In their book Studies in the Economics of Transportation, Beckmann, McGuire and Winsten (BMW) ([2], pp. 11-13, 40-42) proposed a simple queuing model for traffic flow through a fixed-cycle traffic light. Although they derived a relation between the average delay per car and the average length of the queue at the beginning of a red phase of the light, they only indicated some possible numerical schemes for evaluating the latter. Here we shall derive analytic expressions for the average queue length and consequently also the average delay under equilibrium conditions for the BMW model.

Recently there has been much activity dealing with the tabulation of the bivariate normal probability integral. D. B. Owen [3], [4] has summarized many of the properties of the bivariate normal distribution function and tabulated an auxiliary function which enables one to calculate the bivariate normal probability integral. In addition, the National Bureau of Standards [1] has compiled extensive tables of the bivariate normal integral drawn from the works of K. Pearson, Evelyn Fix and J. Neyman, and H. H. Germond. In this same volume, D. B. Owen has contributed an extensive section on applications. It is the purpose of this paper to present three charts, which will enable one to easily compute the bivariate normal integral to a maximum error of $10^{-2}$. This should be sufficient for most practical applications. Owen and Wiesen [5] have also presented charts with a similar objective; however, as pointed out below, we believe the charts presented here lend themselves more easily to visual interpolation. Actually the motivation for these charts came from the Owen and Wiesen work.

A method is given which provides, under conditions satisfied by many common distributions, rules for sampling in two stages so as to obtain an unbiased estimator of a given parameter, having variance equal to, or not exceeding, a prescribed bound. The method is applied to estimation of the means of binomial, Poisson, and hypergeometric distributions; scale-parameters in general and of the Gamma distribution in particular; the variance of a normal distribution; and a component of variance. The use of such estimators to achieve homoscedasticity is discussed. Optimum sampling rules are discussed for some of these estimators, and some tables are given to facilitate their use. The efficiency of the method is shown to be high in many cases.

Using the basic inequalities (1) it is shown that, if, in a sequential probability ratio test, the upper stopping bound is increased and the lower stopping bound decreased, and if the new test is not equivalent to the old one, then at least one of the error probabilities is decreased. This implies the monotonicity result of Weiss [5] in the continuous case, and the uniqueness result of Anderson and Friedman [1] in the general case. The relation of the monotonicity property to the optimum property and the uniqueness of sequential probability ratio tests is discussed. The monotonicity property is a consequence of the following stronger result. Let the old and new tests be given by the stopping bounds $(B', A')$ and $(B, A)$, respectively, with $B < B' < A' < A$; let $(\alpha'_1, \alpha'_2)$ and $(\alpha_1, \alpha_2)$ be the error probabilities and $\Delta\alpha_i = \alpha_i - \alpha'_i$ the changes in the error probabilities; then the vector $(\Delta\alpha_1, \Delta\alpha_2)$ is restricted to a cone consisting of the 3rd quadrant, plus the part of the 2nd quadrant where $-\Delta\alpha_2/\Delta\alpha_1 < B$, plus the part of the 4th quadrant where $-\Delta\alpha_2/\Delta\alpha_1 > A$. Another consequence of this result is that $(\alpha_1, \alpha_2)$ cannot lie in the closed triangle with vertices $(\alpha'_1, \alpha'_2)$, (0, 1) and (1, 0). Finally, the following monotonicity property follows: If the lower stopping bound is fixed and the upper stopping bound increased, then $\alpha_1/(1 - \alpha_2)$ decreases monotonically. The same holds for $\alpha_2/(1 - \alpha_1)$ if the upper stopping bound is held fixed and the lower stopping bound decreased.

In order to solve nonparametric statistical problems, it is often found useful to apply those criteria of optimality which are employed in parametric problems. In the present paper, we are concerned with nonparametric two-sample problems of testing the null hypothesis that two populations have the same distribution against certain nonparametric alternative hypotheses, and generalize the parametric optimality conditions of locally most powerfulness. A rank test for a two-sample problem in this paper is called locally most powerful if it is locally most powerful against a one-parameter family of alternatives. A criterion is constructed by which it is possible to solve the problem whether or not a given rank test is locally most powerful for a two-sample problem in which the set of all possible pairs of cumulative distribution functions is convex and closed (in the weak$^{\ast}$ topology). Let $X_1, \cdots, X_{n_1}$ be sample elements from the first population, $X_{n_1^{+1}}, \cdots, X_n (n = n_1 + n_2)$ from the second population, and the statistic $Z_j = 0$ or 1, according to whether the $j$th smallest observation is from the first population or the second, $j = 1, \cdots, n$. It will be shown that any locally most powerful rank test has the following form: \begin{align*}\tag{*} \begin{cases}\text{Reject the null hypothesis if} \sum_j a_jZ_j > c \\ \text{Accept the null hypothesis if} \sum_j a_jZ_j < c\end{cases}\end{align*} where $a_1, \cdots, a_n$ are constant numbers. For the two-sided two-sample problem, any rank test of the form $(^{\ast})$ is locally most powerful. For the one-sided two sample problem, a non-trivial rank test of the form $(^{\ast})$ is locally most powerful if, and only if, $c_j = \big\lbrack 1\big/\binom{n - 2}{j}\rbrack \sum^{n-2}_{s=j} \binom{s + 1}{j + 1} (a_{s+2} - \bar a),\quad j = 0, 1, \cdots, n - 2,$ where $\bar a = \frac{1}{n} (a_1 + \cdots + a_n),$ have all non-negative Hankel determinants (the precise definition of the Hankel determinants is given in Section 8 below). For the symmetric two-sided two-sample problem, a non-trivial rank test of the form $(^{\ast})$ is locally most powerful if, and only if, $\sum^n_{s=j+1} \binom{s - 1}{j} (a_{n+1-s} - a_s) = 0,\quad\text{for} j = 1, \cdots, n.$ Finally, it will be shown that for the two-sample problem, in which the alternative hypothesis is that the expectation of the first cumulative distribution function with respect to the second distribution is not less than $\frac{1}{2}$, a non-trivial rank test of the form $(^{\ast})$ is locally most powerful if, and only if, $\sum^{\lbrack(n+1)/2\rbrack}_{s=1} \big(\frac{n + 1}{2} - s\big) (a_{n-s} - a_s) \geqq 0.$

Let $X_1, X_2, \cdots, X_n$ be a sample of a one-dimensional random variable $X$ which has the continuous cumulative probability function (cpf) $F$. It has been observed that the distribution-free statistics commonly appearing in the literature can be written in the form $\Phi\lbrack F(X_1), F(X_2), \cdots, F(X_n)\rbrack$ where $\Phi$ is a measurable symmetric function defined on the unit cube. Such statistics are said to have structure (d). Birnbaum and Rubin [12] have proved that for the family $\Omega^{\ast}$, of strictly monotone continuous cpf's, statistics of structure (d) possess a property stronger than that of being distribution-free. The purpose of this paper is to study the extension of the Birnbaum-Rubin (B-R) result to other classes of cpf's and to present a different approach to these results. It is found that a one-sided extension of the B-R result is valid for all properly closed, symmetrically complete classes of cpf's. Then, from the existing literature on completeness, one can conclude that the extension is valid for several other classes of statistical interest. The relation between statistics of structure (d) and strongly distribution-free statistics (Section 1) is of importance for two reasons. First of all, if one is designing distribution-free tests, the results here and in [12] guarantee that if one chooses a statistic of structure (d), one has a strongly distribution-free statistic for several large classes of cpf's. On the other hand if one has a strongly distribution-free statistic, the results guarantee that it is of structure (d). Hence, its cpf can be written as the volume of a polyhedral region in the $n$ dimensional unit cube. Under such circumstances the work of Smirnov [20], Feller [13], Anderson and Darling [4], and Birnbaum [9] indicate that it should be possible to evaluate the cpf explicity; reduce it to a system of recursion formulae; tabulate it with the aid of high-speed computers or at least evaluate its limiting distribution. This article is divided into four sections. In Section 1 distribution-free statistics of various types are introduced. Section 2 contains some preliminary results concerning cpf's. The main theorem is proved in Section 3; and Section 4 contains a survey of the known pertinent completeness results as well as a corollary of the main theorem.

Let \begin{equation*}\tag{1.1}X^{(i)}_1,X^{(i)}_2, \cdots, X^{(i)}_{n_i},\qquad i = 1, 2, \cdots, c,\end{equation*} be samples of $c$ independent random variables $X^{(i)}$ with continuous cumulative distribution functions $F^{(i)}$, and let \begin{equation*}\begin{align*}F^{\ast^{(i)}}(x) &= 0\qquad x &< X^{(i)}_1 \\ \tag{1.2}F^{\ast^{(i)}} (x) &= k/n_i\qquad X^{(1)}_k &\leqq x < X^{(1)}_{k+1}, 1 \leqq k < n_i \\ F^{\ast^{(i)}} (x) &= 1\qquad X^{(i)}_{ni} \leqq x\end{align*}\end{equation*} be the corresponding $c$ emprirical distribution functions. We define the statistics \begin{equation*}\tag{1.3} D(n_1, n_2, \cdots, n_c) = \sup_{\substack{x, i, j\\(i,j=1,2,\cdots,c)}} |F^{\ast(i)} (x) - F^{\ast(j)} (x)|\end{equation*} and \begin{equation*}\tag{1.4} D^+(n_1, n_2, \cdots, n_c) = \sup_{\substack{x,i,j\\(i<j;i,j=1,2,\cdots,c)}} \brack F^{\ast(i)} (x) - F^{\ast(j)} (x)\rbrack.\end{equation*} The well known Kolmogorov-Smirnov statistics $D(m, n)$ and $D^+(m, n)$ are special cases of (1.3) and (1.4), respectively, with $c = 2, n_1 = m, n_2 = n$. The exact small distribution, under the null hypothesis \begin{equation*}\tag{1.5} F^{(i)} = F^{(j)}\quad\text{for all} i,j = 1, 2, \cdots, c,\end{equation*} of the statistics defined by (1.3) and (1.4) for any number $c$ of samples, and for any sample sizes $n_1, n_2,\cdots,n_c$, can be obtained by solving simple difference equations which lend themselves to programming for machine computation. Using this procedure, tables of values of $P\lbrack D(n, n, n) \leqq r\rbrack,\quad P\lbrack D(n, n) \leqq r\rbrack,\quad P\lbrack D^+(n,n) \leqq r\rbrack$ were computed for selected values of $n$ between 1 and 40 and of $r = k/n,k = 1, 2, \cdots, n$. Furhermore, the inequalities \begin{equation*}\begin{align*} P\lbrack D(n, n,\cdots, n) \leqq r\rbrack \geqq 1 - \rbrack c(c - 1)/2\rbrack P\lbrack D(n,n) > r\rbrack \\ P\lbrack D(n, n, \cdots, n) \leqq r\rbrack \geqq 1 - \lbrack c(c - 1)(c - 2)/6\rbrack P\lbrack D(n, n, n) > r\lbrack\end{align*}\end{equation*} are noted, which may be useful for values of $c \geqq 4$ for which tables are not available.

It is well known [3, 4] how, in the case of any general strongly testable [5] linear hypothesis for either ANOVA or MANOVA one can put simultaneous confidence bounds on a particular set of parametric functions, which might be regarded as measures of deviation from the "total" hypothesis and its various components. The parametric functions are such that, in each problem, one of these can be appropriately called the "total" and the rest "partials" of various orders. For each problem the "total" function, (i) in the univariate case, is related to, but not quite the same as, the noncentrality parameter of the usual $F$-test of the "total" hypothesis in ANOVA, and (ii) in the multivariate case, is the largest characteristic root of a certain parametric matrix which is related to, but not quite the same as, another parametric matrix whose nonzero characteristic roots occur as a set of noncentrality parameters in the power function for the test (no matter which of the standard tests we use) of the "total" hypothesis in MANOVA. The same remark applies to "partials" of various orders considered in the proper sense. In this note, for both ANOVA and MANOVA, the hypothesis considered is that of equality of the treatment effects--vector equality in the case of MANOVA. Starting from such a hypothesis, explicit algebraic expressions are obtained for the total and partial parametric functions that go with the simultaneous confidence statements in the case of both ANOVA and MANOVA and for balanced and partially balanced designs. It is also indicated how to obtain, in a convenient form, the algebraic expression for the confidence bounds on each such parametric function, without a derivation of these expressions in an explicit form.

A class of multifactorial designs are defined and analyzed. The designs considered have each a total number of observations that can not be divided equally among the cells of the designs; however, by distributing the observations in a way that is in a certain sense symmetrical, the equations that determine the least squares estimates of the linear parameters become explicitly solvable. The case of two non-interacting factors with arbitrary numbers of levels is treated first. In the $n$-factor case we have to restrict ourselves to factors having equal numbers of levels. After defining the designs, the estimates are computed. Some general discussions of the symmetries and algebraic properties involved conclude the paper.

Roy [8] extended the idea of Group Divisible designs of Bose and Connor [1] to $m$-associate classes, calling such designs Hierarchical Group Divisible designs with $m$-associate classes. Subsequently, no literature is found in this direction. The purpose of this paper is to study these designs systematically. A compact definition of the design, under the name Group Divisible $m$-associate (GD $m$-associate) design is given in Section 2. In the same section the parameters of the design are obtained in a slightly different form than that of Roy. The uniqueness of the association scheme from the parameters is shown in Section 3. The designs are divided into $(m + 1)$ classes in Section 4. Some interesting combinatorial properties are obtained in Section 5. The necessary conditions for the existence of a class of these designs are obtained in Section 7. Finally, some numerical illustrations of these designs are given in the Appendix.

In the case of some incomplete block designs, interesting relations among their blocks have been discovered. For example, Fisher [1] has shown that in the case of a symmetrical BIB (Balanced Incomplete Block) design with parameters $v = b, r = k, \lambda$, any two blocks have exactly $\lambda$ treatments in common. Similarly, Bose [2] has shown that in the case of an affine resolvable BIB design with parameters $v = nk = n^2\{(n - 1)t + 1\}, \quad b = nr = n\{n^2t + n + 1\}, \quad \lambda = nt + 1,$ the blocks can be divided into sets of $n$ blocks, such that each set is a complete replication and any two blocks have $(k^2)/v = (nt - t + 1)$ or 0 treatments in common according as they belong to different groups or the same group. Also see Connor [3] and Bose and Connor [4] for similar results. Confining our attention to PBIB (Partially Balanced Incomplete Block) designs with two or three associate classes, we wish to see how this type of information for blocks of BIB designs can be used to obtain similar information for the blocks of their Kronecker product. In the next section are given a few general properties of the Kronecker product of designs. In Section 3 the main theorems of the paper are proved and their important particular cases are discussed. Some observations on the interconnection between these results and the theorems on inversion of designs (cf. Roy [5], Shrikhande [6]) are made in Section 4.

Shrikhande [9] and Roy [7] have shown that certain Balanced Incomplete Block Designs (BIBDs) can be dualised to give Partially Balanced Incomplete Block Designs (PBIBDs) with exactly two associate classes. Roy and Laha [8] have obtained a necessary and sufficient condition for the dual of a BIBD to be a PBIBD with two associate classes. In this paper, a general result regarding the dual of a BIBD is established and the results of Shrikhande and Roy are obtained as particular cases. An illustration to show the use of the result when the dual is not a 2-associate PBIBD is also given.

The PBIB designs [2] with two associate classes are classified in [3] as 1. Group Divisible, 2. Simple, 3. Triangular, 4. Latin Square type with $i$ constraints, and 5. Cyclic. Group Divisible designs are divided into three types [1]: 1. Singular, 2. Semi-regular, and 3. Regular. It has been proved [1] that every block of a Semi-regular Group Divisible design contains $k/m$ treatments from each of the $m$ groups of the association scheme. In this note we prove analogous results in the case of certain PBIB designs with triangular and $L_2$ association schemes.

Several optimality criteria have been suggested for the efficiency of incomplete block designs. This note surveys these criteria, extends certain results and puts forward a new and simpler criterion.

Let $X_1, X_2, \cdots, X_n$ be a sample of a one-dimensional random variable $X$; let the order statistic $T(X_1, X_2, \cdots, X_n)$ be defined in such a manner that $T(x_1, x_2, \cdots, x_n) = (x^{(1)}, x^{(2)}, \cdots, x^{(n)})$ where $x^{(1)} \leqq x^{(2)} \leqq \cdots \leqq x^{(n)}$ denote the ordered $x's$; and let $\Omega$ be a class of one-dimensional cpf's, i.e., cumulative probability functions. The order statistic, $T$, is said to be a complete statistic with respect to the class, $\{P^{(n)} \mid P \epsilon \Omega\}$, of $n$-fold power probability distributions if $E_p^{(n)}\{h\lbrack T(X_1, \cdots, X_n)\rbrack\} = 0$ for all $P \epsilon \Omega$ implies $h\lbrack T(x_1, \cdots, x_n)\rbrack = 0, a.e., P^{(n)}$, for all $F \epsilon \Omega$. The class $\Omega$ is said to be symmetrically complete whenever the latter condition holds. Since the completeness of the order statistic plays an essential role in nonparametric estimation and hypothesis testing, e.g., Fraser [2] and Bell [1], it is of interest to determine those classes of cpf's for which the order statistic is complete. Many of the traditionally studied classes of cpf's on the real line are known to be symmetrically complete, e.g., all continuous cpf's ([4], pp. 131-134, 152-153); all cpf's absolutely continuous with respect to Lebesgue measure ([3], pp. 23-31); and all exponentials of a certain form ([4], pp. 131-134). The object of this note is to present a different ([4], pp. 131-134, 152-153) demonstration of the symmetric completeness of the class of all continuous cpf's; and to extend this and other known completeness results to probability spaces other than the real line, e.g., Fraser [2], and Lehmann and Scheffe [5], [6]. The paper is divided into four sections. Section 1 contains the introduction and summary. In Section 2 the notation and terminology are introduced. The main theorem is presented in Section 3, and some consequences of the proof of the main theorem and known results are indicated in Section 4.

The paper discusses the general problem of emptiness in the finite dam and considers the probability that, starting with an arbitrary storage, the dam dries up before it fills completely. Some exact results are given both for discrete and continuous inputs. An interesting relation between this probability and the asymptotic distribution function of the dam content has also been obtained.