A number of statistical models, underlying the methods used in the estimation of the sizes and other parameters of animal populations, are set up. The relevant estimation equations are given, with their variances and covariances. For the most part the theory is designed for large populations. In setting up the models, consideration has been given to the desideratum of having them conform as closely as possible to the actual practices of animal sampling. To what extent the models do agree with reality is one of the many open questions which are noted in this paper.

This paper is concerned with a single-sample multiple decision procedure for ranking means of normal populations with known variances. Problems which conventionally are handled by the analysis of variance (Model I) which tests the hypothesis that $k$ means are equal are reformulated as multiple decision procedures involving rankings. It is shown how to design experiments so that useful statements can be made concerning these rankings on the basis of a predetermined number of independent observations taken from each population. The number of observations required is determined by the desired probability of a correct ranking when certain differences between population means are specified.

New methods are introduced for deriving the sampling distributions of statistics obtained from a normal multivariate population. Exterior differential forms are used to represent the invariant measures on the orthogonal group and the Grassmann and Stiefel manifolds. The first part is devoted to a mathematical exposition of these. In the second part, the theory is applied; first, to the derivation of the distribution of the canonical correlation coefficients when the corresponding population parameters are zero; and secondly, to split the distribution of a normal multivariate sample into three independent distributions, (a) essentially the Wishart distribution, (b) the invariant distribution of a random plane which is given by the invariant measure on the Grassmann manifold, (c) the invariant distribution of a random orthogonal matrix. This decomposition provides derivations of the Wishart distribution and of the distribution of the latent roots of the sample variance covariance matrix when the population roots are equal.

R. L. Plackett derived the maximum of the ratio of mean range to the standard deviation as function of the sample size, and gave the initial (symmetrical) distribution for which this maximum is actually reached. On the other hand, Moriguti derived the maximum for the mean largest value under the assumption that the distribution from which the maximum is taken is symmetrical. His mean value turned out to be one half of the value given by Plackett. In the following, these results will be generalized for an arbitrary (not necessarily symmetrical) continuous variate. The mean and the standard deviation of the largest value and the mean range will be given for two distributions: one where the mean largest value is a maximum, and another one where the mean range is a maximum. Obviously, a mean largest value can exist if and only if the initial mean exists. In addition we postulate in both cases the existence of the second moment.

Consider any distribution $f(x)$ with standard deviation $\sigma$ and let $x_1, x_2 \cdots x_n$ denote the order statistics in a sample of size $n$ from $f(x).$ Further let $w_n = x_n - x_1$ denote the sample range. Universal upper and lower bounds are derived for the ratio $E(w_n)/\sigma$ for any $f(x)$ for which $a\sigma \leqq x \leqq b\sigma,$ where $a$ and $b$ are given constants. Universal upper bounds are given for $E(x_n)/\sigma$ for the case $- \infty < x < \infty.$ The upper bounds are obtained by adopting procedures of the calculus of variation on lines similar to those used by Plackett [3] and Moriguti [4]. The lower bounds are attained by singular distributions and require the use of special arguments.

This paper generalizes certain results which are known for balanced incomplete block designs and group divisible designs to partially balanced incomplete block (PBIB) designs with $m$ associate classes. Some of the results are for general $m$ but others are for $m =$ 2, 3, or 4.

For a special class of cumulative distribution functions which are solutions of a given reduced moment problem (cf. paragraph 3, pages 27 and 28, of [4]), the well known expression for the least upper bound of the absolute difference between any two solutions of the same reduced moment problem is improved upon by the introduction of a constant nonnegative multiplier which is smaller than unity in the case of the special class of solutions. Useful properties of the determinantal form of the classical expression for the least upper bound are derived. The numerical value of the constant multiplier is computed in the case of a well known class of cumulative distribution functions. In addition, a simple method is given for constructing, over a finite range, an infinite set of continuous and differentiable cumulative distribution functions which are solutions of the same reduced moment problem when one such solution is known. The new expression for the least upper bound, when applied to members of the constructed class of continuous solutions, may be helpful in deriving general, but crude, inequalities among orthogonal polynomials over a finite interval.

The purpose of this paper is to investigate a problem in the spacing of information in certain applications of polynomial regression. It is shown that for a polynomial of degree $m,$ the variance-covariance matrix of the estimated polynomial coefficients given by a spacing of information at more than $m + 1$ values of the sure variate can always be attained by spacing the same information at only $m + 1$ values of the sure variate, these spacing values being bounded by the first spacing values. The presented results are of use in experimental design involving polynomial regression when a choice of sure variate values is possible but restricted to a specified range. Let the polynomial under consideration be \begin{equation*}\tag{1.1} P(x) = \alpha_1 + \alpha_2x + \cdots + \alpha_{m+1}x^m, m \geqq 1,\end{equation*} and let $P(x_\epsilon) = y(x_\epsilon) + \delta_\epsilon, \epsilon = 1(1)N, N \geqq (m + 1).$ The $y(x_\epsilon)$ are observed uncorrelated variates with random error $\delta_\epsilon$ having mean zero and finite variance $\sigma^2_\epsilon > 0.$ The $x_\epsilon$ are observed variates without error, there being at least $(m + 1)$ distinct $x_\epsilon.$ The following notation is introduced. Let $\overset{\rightarrow}{x} = (1, x, x^2, \cdots, x^m), X = (\overset{\rightarrow}{x}_\epsilon), \epsilon = 1(1)N,$ and let $W$ be the $N \times N$ diagonal matrix with entry $w_\epsilon = 1/\sigma^2_\epsilon$ in the $(\epsilon, \epsilon)$ position. $w_\epsilon$ will henceforth be referred to as the "information" of $y(x_\epsilon),$ and $Q = \sum w_\epsilon, \epsilon = 1(1)N,$ will be referred to as the "total information." The matrix $X'WX$ will be called the "information matrix." The problem is to show that given a spacing of total information $Q$ at locations $x_\epsilon, \epsilon = 1(1)N, N \geqq (m + 1),$ there being at least $(m + 1)$ distinct $x_\epsilon,$ it is always possible to re-space $Q$ at $(m + 1)$ distinct locations $r_j, j = 1(1)(m + 1),$ in such a manner that $\min x_\epsilon \leqq r_j \leqq \max x_\epsilon, \epsilon = 1(1)N, j = 1(1)(m + 1),$ and $X'WX = R'UR$, with $R'UR$ being the information matrix of the re-spacing. The problem is solved by prescribing a method for finding the required $U$ and $R$ which determine the spacing of the total information. The motivation for the problem is as follows. In experimentation in the chemical engineering industry, we most often have control over our sure variates. The sure variate $x$ could be the pressure level of our process equipment, and we would be permitted to choose any operating pressure $x$ in the pressure range $\min x$ to $\max x,$ tolerated by our equipment. Quite often, and in particular with isotopic measurements, laboratory analytical determinations are required for our $y$-variates with the laboratory being the major source of error. With each laboratory determination having variance $\sigma^2,$ we can request $n_x$ determinations on the material sample taken at sure variate $x$. Using the average of the laboratory determinations, the corresponding $y$ variate has variance $\sigma^2/n_x.$ Specifying $Q$ then amounts to specifying total laboratory effort expended on the experiment. It might be set by such usual factors as the dollar allowance on the experiment; if the material is highly radioactive, it might be set by such unusual factors as exposure time allowed the laboratory analysts. Furthermore, in experimentation with fairly large equipment, it is important to minimize the distinct levels of operation, that is, the distinct number of sure $x$'s. The time required to make the change and to reach sufficient equilibrium representing steady-state operation of the process is often long. In any case we lose time, and with production line equipment, we also lose production. These are the reasons for minimizing the distinct number of sure $x$'s in the experiment. The equivalence $X'WX = R'UR$ gives the required minimization. If the functional relationship between $y$ and $x$ is adequately represented by a polynomial of degree $m$, the equivalence assures that only $(m + 1)$ distinct sure $x$'s are required to maintain the same efficiency of statistical evaluation of the experimental results, since most statistical evaluation will require $(X'WX)^{-1},$ which can now be replaced by $(R'UR)^{-1}.$ It may be seen that such experiments, common in physico-chemical industry, present a formulation and require a mathematical model not found in ordinary regression theory, where usually it is not possible to assign various values to the corresponding $y$ variances. With the indicated background in mind, the results of this paper find application in experimental design. The determination of a spacing which optimizes some criteria involving the information matrix is made simpler. A familiar example arising in point estimation is minimizing $\overset{\rightarrow}{p}(X'WX)^{-1}\overset\rightarrow{p}'$ for a specified row vector $\overset\rightarrow{p}.$ An example from interpolation is minimizing the maximum of $\overset{\xi}(X'WX)^{-1}\overset{\Xi}'$ with $\overset{\Xi} = (1, \Xi, \Xi^2 \cdots \xi^m)$ and $\min x_\epsilon \leqq \xi \leqq \max x_\epsilon;$ the extrapolation problem is similar. The advantage of applying the above result to such problems is that the spacing of information is at once reduced to $(m + 1)$ distinct locations, any larger number being unnecessary. The matrix $X$ then is the matrix of a Vandermonde determinant. The properties of these matrices are well known and attractive. These uses will be illustrated by an example given in Section 4.

This paper considers some properties of a two-sample test, suggested by Lehmann [2], against general alternatives. Alternative expressions are given for the test statistic; a general formula for the variance is derived and evaluated for the null case; the expectation is obtained in certain nonnull cases; and the exact distributions in the null case are tabulated for some small samples.

A method of constructing orthogonal arrays of an arbitrary strength $t$ is formulated. This method is a modification of the method based on differences, formulated by R. C. Bose [1] for the purpose of constructing orthogonal arrays of strength 2. It is shown further that each of the multifactorial designs of R. L. Plackett and J. P. Burman [2], in which each factor takes on two levels, provide a scheme for constructing orthogonal arrays of strength 3, consisting of the maximum possible number of rows. An orthogonal array (36, 13, 3, 2) is constructed. The method used for its construction cannot lead to a number of constraints greater than 13. It is known however [3] that 16 is an upper bound for the number of constraints in this case; the problem as to whether this bound can actually be attained remains unsolved.

Explicit characterizations are given of the minimal complete class and a minimal essentially complete class of tests of a simple hypothesis specifying the mean of a uniform distribution of known range. Examples are given of tests which are optimal against various alternatives.

The ratio of two different noncentral $t$ density functions with the same number of degrees of freedom is strictly monotone, with sense depending on the relative values of the two noncentral constants.

An expression (equation 2) suitable for computation is obtained for the probability distribution of $\zeta = (\xi^2 + \eta^2)^{\frac{1}{2}}$ where $\xi$ and $\eta$ are independent Gaussian variables with, in general, unequal means and unequal variances.