Monte Carlo techniques are introduced, using stochastic models which are Markov processes. This material includes the $N$-dimensional Spherical, General Spherical, and General Dirichlet Domain processes. These processes are proved to converge with probability 1, and thus to yield direct statistical estimates of the solution to the $N$-dimensional Dirichlet problem. The results are obtained without requiring any further restrictions on the boundary or the function defined on the boundary, in addition to those required for the existence and uniqueness of the solution to the Dirichlet problem. A detailed study is made for the $N$-dimensional Spherical process; this includes a study of the order of the average number of steps required for convergence. Asymptotic confidence intervals are obtained. When computing effort is measured in terms of the order of the average number of steps required for convergence, the often-made conjecture that the computing effort of a Monte Carlo procedure should be a linear function of the dimensionality of the problem is shown to be true for the cases considered. Comments are included regarding the application of these processes on digital computers, and truncation methods are suggested.

In Section 3 we shall prove a theorem based on a method of A. Birnbaum [1] and E. Lehmann concerning the admissibility of certain tests of simple hypotheses in multivariate exponential families. In Section 4 we compute the supporting hyperplanes of the convex acceptance region in some of the most common applications of Hotelling's $T^2$-test and show that the theorem of Section 3 implies the admissibility of this test. In Section 5 we point out some of the limitations of the method of this paper.

Partially balanced incomplete block designs were introduced by Bose and Nair [1], who described a number of methods of constructing such designs. Among these methods there is one based on incidence properties of finite geometries. This uses the finite geometries associated with the Galois field $GF(p^n)$ with addition and multiplication $(\operatorname{mod} p$). By weakening the geometrical structure (or, equivalently, by weakening the rules of addition and multiplication), it is possible to obtain new designs. A basic feature of a finite projective geometry is that the coordinates are elements of a finite field. What we do here is to allow the coordinates to belong instead to a linear associative algebra $\mathscr{a},$ of finite order $n$ and with modulus, over a finite field $F.$ The procedure is summarized below and explained with more detail in regard to two designs. (For accounts of a similar geometrical theory, using an infinite field, see [7], [8], [9].)

This paper is devoted, in the main, to proving the asymptotic minimax character of the sample distribution function (d.f.) for estimating an unknown d.f. in $\mathscr{F}$ or $\mathscr{F}_c$ (defined in Section 1) for a wide variety of weight functions. Section 1 contains definitions and a discussion of measurability considerations. Lemma 2 of Section 2 is an essential tool in our proofs and seems to be of interest per se; for example, it implies the convergence of the moment generating function of $G_n$ to that of $G$ (definitions in (2.1)). In Section 3 the asymptotic minimax character is proved for a fundamental class of weight functions which are functions of the maximum deviation between estimating and true d.f. In Section 4 a device (of more general applicability in decision theory) is employed which yields the asymptotic minimax result for a wide class of weight functions of this character as a consequence of the results of Section 3 for weight functions of the fundamental class. In Section 5 the asymptotic minimax character is proved for a class of integrated weight functions. A more general class of weight functions for which the asymptotic minimax character holds is discussed in Section 6. This includes weight functions for which the risk function of the sample d.f. is not a constant over $\mathscr{F}_c.$ Most weight functions of practical interest are included in the considerations of Sections 3 to 6. Section 6 also includes a discussion of multinomial estimation problems for which the asymptotic minimax character of the classical estimator is contained in our results. Finally, Section 7 includes a general discussion of minimization of symmetric convex or monotone functionals of symmetric random elements, with special consideration of the "tied-down" Wiener process, and with a heuristic proof of the results of Sections 3, 4, 5, and much of Section 6.

Three test procedures are considered for testing an hypothesis on the mean of a logarithmico-normal distribution with known variance. The first is a normal theory test applied to the logarithms of the original data; the second is a normal theory test applied to the original data; and the third is a test based on the Neyman-Pearson Lemma. The operating characteristics of these tests are developed and some asymptotic properties obtained. It is found that the three procedures give quite different results unless the mean under the null hypothesis is large relative to the standard deviation.

In this paper, two-sample procedures of the type originated by Stein [4] are developed for a number of problems in simultaneous estimation. The results include the construction of simultaneous confidence intervals of prescribed length or lengths and confidence coefficient $1 - \alpha$ for (1) all normalized linear functions of means, (2) all differences between means, and (3) the means of $k$ independent normal populations with common unknown variance. Simultaneous confidence intervals of length $l$ and confidence coefficients known to be not less than $1 - \alpha$ are constructed for all normalized linear functions of the means of a general multivariate normal population. The single sample analogues of these problems have been discussed by Tukey [5], Scheffe [6] and Bose and Roy [7]. Also, a confidence region having prescribed diameter (or volume) and confidence coefficient $1 - \alpha$ is constructed for the mean vector in the general multivariate normal case. The procedures depend only on known and tabulated distributions. Illustrative applications from the analysis of variance are described.

In [1], the author proposed a "rank sum" criterion for testing whether or not a sample was drawn from a population having a completely specified continuous distribution. This criterion under the null hypothesis is distributed as the sum in a random sample from a discrete uniform population; it is otherwise distributed as the sum from a more general discrete population. In this paper we give a method of finding numerically the distribution of such random variables. Tables are given for the distribution of the sums from certain selected discrete uniform distributions. Normal approximations are investigated and applications are briefly discussed.

Let $S$ be the number of successes in $n$ independent trials, and let $p_j$ denote the probability of success in the $j$th trial, $j = 1, 2, \cdots, n$ (Poisson trials). We consider the problem of finding the maximum and the minimum of $Eg(S),$ the expected value of a given real-valued function of $S,$ when $ES = np$ is fixed. It is well known that the maximum of the variance of $S$ is attained when $p_1 = p_2 = \cdots = p_n = p.$ This can be interpreted as showing that the variability in the number of successes is highest when the successes are equally probable (Bernoulli trials). This interpretation is further supported by the following two theorems, proved in this paper. If $b$ and $c$ are two integers, $0 \leqq b \leqq np \leqq c \leqq n,$ the probability $P(b \leqq S \leqq c)$ attains its minimum if and only if $p_1 = p_2 = \cdots = p_n = p,$ unless $b = 0$ and $c = n$ (Theorem 5, a corollary of Theorem 4, which gives the maximum and the minimum of $P(S \leqq c)$). If $g$ is a strictly convex function, $Eg(S)$ attains its maximum if and only if $p_1 = p_2 = \cdots = p_n = p$ (Theorem 3). These results are obtained with the help of two theorems concerning the extrema of the expected value of an arbitrary function $g(S)$ under the condition $ES = np.$ Theorem 1 gives necessary conditions for the maximum and the minimum of $Eg(S).$ Theorem 2 gives a partial characterization of the set of points at which an extremum is attained. Corollary 2.1 states that the maximum and the minimum are attained when $p_1, p_2, \cdots, p_n$ take on, at most, three different values, only one of which is distinct from 0 and 1. Applications of Theorems 3 and 5 to problems of estimation and testing are pointed out in Section 5.

Analyses of variance are sometimes intended to reveal information about means (when tests of significance and, better, confidence procedures are appropriate). At other times analyses of variance have the purpose indicated by their name: to estimate the sizes of the various components contributed to the over-all variance from the corresponding sources. If we make certain assumptions of independence and normality for all of the quantities involved, it is easy to obtain formulas for the variances of the natural estimates of these variance components. The utility of these estimates can be called in question on the grounds of three sorts of assumptions: of certain amounts of independence, of infinite populations, of normality of distribution. This paper treats of the case where the latter two of these assumptions are removed, leaving only the customary (and dangerous) independence assumptions (as do the next two papers in this series). The treatment makes intensive use of polykays (which were introduced in [1], although that name was not used, and discussed in [2]) and is applied specifically to balanced single and double classifications, to Latin squares, and to balanced incomplete blocks. A general definition of balance for an analysis of variance situation is given, and the general application of the technique to balanced situations is set forth. An application to a less simple example of a balanced single classification concludes the paper.

The sampling variance of the least squares estimates of the components of variance in an unbalanced (non-orthogonal) one-way classification and the large sample variances of the maximum likelihood estimates of these quantities are summarized in a paper by Crump [1]. The present paper outlines a method of obtaining these results by the use of matrix algebra, and extends them to the sampling variances of estimates of components of covariance when two variables are considered. The methods are also used to obtain the large sample variance-covariance matrix of the maximum likelihood estimates of the components of variance and covariance.

In a situation in which the observations are frequencies in a multi-way contingency table such that the observations are supposed to be independent and it is only the total number that is supposed to be fixed from sample to sample, a hypothesis on the structure of the probabilities in the different cells or categories is put forward. This hypothesis, by a certain analogy with the customary terminology of analysis of variance, is defined to be the hypothesis of "no interaction" and a large sample test of this hypothesis in terms of $\chi^2$ is offered. Bartlett's results [1] for the case of a $2 \times 2 \times 2$ table and Norton's results [5] for the case of a $2 \times 2 \times t$ table formally turn out to be special cases of the results given here with these differences; (i) Bartlett's and Norton's results refer to "analysis of variance" situations, with marginal frequencies along at least two ways of the table being fixed, while in this paper, for reasons explained elsewhere [7], it is only the total $n$ that is held fixed. (ii) Bartlett's and Norton's papers do not give any indication of the mechanism behind the formulae for the hypothesis of "no interaction," while this paper attempts to give a definite mathematical (and perhaps also physical) mechanism behind the formulae.

This paper is mostly concerned with a modification of the maximum-likelihood estimate of the scale parameter of the extreme-value distribution for which the bias can be explicitly obtained. A formula for computing this bias is derived, and bias factors are tabulated for sample sizes from $n = 2$ to $n = 112.$ A brief comparison is made between this estimator and the optimum linear estimator for a sample of size $n = 6.$ Attention is called to a bias which results from the maximum-likelihood estimate of the second parameter, and formulas for the bias and the variance of this estimate are obtained. In the concluding section, the significance of certain aspects of the maximum-likelihood estimate of the scale parameter in practical applications is briefly discussed.

A stochastic process associated with a queueing system is specified by knowledge of (i) the input, (ii) the queue discipline, and (iii) the service mechanism. A system in which the input is of the "general independent" type and the service times independent and identically distributed according to an arbitrary, general law is given the label $GI/G/s,$ where $s$ is the number of servers (see Kendall [4]). An appointment system for arrivals (or regular service times) is designated by $D$ (deterministic); $M$ describes random arrivals (or negative-exponential service times); and $E_k$ (Erlangian) indicates that a scale-modified $\chi^2$ distribution with 2$k$ degrees of freedom governs the input (or service mechanism). Note that $M$ is equivalent to $E_1.$ The following study was suggested by Kendall in order to extend his description of the system $GI/M/s$ (see [4]) to the system $GI/E_k/s.$ This service time is thought of as the sum of $k$ independent components, identically distributed with negative-exponential distributions. The general system $GI/E_k/s,$ however, appears currently to be intractable in this form, so that we confine ourselves, in this paper, to the system $GI/E_k/1.$ We analyse this with the aid of an embedded Markov chain deriving the stationary distribution for the number of customers in the system at epochs of arrival (equation 1.16) and the distribution of the waiting time for an arbitrary customer (equation 1.21). Lindley [5] has discussed the problem of the waiting time in the system $D/E_k/1,$ solving for this particular example an integral equation governing all systems of the type $GI/G/1:$ the equivalence of our waiting time distribution is demonstrated in Section 2. Pollaczek ([6] and [7]) and Smith [8] have also considered systems of this kind.

For the cdf's of Student's $(t)$ and Thompson's $(\tau)$ distributions, upper and lower bounds are obtained in terms of the normal cdf. It is then shown that, in using the normal approximation for the cdf's of these distributions, the proportional errors are uniformly smaller than $1/n$ for all $n \geqq 8$ and 13, respectively, where $n$ is the number of degrees of freedom. Similar methods may be used to derive bounds for cdf's of similar types. Examples are given.

The WAGR test is a sequential procedure for testing the null hypothesis that the proportion of a normal population greater than a given constant is $p_0$ (given) against the alternative that it is $p_1$ (given). These are equivalent (after a translation) to hypotheses specifying the value of $\mu/\sigma,$ where $\mu$ and $\sigma^2$ are the mean and the variance of the normal population under test. We prove that, with probability one, a decision is reached when the WAGR test is applied. This fact is of importance in its own right; it also has indirect interest because, unless it were true, the standard Wald inequalities on probabilities of error at the two hypothesis points could not be applied.

The problem of plotting on probability paper is extended to continuous distributions which are completely specified except for scale and location parameters. Necessary and sufficient conditions are given to ensure that the plot which is optimal for estimating the scale parameter is also optimal for estimating each of the percentiles.

The unbiasedness of the Tukey Studentized range test for the equality of means of $k$ univariate normal populations with a common variance and of the Hartley $F_{\max}$ ratio test for the equality of variances of $k$ univariate normal populations is proved.

This note discusses some aspects of the estimation of the density function of a univariate probability distribution. All estimates of the density function satisfying relatively mild conditions are shown to be biased. The asymptotic mean square error of a particular class of estimates is evaluated.

For some problems involving a parameter of interest and a nuisance parameter, it is possible to define a statistic sufficient for the parameter of interest. The definition has a number of applications in nonparametric theory. Two theorems are derived and used by way of illustration to prove that the sign test is a uniformly most powerful test for the nonparametric form of the single sample problem of location.

It is a well-known property of the BIB design that all treatment effects are estimated with the same accuracy, i.e., that the variances of the estimates of the treatment effects are all equal and their covariances are also all equal. We show that the converse is also true. If the estimates of the treatment effects in an incomplete block design all have the same variances and the same covariances, then the design is a BIB.

It is shown that the efficiency factor of a design is $r$ times the harmonic mean of the latent roots of the reduced intrablock normal equations exclusing the root which is always zero.

A uniqueness and a characterization theorem are given for the density function over the interval $\lbrack - 1, 1\rbrack$ with a given finite sequence of moments whose square has the smallest possible integral. Extensions are indicated.

Suppose there are $n$ normal populations $N(\mu_i, 1), i = 1, \cdots, n$ and that one random observation from each of these $n$ populations is given. Let $x_1 \leqq x_2 \leqq \cdots \leqq x_n$ be the observations when arranged in order of magnitude and let the corresponding $n$ random variables be denoted by $X_i, i = 1, \cdots, n.$ The following theorem is proved: THEOREM. \begin{equation*}\operatorname{Var}\big(\sum^n_{i = 1} c_i X_i\big), \text{where}\end{equation*}\begin{equation*}\tag{1}\sum^n_{i = 1} c_i = 1,\end{equation*} is minimum when $c_i = 1/n, i = 1, \cdots, n.$ The above theorem may be applied to provide a direct proof of the result that $\sum^n_{i = 1}X_i$ is the best unbiased linear function of order statistics for estimating the sum $\sum^n_{i = 1}\mu_i.$

This note gives an explicit proof of a lemma in matrix theory repeatedly used in [2]; the lemma follows easily from other results, but an explicit proof of it may not be trivial. This note also gives closer confidence bounds for one of the several problems discussed in [2].