class, are related to one another by set inclusion. This result, in turn, yields a formula for the cumulative distribution function of any subset of order statistics from dependent random variables in terms of cumulative distribution functions of subsets of the unordered variables. An important example of dependent random variables, where the variables are jointly distributed as a Dirichlet $D_n(1, 1, \cdots, 1)$, is discussed in detail; various authors' results for this distribution are extended, or rederived as special cases via the formulae presented.

The problem of estimating the compounding distribution of a compound Poisson process from independent observations of the compound process has been analyzed by Tucker (1963). A maximum likelihood method is proposed. The existence, uniqueness and convergence of the resulting estimator are derived. One obtains practical solutions by means of a very simple algorithm which is briefly described. A numerical example is presented in the risk business framework.

Consider $k$ stochastically ordered distributions with $F_{(1)} \leqq\cdots\leqq F_{(k)}$. The present paper deals with distribution-free tolerance intervals for $F_{(j)}$ based on order statistics in samples of same size from each of the $k$ distributions. Two criteria are defined for determining such intervals. These two criteria are extensions of $\beta$-expectation tolerance intervals and $\beta$-content tolerance intervals with confidence coefficient $\gamma$ used in the single population literature. A tolerance interval for the lifetime distribution of a series system is considered as an example.

The paper deals with continuous time Markov decision processes on a fairly general state space. The rewards are continuously discounted at rate $\alpha > 0$. A set of conditions is shown to be necessary and sufficient for a policy to be optimal. For the special case of time independent reward function and under the assumption that the action space is finite a policy improvement algorithm is proposed and its convergence to an optimal policy is proved.

Two people, 1 and 2, are said to have common knowledge of an event $E$ if both know it, 1 knows that 2 knows it, 2 knows that 1 knows is, 1 knows that 2 knows that 1 knows it, and so on. THEOREM. If two people have the same priors, and their posteriors for an event $A$ are common knowledge, then these posteriors are equal.

An improved version of the optimality property of sequential probability ratio tests is described. Alternatives to error probabilities, as measurements of precision, are suggested.

A necessary and sufficient condition for the asymptotic normality of linear combinations of order statistics from the uniform distribution over [0, 1] is derived. The condition implies, that the variances of all weighted extremes are asymptotically zero compared with the total variance of the sum, which conversely, in the case of nonnegative constants, is sufficient for the asymptotic normality.

Progressive censoring schemes are often adopted in clinical trials and life testing problems with a view to monitoring the experiment from the start with the objective of a possible early termination of the experiment depending on the cumulative evidence at its various stages. Along with a basic martingale property, a Wiener process approximation for progressively censored likelihood ratio statistics is established here and the same is incorporated in the formulation of some asymptotic sequential tests for the life testing problem.

In connection with a range of stationary time series models, particularly ARMAX models, recursive calculations of the parameter vector seem important. In these the estimate, $\theta(n)$, from observations to time $n$, is calculated as $\theta(n) = \theta(n - 1) + k_n$ where $k_n$ depends only on $\theta(n - 1), \theta(n - 2), \cdots$ and the data to time $n$. The convergence of two recursions is proved for the simple model $x(n) = \varepsilon(n) + \alpha\varepsilon(n - 1), |\alpha| < 1$, where the $\varepsilon(n)$ are stationary ergodic martingale differences with $E\{\varepsilon(n)^2\mid\mathscr{F}_{n-1}\} = \sigma^2$. The method of proof consists in reducing the study of the recursion to that of a recursion involving the data only through the $\theta(n)$. It seems that many of the recursions introduced for ARMAX models may be treated in this way and the nature of the extensions of the theory is discussed.

The Remez exchange procedures of approximation theory are used to find the optimal design for the problem of estimating $c'\theta$ in the regression model $EY(x) = \theta_1f_1(x) + \theta_2f_2(x) + \cdots + \theta_kf_k(x)$, when $c$ is not a linear combination of less than $k$ vectors of the form $f(x)$. A geometric approach is given first with a proof of convergence. When the design space is a closed interval, the Remez exchange procedure is illustrated by two examples. This type procedure can be used to find the optimal design very efficiently, if there exists an optimal design with $k$ support points.

Bivariate distributions with minimum and maximum correlations for given marginal distributions are characterized. Such extremal distributions were first introduced by Hoeffding (1940) and Frechet (1951). Several proofs are outlined including ones based on rearrangement theorems. The effect of convolution on correlation is also studied. Convolution makes arbitrary correlations less extreme while convolution of identical measures on $R^2$ makes extreme correlations more extreme. Extreme correlations have applications in data analysis and variance reduction in Monte Carlo studies, especially in the technique of antithetic variates.

Consider a modified version of the empirical Bayes decision problem where the component problems in the sequence are not identical in that the sample size may vary. In this case there is not a single Bayes envelope $R(\bullet)$, but rather a sequence of envelopes $R^{m(n)}(\bullet)$ where $m(n)$ is the sample size in the $n$th problem. Let $\mathbf{\theta} = (\theta_1, \theta_2, \cdots)$ be a sequence of i.i.d. $G$ random variables and let the conditional distribution of the observations $\mathbf{X}_n = (X_{n,1}, \cdots, X_{n,m(n)})$ given $\mathbf{\theta}$ be $(P_{\theta_n})^{m(n)}, n = 1, 2, \cdots$. For a decision concerning $\theta_{n+1}$, where $\theta$ indexes a certain discrete exponential family, procedures $t_n$ are investigated which will utilize all the data $\mathbf{X}_1, \mathbf{X}_2, \cdots, \mathbf{X}_{n+1}$ and which, under certain conditions, are asymptotically optimal in the sense that $E|t_n - \theta_{n+1}|^2 - R^{m(n+1)}(G) \rightarrow 0$ as $n \rightarrow \infty$ for all $G$.