A sequential problem is considered in which independent observations are taken on a chance variable $X$ whose distribution can be represented by \begin{equation*}\tag{1}dG_\theta(x) = \psi(\theta)e^{\theta x} d\mu(x),\end{equation*} where the parameter $\theta$ belongs to a given interval $\Omega$ of the real line but is otherwise unknown. The problem is to test $H_1:\theta \leqq \theta^\ast$ against $H_2:\theta > \theta^\ast$, where $\theta^\ast$ is a given point in $\Omega$. Under certain assumptions the following class $A$ is shown to be essentially complete relative to the class of decision rules with bounded risk functions. The decision rule $\delta \varepsilon A$ if and only if after taking $n$ observations (i) $\delta$ depends on the observations only through $n$ and $v_n = \sum^n_{i = 1} x_i$ and (ii) $\delta$ specifies a closed interval $J_n:\lbrack a_{1n}, a_{2n} \rbrack$ for each $n$ and the following rule of action (a) Stop experimentation as soon as $v_n \not\varepsilon J_n$ and (1) accept $H_1 \text{if} v_n < a_{1n}$ (2) accept $H_2 \text{if} v_n > a_{2n}$. (b) If $a_{1n} < a_{2n}$ take another observation if $a_{1n} < v < a_{2n}$. (c) If $a_{1n} < a_{2n}$ and $v = a_{in},$ accept $H_i$ or take another observation or randomize between these two $(i = 1, 2)$. The Koopman-Darmois family of probability laws given above contains discrete members such as the binomial and Poisson distributions as well as absolutely continuous members such as the normal and exponential. It is interesting to note that the members of the class A can be obtained by starting with the sequential probability ratio test for testing some point $\theta^\ast_1 \leqq \theta^\ast$ against another point $\theta^\ast_2 > \theta^\ast$, namely, continue as long as $B < \frac{\prod^n_{i = 1} \psi(\theta^\ast_2)e^{\theta^\ast_2 x_i}}{\prod^n_{i = 1} \psi(\theta^\ast_1)e^{\theta^\ast_1 x_i}} < A$ and replacing the constants $B, A$ by two arbitrary sequences $B_n, A_n$ such that $B_n \leqq A_n (n = 1, 2, \cdots).$

The stochastic processes which occur in the theory of queues are in general not Markovian and special methods are required for their analysis. In many cases the problem can be greatly simplified by restricting attention to an imbedded Markov chain. In this paper some recent work on single-server queues is first reviewed from this standpoint, and the method is then applied to the analysis of the following many-server queuing-system: Input: the inter-arrival times are independently and identically distributed in an arbitrary manner. Queue-discipline: "first come, first served." Service-mechanism: a general number, $s$, of servers; negative-exponential service-times. If $Q$ is the number of people waiting at an instant just preceding the arrival of a new customer, and if $w$ is the waiting time of an arbitrary customer, then it will be shown that the equilibrium distribution of $Q$ is a geometric series mixed with a concentration at $Q = 0$ and that the equilibrium distribution of $w$ is a negative-exponential distribution mixed with a concentration at $w = 0$. (In the particular case of a single server this property of the waiting-time distribution was first discovered by W. L. Smith.) The paper concludes with detailed formulae and numerical results for the following particular cases: Numbers of servers: s = 1, 2 and 3. Types of input: (i) Poissonian and (ii) regular.

We shall be concerned with an irreducible Markov chain, which we shall call "the system." For simplicity we shall assume that the system is aperiodic, but this is not essential. The reader is referred to [1] for explanations of the terminology used. We first state some general theorems which provide criteria for determining whether the system is transient, recurrent-null or ergodic (recurrent-nonnull). These are then applied to the Markov chains associated with certain queuing processes recently studied by D. G. Kendall [4], [5]; most of the results have already been obtained by Kendall using direct methods, and the main purpose of the present paper is to illustrate the application of general theorems to this type of problem.

In this paper we have studied certain combinatorial properties of incomplete block designs and efficient necessary conditions for the existence of affine resolvable balanced incomplete block (b.i.b., for abbreviation) designs. Two theorems give combinatorial properties of certain b.i.b. designs. The well known inequality of Fisher between the number of varieties and number of blocks is shown in this paper to hold under very general conditions. An intrinsic characteristic property of symmetrical b.i.b. designs is given in another theorem. In the last two theorems we have deduced efficient necessary conditions for the existence of affine resolvable b.i.b. designs. Besides these there are some minor results. Utilizing the simple yet very fruitful idea of associating an incidence matrix with a design, all the results are deduced with the help of arguments of algebra of matrices and linear equations. The last theorem requires the use of the celebrated four square theorem of Lagrange and a result due to Legendre in number theory.

For two normal populations with unknown variances and means depending linearly on $p + q$ regression variables, a Behrens-Fisher generalization is to test the equality of $q$ regression coefficients in one population with a corresponding set in the second population. When $q = 1$ a general class of similar regions is obtained for the hypothesis, and for regions restricted to this class a most powerful or most powerful unbiased test is found. When $q > 1$ several tests are presented and discussed.

In a recent paper [1] the authors began the study of the theory of sequential decision functions for stochastic processes with a continuous time parameter. This paper treated the standard problem of testing hypotheses, and the advantage of being able to stop at an arbitrary time point (not necessarily a multiple of some unit given in advance) was demonstrated in several cases, notably in that of deciding between two Poisson processes. The optimal tests were Wald probability ratio tests and thus truly sequential. In the present paper we treat the problem of estimation, and study in detail the Poisson, Gamma, Normal and Negative Binomial processes. It turns out for these processes that, with a proper weight function, the minimax (sequential) rule reduces to a fixed-time rule. Though we confine ourselves to point-estimation it is clear that similar methods apply to interval estimation. It may also be remarked that the case when the time-parameter is discrete need not be treated separately. For example, as described in Section 6.1, the results of Sections 2 and 3 imply analogous results in the case of discrete time, which in turn imply certain results proved in [3] and (in the nonsequential case) in [2] by other methods. The treatment of some other problems in estimation is discussed in Section 6. This paper may be read independently of [1].

Let the random variable $X = (X_1, X_2, \cdots, X_n)$ have the probability density $p(x) = \frac{\det^{\frac{1}{2}} \Omega}{(2\pi)^{n/2}} e^{-\frac{1}{2}x\Omega x'}$ where $x\Omega x'$ is positive definite. The present article solves, by means of Laguerian expansions, the problem of finding the distribution of any nonnegative quadratic form $XPX'$. If the semimoments (defined below) are known, it also solves, by means of Laguerrian expansions the problems of finding the distribution of any indefinite quadratic form, and the distribution of the ratio of any indefinite quadratic form to any nonnegative quadratic form. For an outline of the procedure, see Section 2. If the distribution of the indefinite form is symmetric, the semimoments are easily found, but often, especially for the technique described below for ratios, the semimoments are difficult to obtain. In view of this, a new system of orthogonal polynomials is proposed, which is analogous to the Laguerre system, but which obviates the need of semimoments.

There is presently available a large number of techniques purporting to accomplish the inversion of matrices. While the purely mathematical aspects of this problem, on one hand, are thus well recognized, the computational ones, on the other hand, are not. The growth of the rounding error, in particular, may be so rapid as to make some inversion procedures altogether unstable. It is from this point of view that the partitioning method seems to be capable of yielding more accurate results than do other methods. By stopping, at any desired step, to improve the intermediate inverses until satisfactory accuracy is attained, the growth of the rounding error may be kept in check. The following sections, then, give a brief description of the partitioning method and treat in some detail an effective scaling scheme permitting the inversion routine to be carried out by high speed computing machinery. Next a careful examination is carried out of the accuracy attainable by the proposed scheme; together with an error squaring iteration procedure it is found capable of yielding accuracies sufficient for most practical purposes.

The paper considers classes of decision procedures which in certain ways put a bound on the associated losses of incorrect terminal decision or cost of experimentation. Conditions are given under which these classes fulfill the conditions that Wald imposes on classes of decision procedures in his general theory.

Three different testing procedures which involve a minimum of modification of the usual single sample tests of the hypotheses considered are given here. Tests are made by taking samples at two stages for testing the mean of a normal distribution. A known standard deviation is assumed, but an extension to the case where the standard deviation is unknown is also given. Special examples show that tests can be chosen so that the expected number of observations is less than the number required for the ordinary single sample test and indeed can give considerable savings. The tests in Sections 3 and 4 give the greater savings, but the powers are more difficult to evaluate than the power for the test of Section 2. Also, it is a little more work to apply the test in Section 4. Wald in [9] has discussed a sequential test where the observations are taken in groups. The tests given here could be considered very special cases of this where the number of observations is truncated after two groups. Roming in [7] has set up a double sampling procedure for sampling from a finite population that is approximately normal where the rejection points are determined by preassigned engineering or specifications limits and not by the normal distribution itself as is done for the first sample of the double sample tests given below. Bowker and Goode in [1] give tests similar to those given by Romig. Chapman in [2] and Stein in [8] have discussed two sample tests where the object is to obtain tests with the power independent of an unknown variance and where there is no upper limit on the number of observations required. There is a definite ceiling on the number of observations required for the tests presented here and they have many interesting properties that make them very desirable from the standpoint of saving of observations and simplicity.

Power functions are tabulated for the sign test for various sample sizes and $\alpha$ near .05 and .01. Several of these power functions are compared with the power function of the $t$-test for samples from normal populations by means of a power efficiency function. The results indicate decreasing power efficiency for increasing sample size, for increasing level of significance and for increasing alternative.

In his first continuous sampling plan [1], H. F. Dodge showed that his procedure guarantees an Average Outgoing Quality Limit (AOQL) with the assumption that the process is in a state of statistical control. It is proved in this paper that the Dodge procedure, without the assumption of control, guarantees an AOQL, although different from that specified by Dodge.

If $F(x)$ is a continuous distribution function of a random variable $X$, and $F_n(x)$ the empirical distribution function determined by a sample $X_1, X_2, \cdots, X_n$, then the probability $\mathrm{Pr}\{F(x) \leqq F_n(x) + \epsilon \text{for all} x\}$ is known [1] to be a function $P_n(\epsilon)$, independent of $F(x)$. A closed expression for $P_n(\epsilon)$ and a table of some of its values were presented in [2]. In the present paper $P_n(\epsilon)$ is used to test a hypothesis $F(x) = H(x)$ against an alternative $F(x) = G(x)$. The power of this test is studied and sharp upper and lower bounds for it are obtained for alternatives such that $\sup_{-\infty < x < + \infty}\{H(x) - G(x)\} = \delta$, with preassigned $\delta$. The results of [2] are assumed known.