The work of Neyman on confidence limits and of Fisher on fiducial limits is well known. However, in most applications the interval or limits for only a single parameter or a single function of the parameters has been considered. Recently Scheffe [2] and Tukey [3] have considered special cases of what may be called problems of simultaneous estimation, in which one is interested in giving confidence intervals for a finite or infinite set of parametric functions such that the probability of the parametric functions of the set being simultaneously covered by the corresponding intervals is a preassigned number $1 - \alpha(0 < \alpha < 1).$ In this paper we discuss in Section 1, a set of sufficient conditions under which such simultaneous estimation is possible, and bring out the connection of this with a method of test construction considered by one of the authors in a previous paper [1]. In Section 2 some univariate examples (including the ones due to Scheffe and Tukey) are considered from this point of view. Sections 3 to 6 are concerned with multivariate applications, giving results which are believed to be new. The associated tests all turn out to be the same as in [1] except for the example in Section 4.3 which, in a sense, is a multivariate generalization of Tukey's example (Section 2.2). Section 3 gives the notation and preliminaries for multivariate applications. Section 4 gives confidence bounds on linear functions of means for multivariate normal populations. Sections 5 and 6 give respectively confidence bounds on certain functions of the elements of population covariance matrices and population canonical regressions, from which a chain of simpler consequences would follow by the application of a set of matrix theorems. This has been partly indicated in the present paper and will be more fully discussed in a later paper.

We consider time series which are realizations of a stochastic process. From the time series we construct various estimates of the spectral distribution function of the process (Section 3) and we study the sampling distributions of some functionals of these estimates (Sections 4-7). We then obtain confidence regions for the spectral distribution function and various tests of hypotheses in the normal case (Sections 8-10).

A stochastic model designed for analyzing data with changing probabilities is presented. On each of a series of trials one of two alternatives occurs and the probabilities of occurrence are changed from time to time by events. Corresponding to each class of events is an operator which represents a linear transformation on the probabilities of the two alternatives. Cases of fixed event probabilities and of changing event probabilities are considered. Recurrence formulas for moments of the resulting distributions of probabilities are provided. These formulas are often tedious to apply, but for the first and second moments several bounds are provided; these bounds are relatively easy to compute. The problem of estimating the parameters of the model is discussed. No general solution is obtained but simplifying assumptions lead to interesting special cases for which detailed procedures of parameter estimation are presented. One such special case arises when there are two event operators which commute, implying that the operators have equal limit points or that one operator is the identity operator. The method of maximum likelihood is applied to this case. Another special case, which arises when the slope parameters of the two operators are equal, is discussed in Section 8. Applications of the model and estimation procedures to certain kinds of data on animal and human learning are described. The examples given are experiments on verbal learning, the avoidance training of dogs, the reward training of rats in a simple T-maze, and the behavior of human subjects in a two-choice situation.

It is desired to estimate $s$ parameters $\theta_1, \theta_2, \cdots, \theta_s.$ There is available a set of experiments which may be performed. The probability distribution of the data obtained from any of these experiments may depend on $\theta_1, \theta_2, \cdots, \theta_k, k \geqq s.$ One is permitted to select a design consisting of $n$ of these experiments to be performed independently. The repetition of experiments is permitted in the design. We shall show that, under mild conditions, locally optimal designs for large $n$ may be approximated by selecting a certain set of $r \leqq k + (k - 1) + \cdots + (k - s + 1)$ of the experiments available and by repeating each of these $r$ experiments in certain specified proportions. Examples are given illustrating how this result simplifies considerably the problem of obtaining optimal designs. The criterion of optimality that is employed is one that involves the use of Fisher's information matrix. For the case where it is desired to estimate one of the $k$ parameters, this criterion corresponds to minimizing the variance of the asymptotic distribution of the maximum likelihood estimate of that parameter. The result of this paper constitutes a generalization of a result of Elfving [1]. As in Elfving's paper, the results extend to the case where the cost depends on the experiment and the amount of money to be allocated on experimentation is determined instead of the sample size.

A method is developed for the evaluation of the probability density function of the statistic: $w_r = x_{n-r} - x_{r+1}$ where $x_1, x_2, \cdots, x_n$ are ordered values in a sample of $n$ from a normal population. It is shown that, up to $n = 17, w_0$ is the most efficient statistic of this type for the estimation of population standard deviation. Beyond this point $w_1$ is optimum up to $n = 31,$ where $w_2$ becomes better. Tables of moment constants and percentage points are given for $w_1$ over the range 10 to 30. Similar methods are used to determine the efficiencies of two estimates of the form $w_r + \lambda w_s.$ The approximation used is compared with three other published approximations in the case of range $(r = 0).$ Godwin [5] and Nair [11] have discussed problems of this kind for sample sizes up to 10, using exact values of the first two moments. Karl Pearson [12], Mosteller [10] and Jones [9] have considered the large sample case. The methods of the present paper go some way towards filling the gap between these approaches. Moreover, they are not restricted to consideration of mean and variance only.

Sequential tests of linear hypotheses in the systematic linear model are studied. Methods of overcoming difficulties in the construction of tests when there is a random model are considered.

We give in this paper the solution to the first passage problem for a strongly continuous temporally homogeneous Markov process $X(t).$ If $T = T_{ab}(x)$ is a random variable giving the time of first passage of $X(t)$ from the region $a > X(t) > b$ when $a > X(0) = x > b,$ we develop simple methods of getting the distribution of $T$ (at least in terms of a Laplace transform). From the distribution of $T$ the distribution of the maximum of $X(t)$ and the range of $X(t)$ are deduced. These results yield, in an asymptotic form, solutions to certain statistical problems in sequential analysis, nonparametric theory of "goodness of fit," optional stopping, etc. which we treat as an illustration of the theory.

This paper gives a method for determining the size of the first part of a two-stage sample for estimating the population mean with a given accuracy. The method is based on a scheme of Stein [4]. The tables necessary for application of this method have been given. A more detailed summary will be found at the end of the Introduction.

This paper considers the distribution of the measure of a special random two-dimensional set. Related work, usually motivated by a search for principles for bombing operations, deals exclusively with moment problems and appears in [1], [3], [4], [5], [6], [7], [8]. A one-dimensional distribution problem appears in [2]. The random set considered is the intersection of a fixed circle with the union of $N$ random circles. Centers of the random circles are subject to the variability imposed by the bivariate normal distribution with circular symmetry and means not necessarily coincident with the coordinates of the center of the fixed circle. The measure of interest is the ratio of the area of the intersection ("covered area") to the total area of the fixed circle. For $N = 1,$ the distribution is determined and its use facilitated by the graphs in Fig. 1 and Fig. 2. A procedure for obtaining upper and lower bounds of the distribution for $N = 2$ is given. Tables I, II, III, and IV give upper and lower bounds for the percentage points of the distribution for $N = 2$ for some special illustrative situations. For $N = 1$ in all situations, and for $N = 2$ in many situations; the graphs and tables demonstrate that a realistic decision can be made rather easily without resorting to the usual practice of random number "Monte Carlo" devices for each ad hoc situation of interest.

In many practical problems, the experimenter is faced with the task of deciding if the variability within several classes is uniform throughout the classes, or if not, which class exhibits the greatest amount of variability. This type of problem arises when the data relate to several processes, to the same process at different times, to several different products, or to the same products from different sources. If the variability is not uniform throughout the classes, then misleading results would be obtained in comparing the classes in other respects. If the experimenter expects the variability to be uniform throughout the different classes, and if the variability is large in a particular class, he will consider the situation to be "out of control" and take measures to locate the source of the large variability. The problem we will consider here is that of comparing the variances of $k$ populations, $\Pi_1, \Pi_2, \cdots, \Pi_k,$ on the basis of $n$ observations $x_{i1}, x_{i2}, \cdots, x_{in}$ from the $i$th population. We will assume that these observations are normally and independently distributed with unknown mean $m_i$ and unknown standard deviation $\sigma_i$ for $i = 1, 2, \cdots, k.$ Our problem is to find a statistical procedure which will, on the basis of these observations, decide if all the populations have equal variances, and if not, which has the largest variance. We would like the procedure to be in some sense "optimum." We will say that our procedure is optimum if, subject to certain restrictions, it maximizes the probability of making the correct decision. A similar problem dealing with the means of several normal distributions has been studied by Paulson [1]. Let $D_0$ be the decision that all $k$ variances are equal, and let $D_j$ be the decision that $D_0$ is false and $\sigma^2_1 = \max (\sigma^2_1, \cdots, \sigma^2_k)$ for $j = 1, 2, \cdots, k.$ Our problem now is to find a statistical procedure for selecting one of these $k + 1$ decisions. Let $x_{i\alpha}$ denote the $\alpha$th observation from the $i$th population, and let $\bar{x}_i = \sum^n_{\alpha=1} x_{i\alpha}/n.$ Let $s^2_i = \sum^n_{\alpha=1}(x_{i\alpha} - \bar{x})^2/(n - 1)$ denote the unbiased estimate of the variance of the $i$th population. We will say that $\Pi_i$ has "slipped to the right" if $\sigma^2_1 = \cdots = \sigma^2_{i-1} = \sigma^2_{i+1} = \cdots = \sigma^2_k$ and $\sigma^2_i = \lambda^2\sigma^2_1$ where $|\lambda| > 1.$ In our first formulation of the problem we will want to find a statistical procedure which will select one of the $k + 1$ decisions $D_0, D_1, \cdots, D_k$ so that (a) when all the variances are equal, $D_0$ should be selected with probability $1 - \alpha,$ where $\alpha$ is a small positive number fixed prior to the experiment. Since the class of possible decision procedures seems to be too large to admit an optimum solution we will impose the following restrictions which seem to be reasonable: (b) the procedure should be symmetric, that is, the probability of selecting $D_i$ when $\sigma^2_1 = \cdots = \sigma^2_{i-1} = \sigma^2_{i+1} = \cdots = \sigma^2_k$ and $\sigma^2_i = \lambda^2\sigma^2_1$ should be the same for all $i$; (c) the procedure should be invariant if all the observations are multiplied by the same positive constant; and (d) the procedure should be invariant if some constant $b_i$ is added to all the observations in the $i$th population. We will now reformulate the problem as follows. We want a statistical procedure for selecting one of the $k + 1$ decisions $D_0, D_1, \cdots, D_k$ which, subject to conditions (a), (b), (c), and (d) will maximize the probability of making the correct decision when one of the populations has slipped to the right. We shall prove that the optimum solution is the following: \begin{equation*}\begin{split} \text{if} s^2_M\big/\sum^k_{i=1} s^2_i > L_\alpha \text{select} D_M; \\ \text{if} s^2_M\big/\sum^k_{i=1} s^2_i \leqq L_\alpha \text{select} D_0,\end{split} \end{equation*} where $M$ denotes the population yielding the largest sample variance. $L_\alpha$ is a constant whose value is determined by restriction (a). This statistic has been suggested, on intuitive grounds, by Cochran [2], and a good tabulation of $L_\alpha$ for several values of $\alpha, n,$ and $k$ is available [3].