An operational method is used to obtain known formulas [2] for the expected values of the mean squares and the variances of estimates of variance components obtained from the analysis of variance of an $r$-way crossed classification. The results are independent of normality assumptions.

Vartak [6] has considered the construction of experimental designs with the help of Kronecker products of matrices. The method is equivalent to the replacement of two elements, 0 and 1, by two matrices. A generalisation of the above idea is given by the author [4], using only the incidence matrices of balanced incomplete block (BIB) designs for substitution. In the present paper the same idea is extended to the case where substitution is by the incidence matrices of partially balanced incomplete block (PBIB) designs and factorial experiments. In Sections 2 and 3 some ideas regarding canonical vectors and PBIB designs are introduced. Section 4 deals with associable designs and their properties. In Section 5 balanced matrices are defined and in Section 6 a method is given for constructing designs by substituting for the elements of a balanced matrix, the incidence matrices of associable designs. The application of this method to the construction of factorial experiments is considered in Section 7.

Let $p$ be given, $0 < p < 1$. Let $n$ and $k$ be positive integers such that $np \leqq k \leqq n$, and let $B_n(k) = \sum^n_{r=k} \binom{n}{r} p^rq^{n-r}$, where $q = 1 - p$. It is shown that $B_n(k) = \big\lbrack\binom{n}{k} p^kq^{n - k}\big\rbrack qF(n + 1, 1; k + 1; p),$ where $F$ is the hypergeometric function. This representation seems useful for numerical and theoretical investigations of small tail probabilities. The representation yields, in particular, the result that, with $A_n(k) = \big\lbrack\binom{n}{k}p^kq^{n - k + 1}\big\rbrack \lbrack(k + 1)/(k + 1 - (n + 1)p)\rbrack$, we have $1 \leqq A_n(k)/B_n(k) \leqq 1 + x^{-2}$, where $x = (k - np)/(npq)^{\frac{1}{2}}$. Next, let $N_n(k)$ denote the normal approximation to $B_n(k)$, and let $C_n(k) = (x + \sqrt{q/np}) \sqrt{2\pi} \exp \lbrack x^2/2 \rbrack$. It is shown that $(A_nN_nC_n)/B_n \rightarrow 1$ as $n \rightarrow \infty$, provided only that $k$ varies with $n$ so that $x \geqq 0$ for each $n$. It follows hence that $A_n/B_n \rightarrow 1$ if and only if $x \rightarrow \infty$ (i.e. $B_n \rightarrow 0$). It also follows that $N_nN_n \rightarrow 1$ if and only if $A_nC_n \rightarrow 1$. This last condition reduces to $x = o(n^{1/6})$ for certain values of $p$, but is weaker for other values; in particular, there are values of $p$ for which $N_n/B_n$ can tend to one without even the requirement that $k/n$ tend to $p$.

If $\mathcal{F} = \{F\}$ is a family of distribution functions and $\mu$ is a measure on a Borel Field of subsets of $\mathcal{F}$ with $\mu(\mathcal{F}) = 1$, then $\int F(\cdot) d\mu (F)$ is again a distribution function which is called a $\mu$-mixture of $\mathcal{F}$. In Section 2, convergence questions when either $F_n$ or $\mu_k$ (or both) tend to limits are dealt with in the case where $\mathcal{F}$ is indexed by a finite number of parameters. In Part 3, mixtures of additively closed families are considered and the class of such $\mu$-mixtures is shown to be closed under convolution (Theorem 3). In Section 4, a sufficient as well as necessary conditions are given for a $\mu$-mixture of normal distributions to be normal. In the case of a product-measure mixture, a necessary and sufficient condition is obtained (Theorem 7). Generation of mixtures is discussed in Part 5 and the concluding remarks of Section 6 link the problem of mixtures of Poisson distributions to a moment problem.

In a priority queue different types of items (individuals or elements) arrive at a service mechanism and each item has a relative priority for order of service. Let there be $K$ classes of items, $1, 2, \cdots, K$. If the service mechanism is to select an item for service, a type $i$ item will be selected in preference to a type $j$ item for $i < j$ even if the type $j$ item arrived before the type $i$ item, and within each class the "first come, first served" policy determines the order of service. When a type $j$ item is in service and a type $i$ item arrives $(i < j)$, there are two primary disciplines for handling the priority demand. The "head-of-the-line" discipline allows the type $j$ item to complete service but places the type $i$ item ahead of any other lower priority items. The "preemptive" discipline withdraws the type $j$ item from service and replaces it by the type $i$ item. Under the preemptive scheme the only time at which a type $j$ item $(1 < j)$ can be in service is when there are no items of types $1, \cdots, j - 1$ in the queue. When a lower priority item which has been preempted returns to service, the preemptive discipline must distinguish two cases. The "preemptive resume" policy allows the preempted item to resume service at the point at which it was preempted so that its service time upon reentry has been reduced by the amount of time the item has already spent in service. The "preemptive repeat" policy requires the preempted item to commence service again at the beginning. A priority queue with an indifferent server is of course a special case of the preemptive resume discipline. In the special case $K = 2$ the type 1 items will be referred to as priority items and the type 2 items as non-priority items. It will be assumed throughout this paper that the input process for type $i$ items, $i = 1, \cdots, K$, is Poisson with arrival rate $\lambda_i$ and the input processes operate independently. The service time distribution for a type $i$ item (in isolation) will be denoted by $F_{s_i}$ and unless explicitly stated to the contrary will be assumed to be general subject only to the restrictions $F_{s_i}(0+) = 0$ and $E(S_i) < \infty$. Let $\rho_i = \lambda_iE(S_i)$, and let $\tilde S_i$ be the Laplace-Stieltjes transform of $F_{s_i}$. The service mechanism consists of a single channel or server. A. Cobham [1], [2] introduced the head-of-the-line priority queue and derived equilibrium expected waiting times. Subsequent contributions have been made by Holley [3], Kesten and Runnenburg [4], [5], and Morse [6]. The first published results for the preemptive discipline were by H. White and L. S. Christie [7], and additional results have been presented by Stephan [8]. Koenigsberg [9] has generalized the priority model to a continuous number of priority types with application to machine breakdown problems. Under various assumptions in this paper the following quantities have either been obtained explicitly or characterized as the unique (subject to regularity conditions) solution to a functional equation: the generating function for the stationary probabilities on the number of priority and non-priority items $(K = 2)$ in the queue, the Laplace-Stieltjes transforms of the waiting time distributions, the Laplace-Stieltjes transform of the distribution of a busy period, and the generating function for the probabilities on the number of items serviced during a busy period. For most of the distributions mentioned the first two moments are computed.

The distribution of the latent roots of the covariance matrix calculated from a sample from a normal multivariate population, was found by Fisher [3], Hsu [6] and Roy [10] for the special, but important case when the population covariance matrix is a scalar matrix, $\Sigma = \sigma^2I$. By use of the representation theory of the linear group, we are able to obtain the general distribution for arbitrary $\Sigma$.

$X(1), X(2), \cdots, X(m), Y(1), Y(2), \cdots, Y(n)$ are independent $k$-variate random variables. The distribution of $X(i)$ has pdf $f(x)$, say, where $x$ denotes a $k$-dimensional vector throughout this paper, and the distribution of $Y(j)$ has pdf $g(x)$, say. We assume that $f(x)$ and $g(x)$ are piecewise continuous, and that each has a finite upper bound, which it is not necessary to specify. Denote by $2R_i$ the distance from $X(i)$ to the nearest of the points $X(1), \cdots, X(i - 1), X(i + 1), \cdots, X(m)$, and denote by $S_i$ the number of points $Y(1), \cdots, Y(n)$ contained in the open sphere $\{x: | x - X(i) | < R_i\}$. Clearly, the joint distribution of $S_i, S_j$ is the same as the joint distribution of $S_{i'}, S_{j'}$, for any subscripts with $i \neq j, i' \neq j'$. Let $r$ be a non-negative integer, and $\alpha$ any fixed positive value. $Q(r)$ denotes the Lebesgue integral $\int_{E_k} \frac{2^k \alpha f^2 (x)\lbrack g(x) \rbrack^r}{\lbrack g(x) + 2^k\alpha f(x) \rbrack^{r + 1}} dx,$ where $E_k$ denotes Euclidean $k$-space. We will show that $\lim_{m \rightarrow \infty, m/n = \alpha} P_{m, n}\lbrack S_1 = s_1, S_2 = s_2\rbrack = Q(s_1)Q(s_2),$ for any non-negative integers $s_1,s_2$, the approach being uniform in $s_1,s_2$. Thus, in the limit $S_1, S_2$ are independently distributed, with $\lim_{m \rightarrow \infty, m/n = \alpha} P_{m, n}\lbrack S_1 = s_1\rbrack = Q(s_1).$ In [1], which discussed the univariate case, $S_i$ was defined as the number of $Y$'s closer to $X(i)$ than to any other $X$ to their right. In the present paper, $S_i$ is defined as the number of $Y$'s in another neighborhood of $X(i)$. Our present definition of $S_i$ does not become for $k = 1$ the same as the definition of $S_i$ in [1]. Rather, in the univariate case, our present definition of $S_i$ is the number of $Y$'s lying within a distance $R_i$ on either side of $X(i)$. However, if $\lim_{m \rightarrow \infty, m/n = \alpha} P_{m, n}\lbrack S_1, = s_1, S_2 = s_2\rbrack$ is computed for the univariate case using the definition of $S_i$ given in [1], the only way in which it differs from $Q(s_1)Q(s_2)$ is that $\alpha$ is replaced by $\alpha/2$. Thus it seems reasonable to treat the $S_i$ as defined here as $k$-dimensional analogues of the $S_i$ as defined in [1], at least for large samples. An intuitive reason for $\alpha$ being replaced by $\alpha/2$ is that in our present case, $\sum^m_{i = 1} S_i$ may be less than $n$, whereas in [1] this sum must always equal $n$. Thus in our present case, we are in a sense discarding some of the $Y$'s, which lowers $n$ relative to $m$ and thus raises $\alpha$ by a certain factor (2, as it happens). In our present case, $\sum S_i$ may be less than $n$ because the $R_i$ are chosen to make the spheres around the $X$'s non-overlapping, thus simplifying the analysis. The $R_i$ were chosen to give the largest possible non-overlapping spheres because it would seem intuitively that the larger the spheres, the more rapid the approach of the probabilities to their limiting values.

The sequential probability ratio test is constructed as a sequential test of one simple hypothesis against another. In many instances a parametric form is assumed for the density or (discrete) probability function, and the two simple hypotheses are specified by two values of the parameter. The sequential probability ratio test has an optimum property for these two hypotheses, namely, given such a test there is no other test with at least as low probabilities of Type I and Type II errors and with smaller expected sample sizes under either or both of the two hypotheses. Usually, however, one is interested in the performance of the procedure for more values of the parameter than these two. A disadvantage of the sequential probability ratio test is that in general the expected sample size is relatively large for values of the parameter between the two specified ones; that is, in cases in which one does not care greatly which decision is taken, a large number of observations is expected. The question is how to reduce the expected sample size for values of the parameter when this tends to be large. In this paper we consider a special case of the problem, when the distribution is normal with known variance and the parameter of interest is the mean. The sequential probability ratio test in this case consists in taking observations sequentially and after each observation is taken comparing the sum of the observations (referred to a suitable origin) with two constants. In this study the two constants are replaced by two linear functions of the number of observations taken, and the taking of observations is truncated (Section 2). Approximations to the operating characteristic (or power function) and the average sample size number are given (Section 4 and 5). Computations for two cases of special interest show a considerable decrease in average sample size at parameter values between the two specified ones (Section 3). The problem is studied by replacing the sum of observations by the Wiener stochastic process (of a continuous time parameter); this can be thought of intuitively as interpolating between observations in a manner consistent with the addition of independent random variables. For this procedure we calculate exactly the operating characteristic, the distribution of observation time, the expected observation time, and related probabilities.

Consider a measurable space with a linear ordering on the space and the family of all probability measures which assign measure zero to each equivalence class induced by the ordering. For such a space and family of probability distributions sequential tolerance regions are defined. The procedure assigns for each finite sample a Borel set with boundaries determined by the order observations. The sampling terminates when the region remains unchanged for a certain number of observations. The coverage of the region thus sequentially determined is distribution free with respect to that family of distributions. Some relationships are derived between the distribution of the coverage and the generating function of the random sample size, which permit the determination of one in terms of the other. This paper includes as a special case the previous results of Jirina on the distribution of coverage for his sequential procedure. Also, formulae are obtained for the expected sample sizes of the Jirina procedure which were previously unknown. The results of Wilks for fixed sample tolerance limits are obtained as a limiting case and comparisons are made with sequential procedures in terms of coverage and expected sample size. For example it is shown that for one-sided tolerance limits no sequential procedure is as good as Wilks fixed sample procedure in the sense that if the expected sample sizes are the same the coverage of the Wilks procedure is stochastically greater than the coverage of the sequential procedure.

Let $(x_{nk}), k = 1, 2, \cdots, k_n; n = 1, 2, \cdots$ be a double sequence of infinitesimal random variables which are rowwise independent (i.e. $\lim_{n \rightarrow \infty} \max_{1 \leqq k \leqq k_n} P(|x_{nk}| > \epsilon) = 0$ for every $\epsilon > 0$, and for each $n, x_{n1}, \cdots, x_{nk_n}$ are independent). Let $S_n = x_{n1} + \cdots + x_{nk_n} - A_n$ where the $A_n$ are constants and let $F_n(x)$ be the distribution function of $S_n$. In a previous paper [3] the system of infinitesimal, rowwise independent random variables $(|x_{nk}|^r)$ was studied for $r \geqq 1$. Specifically, let $S^r_n = |x_{n1}|^r + \cdots + |x_{nk_n}|^r - B_n(r),$ where the $B_n(r)$ are suitably chosen constants. Let $F_n^r(x)$ be the distribution function of $S^r_n$. Necessary and sufficient conditions for $F_n^r(x)$ to converge $(n \rightarrow \infty)$ to a distribution function $F^r(x)$ and for $F^r(x)$ to converge $(r \rightarrow \infty)$ to a distribution function $H(x)$ were given, together with the form that $H(x)$ must take. In Section 2 of this paper we consider the system $(|x_{nk}|^r)$ for $0 < r < 1$. Results similar to the above are found, replacing $(r \rightarrow \infty)$ by $(r \rightarrow 0^+)$. However different assumptions must be made at certain points. Various remarks are made in this paper to show where the results here differ from [3]. In particular it is shown that, if $F^r(x)$ converges $(r \rightarrow 0^+)$ to a distribution function $H(x)$, then $H(x)$ will be the distribution function of the sum of two independent random variables, one Poisson and the other Gaussian. Furthermore, while the Gaussian summand may or may not be degenerate, the Poisson summand will be nondegenerate in all but one special case.