The history of mathematical statistics in the United States prior to 1885 is reviewed, with emphasis upon the works of Robert Adrain, Benjamin and Charles Peirce, Simon Newcomb, and Erastus De Forest. While the period before 1850 produced little of substance, the years from 1850 to 1885 saw such innovations as an outlier rejection procedure, randomized design of experiments, elicitation of personal probabilities, kernel estimation of density functions, an anticipation of sufficiency, a runs test for fit, a Monte Carlo study, optimal linear smoothing, and the fitting of gamma distributions by the method of moments. Reasons for the rapid acceleration in the growth of the field are explored.

We study the asymptotic power functions of tests for heteroscedasticity and nonlinearity in the linear model which were proposed by Anscombe and introduce and study some competitors robust against gross errors.

A nonparametric minimum Hellinger distance estimator of location is introduced and shown to be asymptotically efficient at every symmetric density with finite Fisher information. Under small, possibly asymmetric, perturbations in such a density, the estimator is asymptotically robust in a technical sense which extends Hajek's concept of "regularity." A numerical example illustrates the computational feasibility of the estimator and its resistance to an arbitrary single outlier.

Let $T_n$ be an $M$-estimator with defining function $\psi$ and preliminary estimate of scale $s_n$. Without loss of generality, let $s_n \rightarrow 1$ and take $E\psi(X/\xi) = 0$. Under various conditions, it is shown that any consistent version of $T_n$ is almost surely to order $O(n^{-1} \log_2 n)$ a linear combination of $n^{-1} \sum^n_1 \psi(X_i)$ and $s_n$. Only in the case $EX_1\psi'(X_1) = 0$ does the contribution of $S_n$ vanish; it is shown how this affects the estimation of the asymptotic variance of $T_n$.

This paper lays the foundation for a general theory of weakly admissible tests, where a test consists of both an experiment (a $\sigma$-field of information) and a decision function based on the experiment. A test is weakly admissible unless there is another test with as good a decision function, which uses no more information, and is strictly better in one of these respects. Weakly admissible tests are completely characterized. In the case where the experiment consists of a sequential sampling procedure, a method for constructing such tests is given.

A decision maker is seen to be coherent in the sense of de Finetti if, and only if, his probabilities are computed in accordance with some finitely additive prior. If a bounded loss function is specified, then a decision rule is extended admissible (i.e., not uniformly dominated) if and only if it is Bayes for some finitely additive prior. However, if an improper countably additive prior is used, then decisions need not cohere and decision rules need not be extended admissible. Invariant, finitely additive priors are found and their posteriors calculated for a class of problems including translation parameter problems.

The asymptotic form of the Bayes acceptance region is derived for testing simple null-hypotheses in multiparameter exponential families. This result suggests a reasonable definition for tests which might be called "almost-Bayes". The rate at which the risk of the Bayes test converges to zero is obtained, showing the nature of its dependence on the prior distribution and providing a basis for comparison of almost-Bayes procedures. Concluding remarks contain a brief discussion of some asymptotic consequences of poor prior guessing.

There are two important spaces connected with every multivariate exponential family, the natural parameter space and the expectation parameter space. We describe some geometric results relating the two. (In the simplest case, that of a normal translation family, the two spaces coincide and the geometry is the familiar Euclidean one.) Maximum likelihood estimation, within one-parameter curved subfamilies of the multivariate family, has two simple and useful geometric interpretations. The geometry also relates to the Fisherian question: to what extent can the Fisher information be replaced by $-\partial^2/\partial\theta^2\lbrack\log f_\theta(x)\rbrack\mid_{\theta=\hat{\theta}}$ in the variance bound for $\hat{\theta}$, the maximum likelihood estimator?

Families of minimax estimators are found for the location parameter of a $p$-variate $(p \geqq 3)$ spherically symmetric unimodal distribution with respect to general quadratic loss. The estimators of James and Stein, Baranchik, Bock and Strawderman are all considered for this general problem. Specifically, when the loss is general quadratic loss given by $L(\delta, \theta) = (\delta - \theta)'D(\delta - \theta)$ where $D$ is a known $p \times p$ positive definite matrix, one main result, for one observation, $X$, on a multivariate s.s.u. distribution about $\theta$, presents a class of minimax estimators whose risk dominate the risk of $X$, provided $p \geqq 3$ and trace $D \geqq 2d_L$ where $d_L$ is the maximum eigenvalue of $D$. This class is given by $\delta_{a,r}(X) = (1 - a(r(\|X\|^2)/\|X\|^2)) X$ where $0 \leqq r(\bullet) \leqq 1, r(\|X\|^2)$ is nondecreasing, $r(\|X\|^2)/\|X\|^2$ is nonincreasing, and $0 \leqq a \leqq (c_0/E_0(\|X\|^{-2}))((\operatorname{trace} D/d_L) - 2)$, where $c_0 = 2p/((p + 2)(p - 2))$ when $p \geqq 4$ and $c_0 \approx .96$ when $p = 3$.

Assuming only the existence of the third absolute moment we prove that $\sup_x |P(\sigma_n^{-1} U_n \leqq x) - \Phi (x)| \leqq C_{\nu_3\sigma_g}^{-3}n^{-\frac{1}{2}}$ where $U_n$ is a $U$-statistic. This concludes a series of investigations on the Berry-Esseen theorem for $U$-statistics by Grams and Serfling, Bickel, and Chan and Wierman.

Limit processes are obtained for the sequences of partial sums of polynomial regression residuals. Properties of linear and quadratic functionals on the sequences are discussed. Distribution theory for Cramer-von Mises type functionals is obtained. An indication is given of the relevance of these results to the problem of testing for change of regression at unknown times.

Let $\{Y_n\}_{n\geqq 1}$ be a sequence of i.i.d. $m$-dimensional random vectors, and let $f_1,\cdots, f_k$ be real-valued Borel measurable functions on $R^m$. Assume that $Z_n = (f_1(Y_n),\cdots, f_k(Y_n))$ has finite moments of order $s \geqq 3$. Rates of convergence to normality and asymptotic expansions of distributions of statistics of the form $W_n = n^{\frac{1}{2}}\lbrack H(\bar{Z}) - H(\mu)\rbrack$ are obtained for functions $H$ on $R^k$ having continuous derivatives of order $s$ in a neighborhood of $\mu = EZ_1$. This asymptotic expansion is shown to be identical with a formal Edgeworth expansion of the distribution function of $W_n$. This settles a conjecture of Wallace (1958). The class of statistics considered includes all appropriately smooth functions of sample moments. An application yields asymptotic expansions of distributions of maximum likelihood estimators and, more generally, minimum contrast estimators of vector parameters under readily verifiable distributional assumptions.

A new method of construction of generalized Youden designs for $\nu = s^m, s$ a power of a prime is introduced here. This generalizes the construction of Ruiz and Seiden which could be applied only to even powers of a prime. The number of experimental units required to carry out the design in the corresponding cases is the same. However, the present method can be used for construction of designs which could not be constructed previously even in the case of even powers. Moreover the present method presents a unified construction for even and odd powers of primes. For a fixed value of a prime it is noticed here that one can construct an infinite number of designs. This provides the experimenter with a choice of designs which may prove very useful in applications. A simpler method of construction is also presented. The price one has to pay for the simplicity is that more experimental units are required for carrying out the design.

The problem of selecting one of a number of models of different dimensions is treated by finding its Bayes solution, and evaluating the leading terms of its asymptotic expansion. These terms are a valid large-sample criterion beyond the Bayesian context, since they do not depend on the a priori distribution.

The main result of this paper is that $g$-inverses are not needed for computing optimum designs when the singularity of the information matrix is unavoidable. They are, of course, needed for the analysis. It will be shown that it is possible to augment the experimental region so that procedures for computing optimum designs for $s$ out $k$ parameters $(s < k)$ which are developed for the nonsingular case may also be used for the singular case.

A necessary condition for the $\alpha$-resolvability of an incomplete block design is obtained. Further, a necessary and sufficient condition for an $\alpha$-resolvable incomplete block design to be affine $\alpha$-resolvable is obtained in terms of the largest characteristic root of $NN'$ other than $rk$, where $N$ is the incidence matrix of the design.