Fisher's contributions to statistics are surveyed. His background, skills, temperament, and style of thought and writing are sketched. His mathematical and methodological contributions are outlined. More attention is given to the technical concepts he introduced or emphasized, such as consistency, sufficiency, efficiency, information, and maximum likelihood. Still more attention is given to his conception and concepts of probability and inference, including likelihood, the fiducial argument, and hypothesis testing. Fisher is at once very near to and very far from modern statistical thought generally.

F. Y. Edgeworth's 1908-9 investigation is examined for its contribution to knowledge of the sampling properties of maximum likelihood and related estimates, especially asymptotic efficiency. The nature and extent of his progress and anticipation of R. A. Fisher are described. Fisher's relevant work is briefly examined in relation to Edgeworth's and to the Cramer-Rao inequality.

Consider the class of balanced fractional $2^m$ factorial designs of resolution VII. Within this class, optimal designs with respect to the trace criterion are given for any fixed $N$ assemblies, which satisfy (i) $m = 6, 42 \leqq N \leqq 64$, (ii) $m = 7, 64 \leqq N \leqq 90$ and (iii) $m = 8, 93 \leqq N \leqq 128$. The covariance matrices of the estimates of the effects are also given for such designs.

Suppose that for each number $t$ in [0, 1] there is a distribution with distribution function $F_t(\bullet)$ which has $p$th percentile $\xi(t)$. Consider the problem of estimating $\xi(\bullet)$ under the assumption that $\xi(\bullet)$ is monotone. An estimator which is analogous to the median regression estimator considered in Cryer, Robertson, Wright and Casady (1972), is studied. Asymptotic properties including consistency and law of the iterated logarithm results are obtained under various assumptions.

This paper deals with distribution-free tests for independence under the constraint that the population has a bivariate discrete distribution. The locally most powerful conditional test, given the marginal empirical distributions, is derived. The unconditional asymptotic distribution of the conditional test statistic standardized by the conditional mean and variance is also given under the hypothesis of independence and under contiguous alternatives. Furthermore, some discussions on asymptotic relative efficiency are made. Two competitive test statistics having asymptotically chi-square distributions with different degrees of freedom are compared by means of the local asymptotic relative efficiency.

A sequential version of Chernoff-Savage linear rank statistics is introduced as a basis for inference. The principal result is an invariance principle for two-sample rank statistics, i.e., under a fixed alternative the sequence of sequential linear rank statistics converges weakly to a Wiener process. The domain of application of the theorem is quite broad and includes score functions which tend to infinity at the end points much more rapidly than that of the normal scores test. The method of proof involves new results in the theory of multiparameter empirical processes as well as some new probability bounds on the joint behavior of uniform order statistics. Applications of weak convergence are explored; in particular, the extension of the theory of Pitman efficiency to the sequential case.

The purpose of this paper is to obtain weaker sufficient conditions than those given by Robbins [6] for the asymptotic optimality of empirical Bayes procedures. In so doing a general method for providing asymptotic optimality is given which because of its simplicity should prove quite useful not only for the empirical Bayes problem but for other related asymptotic problems as well. Three areas of application are indicated herein.

A sequence of empirical Bayes estimators is defined for estimating a distribution function. The sequence is shown to be asymptotically optimal relative to a Ferguson Dirichlet process prior. Exact risk expressions are derived and the rate, at which the overall expected loss approaches the minimum Bayes risk, is exhibited. The empirical Bayes approach, based on the Dirichlet process, is also applied to the problem of estimating the mean of a distribution.

This paper provides a proof of the asymptotic sufficiency of the ranks in the two sample situation when the observations are censored at the $r$th order statistic. Several important conclusions may then be drawn concerning optimal tests.

In the present paper, an integer-valued version $(T_N)$ of the Kendall rank correlation coefficient is considered. Under the hypothesis of independence, a local limit theorem with the Edgeworth expansion for $T_N$ is proved and an asymptotic expansion of the distribution function of $T_N$ is derived.

For univariate distributions there is a generally accepted definition of unimodality due to Khintchine which requires the existence of a number $a$, called vertex, such that the distribution function is convex on $(-\infty, a)$ and concave on $(a, \infty)$. For multivariate distributions, however, unimodality can be defined in several different ways. Anderson (1955) and Olshen and Savage (1970) have given such definitions. This paper first examines the definition which calls a random vector $(X_1, \cdots, X_n)$ unimodal if all linear combinations $\sum a_iX_i$ are univariate unimodal. We show that this definition is somewhat unnatural because the density of such a "unimodal" distribution may not become maximum at the vertex of unimodality. The paper also examines two other definitions based on the results of Sherman (1955). One of these looks at the closed convex hull of the set of all uniform distributions on symmetric convex bodies and the other requires that the probability carried by a symmetric convex set decreases as the set is moved away from the origin in a fixed direction. The equivalence of these definitions was conjectured by Sherman and this paper gives some results having a bearing on this conjecture.

Consider $k(\geqq 2)$ independent $p$-variate $(p \geqq 2)$ normal distributions $N(\mathbf{\mu}_i, \Sigma_i), i = 1,2, \cdots, k$, where the mean vectors $\mathbf{\mu}_i$ and the covariance matrices $\Sigma_i$ are all unknown. Let $\theta_i$ denote for the $i$th distribution the squared population multiple correlation coefficient between the first variate and the set of $(p - 1)$ variates remaining. A procedure based on the natural ordering of the $k$ sample squared multiple correlation coefficients, each computed from a random sample of size $n(\geqq p + 2)$, is considered for the problem of selection of the $t(< k)$ largest $\theta_i$'s. Given $(1 - \theta_{\lbrack k - t\rbrack}) \geqq \delta(1 - \theta_{\lbrack k - t + 1\rbrack})$ and $\theta_{\lbrack k - t + 1\rbrack} \geqq \gamma\theta_{\lbrack k - t\rbrack}$, where $\theta_{\lbrack i\rbrack}$ denotes the $i$th smallest $\theta$ and $\delta > 1$ and $\gamma > 1$ are preassigned constants, it is shown that the probability of a correct selection is minimized for $\theta_{\lbrack i\rbrack} = (\delta - 1)/(\delta\gamma - 1), i = 1, \cdots, k - t$ and $\theta_{\lbrack i\rbrack} = \gamma(\delta - 1)/(\delta\gamma - 1), i = k - t + 1, \cdots, k$. For a given $P^\ast (< 1)$, the exact common sample size $n$ is then determined so that the infimum of the probability of a correct selection is not smaller than $P^\ast$. For $p = 2$, the problem reduces to selecting $t$ largest correlation coefficients from the $k$ bivariate normal distributions.

The sample paths of cumulative sums of induced order statistics obtained from $n$ independent two-dimensional random vectors, when appropriately normalized, converge weakly (as $n$ increases indefinitely) to the sum of a Brownian motion with time change and an integrated Brownian bridge which is independent of the Brownian motion. Applications in regression analysis are given.

In recent work by O'Reilly, a necessary and sufficient condition for the existence of an unbiased estimate of the distribution function of a "future" observation was given. The result was obtained under the condition that the model had full rank. Here, the result is generalized to any number of future observations and the full rank condition is relaxed. The corresponding uniformly minimum variance unbiased estimator is identified from the density estimates given by Ghurye and Olkin.

A multivariate extension is made of Stein's result (1964) on the estimation of the normal variance. Here the generalized variance $|\Sigma|$ is being estimated from a Wishart random matrix $S: p \times p \sim W(n, \Sigma)$ and an independent normal random matrix $X: p \times k \sim N(\xi, \Sigma \otimes 1_k)$ with $\xi$ unknown. The main result is that the minimax, best affine equivariant estimator $((n + 2 - p)!/(n + 2)!)|S|$ is dominated by $\min\{((n + 2 - p)!/(n + 2)!)|S|, ((n + k + 2 - p)!/(n + k + 2)!)|S + XX'|\}$. It is obtained by a variant of Stein's method which exploits zonal polynomials.

Some generalizations of Milliken's necessary and sufficient condition for estimability of linear parametric functions in linear models are established. The more universal character of the present theorems consists in avoiding the assumption of the linear independence of examined functions, in using any generalized inverse instead of the Moore-Penrose inverse, and in extending the criterion on more general linear models.

Let $X = (X_1, \cdots, X_p)^t$ be a $p$-variate normal random vector with unknown mean $\theta = (\theta_1, \cdots, \theta_p)^t$ and unknown positive definite diagonal covariance matrix $A$. Assume that estimates $V_i$ of the variances $A_i$ are available, and that $V_i/A_i$ is $\chi^2_{n_i}$. Assume also that all $X_i$ and $V_i$ are independent. It is desired to estimate $\theta$ under the quadratic loss $\lbrack\sum^p_{i=1} q_i(\delta_i - \theta_i)^2\rbrack/\lbrack\sum^p_{i=1} q_i A_i\rbrack,\quad\text{where} q_i > 0, i = 1, \cdots, p.$ Defining $W_i = V_i/(n_i - 2), W = (W_1, \cdots, W_p)^t$, and $\|X\|_{W^2} = \sum^p_{j=1} \lbrack X_{j^2}/(q_jW_j^2)\rbrack$, it is shown that under certain conditions on $r(X, W)$, the estimator given componentwise by $\delta_i(X, W) = (1 - r(X, W)/\lbrack\|X\|_{W^2}q_i W_i\rbrack)X_i$ is a minimax estimator of $\theta$. (The conditions on $r$ require $p \geqq 3$.) A good practical version of this estimator is also given.

Let $X$ be a normal random variable with mean $\theta$ and variance 1 and consider the problem of estimating $\theta$ with squared error loss. If $\delta(x) = ax + b$ is a linear estimate with $0 \leqq a \leqq 1$ then it is well known that $\lambda\delta$ is an admissible proper Bayes estimate for $\lambda \in (0, 1)$. That is, all contractions of $\delta$ are proper Bayes estimates. In this note we show that no other estimates have this property.

If $m$ objects $x_1, x_2, \cdots, x_m$ are compared pairwise, then let $s_{ij}$ denote the number of times $x_i$ beats $x_j$ in $n_{ij}$ independent comparisons. In a ranking, if $x_i$ precedes $x_j$ then one may require that the probability of $x_i$ beating $x_j$ be at least $\frac{1}{2}$. Such a ranking is called weak stochastic ranking. Let $I(R)$ be the set of all pairs $(i, j)$ such that $x_i$ precedes $x_j$ in the ranking $R$ in spite of the paired comparison outcomes resulting in $s_{ij} < s_{ji}$. A statistic $D(R) = \sum_{I(R)} (S_{ij} - S_{ji})^2/n_{ij}$ is derived and proposed for estimating a weak stochastic ranking. Since $D(R)$ is seen to be the sum of a random number of asymptotically distributed chi-square variates, a ranking is called minimum chi-square weak stochastic if $D(R) \leqq D(R_t)$, for $t = 1,2, \cdots, m$! It is proved that minimum chi-square rankings share at least two properties with the maximum likelihood rankings. That is, every minimum chi-square ranking is Hamiltonian ranking and when in particular $n_{ij} = 1$, every minimum chi-square ranking minimizes the violations of observed outcomes. Moreover, the branch and bound algorithm can be used for estimating the minimum chi-square rankings.

The concept of resolvability of a balanced incomplete block (BIB) design, introduced by Bose (1942), was generalized to $\mu$-resolvability of an incomplete block design by Shrikhande and Raghavarao (1964). As a further generalization of these, the concept of $(\mu_1, \mu_2, \cdots, \mu_t$)-resolvability is here introduced for an incomplete block design. This concept may be useful from both combinatorial and practical points of view. Furthermore, some methods of constructing BIB designs with the generalized concept are discussed with illustrations. One method is based on a finite geometry over a Galois field.

For the minimum weight in $s$-ary $(n, p)$ linear codes (or for the maximum resolution in $s^{n-p}$ designs), an upper bound has been obtained by Plotkin [4] where $s$ is a prime power. The purpose of this note is to obtain an improvement of the Plotkin's upper bound. Main result is as follows: when $p \geqq 2$, the maximum resolution $R_p(n, s)$ of any $s^{n-p}$ design satisfies the following: \begin{equation*}\begin{split}R_p(n, s) = s^{p-1}q\quad\text{if} m = 0, 1 \\ &\leqq s^{p-1}q + \lbrack s^{p-2}(s - 1)(m - 1)/(s^{p-1} - 1)\rbrack \\ \text{if} m = 2, 3, \cdots, s^{p-1} \\ &\leqq s^{p-1}q + \lbrack(s - 1)m/s\rbrack\quad\text{if} m = s^{p-1} + 1, \cdots, N - 1,\end{split}\end{equation*} where $\lbrack x\rbrack$ is the greatest integer not exceeding $x, n = qN + m$ and $N = (s^p - 1)/(s - 1)$.

Let $X_1, \cdots, X_n$ be i.i.d. rv's. Let further $\bar{X} = \sum X_i/n, S^2 = \sum(X_i - \bar{X})^2$, and $U = ((X_1 - \bar{X})/S, \cdots, (X_n - \bar{X})/S)$. If the variables $X_i$ are normally distributed or distributed as linearly transformed Gamma variables, $\bar{X}$ and $U$ are independent. In this paper we show that also the converse must hold.