Problems involving dependent pairs of variables $(X, Y)$ have been studied most intensively in the case of bivariate normal distributions and of $2 \times 2$ tables. This is due primarily to the importance of these cases but perhaps partly also to the fact that they exhibit only a particularly simple form of dependence. (See Examples 9(i) and 10 in Section 7.) Studies involving the general case center mainly around two problems: (i) tests of independence; (ii) definition and estimation of measures of association. In most treatments of these problems, there occurs implicitly a concept which is of importance also in other contexts (for example, the evaluation of the performance of certain multiple decision procedures), the concept of positive (or negative) dependence or association. Tests of independence, for example those based on rank correlation, Kendall's $t$-statistic, or normal scores, are usually not omnibus tests (for a discussion of such tests see [4], [15] and [17], but designed to detect rather specific types of alternatives, namely those for which large values of $Y$ tend to be associated with large values of $X$ and small values of $Y$ with small values of $X$ (positive dependence) or the opposite case of negative dependence in which large values of one variable tend to be associated with small values of the other. Similarly, measures of association are typically designed to measure the degree of this kind of association. The purpose of the present paper is to give three successively stronger definitions of positive dependence, to investigate their consequences, explore the strength of each definition through a number of examples, and to give some statistical applications.

The paper studies hypothesis testing problems for the mean of a vector variate having a multivariate normal distribution, in cases where the alternative is restricted by a number of linear inequalities. A new criterion which can be regarded as a generalization of the "maximin-$r^2$" criterion of Abelson and Tukey (cf. [1]) is introduced: we try to obtain tests which are "most stringent" among the "somewhere most powerful" tests, (Section 2). For an important class of testing problems (Section 3) such tests can be characterized by a (half-) line $l_0$ minimizing a maximum angle (Sections 4, 6 and 7). This (half-) line $l_0$ can be obtained by means of a method described by Abelson and Tukey (Section 5). The theory of this paper can be applied to a large number of actual testing problems. Such applications, (one of which is treated in Section 8), generalizations and details of the theory of this paper will be considered in the thesis [5].

The present study is concerned with the properties of a test statistic proposed by H. Chernoff and S. Zacks [1] to detect shifts in a parameter of a distribution function, occurring at unknown time points between consecutively taken observations. The testing problem we study is confined to a fixed sample size situation, and can be described as follows: Given observations on independent random variables $X_1, \cdots, X_n$, (taken at consecutive time points) which are distributed according to $F(X; \theta_i); \theta_i \varepsilon \Omega$ for all $i = 1, \cdots, n$, one has to test the simple hypothesis: $H_0 : \theta_1 = \cdots = \theta_n = \theta_0$ ($\theta_0$ is known) against the composite alternative: $H_1 : \theta_1 = \cdots = \theta_m = \theta_0 \\ \theta_{m + 1} = \cdots = \theta_n = \theta_0 + \delta;\quad\delta > 0,$ where both the point of change, $m$, and the size of the change, $\delta$, are unknown $(m = 1, \cdots, n - 1), 0 < \delta < \infty$. A Bayesian approach led Chernoff and Zacks in [1] to propose the test statistic $T_n = \sum^{n - 1}_{i = 1} iX_{i + 1}$, for the case of normally distributed random variables. A generalization for random variables, whose distributions belong to the one parameter exponential family, i.e., their density can be represented as $f(x; \theta) = h(x) \exp \lbrack\psi_1(\theta)U(x) + \psi_2(\theta)\rbrack, \theta \varepsilon \Omega$ where $\psi_1(\theta)$ is monotone, yields the test statistic $T_n = \sum^{n - 1}_{i = 1} iU(x_{i + 1})$. In the present paper we study the operating characteristics of the test statistic $T_n$. General conditions are given for the convergence of the distribution of $T_n$, as the sample size grows, to a normal distribution. The rate of convergence is also studied. It was found that the closeness of the distribution function of $T_n$ to the corresponding normal distribution is not satisfactory for the purposes of determining test criteria and values of power functions, in cases of small samples from non-normal distributions. The normal approximation can be improved by considering the first four terms in Edgeworth's asymptotic expansion of the distribution function of $T_n$ (see H. Cramer [2] p. 227). Such an approximation involves the normal distribution, its derivatives and the semi-invariants of $T_n$. The goodness of the approximations based on such an expansion, and that of the simple normal approximation, for small sample situations, were studied for cases where the observed random variables are binomially or exponentially distributed. In order to compare the exact distribution functions of $T_n$ to the approximations, the exact forms of the distributions of $T_n$ in the binomial and exponential cases were derived. As seen in Section 4, these distribution functions are quite involved, especially under the alternative hypothesis. Tables of coefficients are given for assisting the determination of these distributions, under the null hypothesis assumption, in situations of samples whose size is $2 \leqq n \leqq 10$. For samples of size $n \geqq 10$ one can use the normal approximation to the test criterion and obtain good results. The power functions of the test statistic $T_n$, for the binomial and exponential cases, are given in Section 5. The comparison with the values of the power function obtained by the normal approximation is also given. As was shown by Chernoff and Zacks in [1], when $X$ is binomially distributed the power function values of $T_n$ are higher than those of a test statistic proposed by E. S. Page [5], for most of the $m$ values (points of shift) and $\delta$ values (size of shift). A comparative study in which the effectiveness of test procedures based on $T_n$ relative to those based on Page's and other procedures will be given elsewhere for the exponential case, and other distributions of practical interest.

The main result proved here (Theorem 3) is a generalization of a formula of Goldstein [2], who showed that if the estimate $\hat S(\omega)$ for the spectral density is computed by the use of the function $y(x) = \operatorname{sgn} (x)$, and the spectrum is flat, then the dominant term in the variance of $\hat S(\omega)$ is $\frac{1}{2}\pi^2K/N$. Theorem 3 evaluates this term for nonflat spectra and for more general functions $y(x)$. This analysis shows that the loss in accuracy caused by working with $y(x)$ instead of $x$ itself can be decreased considerably by using for $y(x)$ a step function with more than two values. Some results on Gaussian process, interesting in their own right, are proved along the way.

The work presented here is a continuation of that presented in an earlier paper, M. Rosenblatt and J. Van Ness [15], in which the basic properties (unbiasedness and consistency) of certain estimates of the bispectrum and bispectral density are discussed. (The bispectrum can be thought of as the Fourier transform of the third-order moment function of the process.) The present paper is concerned with the asymptotic distribution of these estimates. One would expect that under certain regularity conditions these estimates would have distributions which tend to complex normal distributions. The following develops two different sets of such conditions either of which suffice. The first set involves a uniform summability condition on the first six cumulants of a sequence of processes obtained from the original process by projecting it onto a certain sequence of Borel fields. The second, and much more intuitively meaningful, set involves the strong mixing condition (see Rosenblatt [11], [12]; Kolmogorov and Rozanov [6], and Volkonskii and Rozanov [18] e.g.). The calculations in the earlier paper were carried out for the continuous parameter case; here we restrict ourselves to the discrete parameter case. For reasons for interest in polyspectra see, for example, Rosenblatt and Van Ness [15]; Hasselmann, Munk and MacDonald [4]; and Brillinger [2].

In an attempt to refine the axiomatic model of a probability space introduced by Kolmogorov [11], Gnendenko and Kolmogorov [5] introduced the concept of a perfect probability measure. The desirability of some sort of refinement has been pointed out by several well known examples [2], [3], [8], which display a certain amount of pathology inherent in Kolmogorov's theory. It is known that if $(X, \mathscr{S}, \mu)$ is a probability space and $\mu$ is perfect, then each of the examples mentioned is ruled out. There have been attempts (e.g., [1], [10]) at characterizing all those measurable spaces $(X, \mathscr{S})$ having the property that every probability measure $\mu$ on $\mathscr{S}$ is perfect. There have also been investigations [12], [13], [14] of the measure-theoretic properties of perfect measures and their relationships to other concepts in measure theory and probability theory. In this paper, we consider mixtures of perfect probability measures and their relationship to regular conditional probabilities. In Section 2, mixtures are defined and some interesting special cases are considered. Section 3 is a brief study of the imbedding of mixtures in a regular conditional probability space. Using known results and some results from Section 2, the perfectness of the underlying probability measure of a regular conditional probability space is characterized. Unless explicit mention is made to the contrary, the notation and terminology of [6] will be used throughout; however, the word "function," used without qualification, will always mean "real-valued function." If $(X, \mathscr{S}, \mu)$ is a probability space, then $\mu$ is called perfect if for every $\mathscr{S}$-measurable function $f$ on $X$ and every set $A$ on the real line for which $f^{-1}(A)$ belongs to $\mathscr{S}$, there is a linear Borel set $B$ contained in $A$ such that $\mu(f^{-1}(A)) = \mu(f^{-1} (B))$. It is known (see [13]): (1) that a measure is perfect if and only if its restriction to every countably-generated sub-sigma-algebra is perfect, and (2) that the restriction to any sub-sigma-algebra of a perfect measure is perfect. Since the following characterization of perfectness will be used frequently in the sequel, we quote it here as a lemma. A proof may be found in [14]. Lemma 1. A measure $\mu$ on a measurable space $(X, \mathscr{S})$ is perfect if and only if for every $\mathscr{S}$-measurable function $f$ on $X$ there is a linear Borel set $B(f)$ contained in $f(X)$ such that $\mu(f^{-1}(B(f))) = \mu(X)$. The following definitions are those of Jirina [9]. Let $(X, \mathscr{S}, \mu)$ be a probability space and let $\mathscr{S}_1$ and $\mathscr{S}_2$ be sub-sigma-algebras of $\mathscr{S}$. Any function $\mu(\cdot, \cdot \mid \mathscr{S}_1, \mathscr{S}_2)$ defined on $\mathscr{S}_1 \times X$ will be called a conditional probability (c.p.) if it satisfies: CP1. for fixed $S$ in $\mathscr{S}_1, \mu(S, \cdot \mid \mathscr{S}_1, \mathscr{S}_2)$ is $\mathscr{S}_2$-measurable, and CP2. for every $S$ in $\mathscr{S}_1$, and every $T$ in $\mathscr{S}_2$, $\mu(S \cap T) = \int_T \mu(S_1, x \mid \mathscr{S}_1, \mathscr{S}_2) d(\mu \mid \mathscr{S}_2),$ where $(\mu \mid \mathscr{S}_2)$ is the restriction of $\mu$ to $\mathscr{S}_2$. If the c.p. $\mu(\cdot, \cdot \mid \mathscr{S}_1, \mathscr{S}_2)$ also satisfies: CP3. for each fixed $x$ in $X, \mu(\cdot, x \mid \mathscr{S}_1, \mathscr{S}_2)$ is a probability measure on $\mathscr{S}_1$, then it will be called a regular conditional probability (r.c.p.). It is well known that if $(X, \mathscr{S}, \mu)$ is a probability space, $\mathscr{S}_1$ is any sub-algebra of $\mathscr{S}$, and $\mathscr{S}_2$ is any sub-sigma-algebra of $\mathscr{S}$, then there is a c.p. $\mu(\cdot, \cdot \mid \mathscr{S}_1, \mathscr{S}_2)$. We shall denote by $\mu(\cdot, \cdot \mid \mathscr{S}, \mathscr{S}_2)$ any function satisfying CP1 and CP2, with $\mathscr{S}_1 = \mathscr{S}$.

Let $X_1, \cdots, X_n$ be independent random variables having a common, continuous distribution function (df) $F$ and define \begin{align*}\tag{1.1a}S_k &= S_k(X) = \sum^k_{j = 1} X_j,\quad k = 1, \cdots, n, \\ \tag{1.1b}S_V &= S_V(X) = \sum_{j\varepsilon V} X_j,\quad \varnothing \neq V \subseteq \{1, \cdots, n\}, \\ \tag{1.1c}M_n &= M_n(X) = \sum^n_{k = 1} e(S_k), \\ \tag{1.1d}N_n &= N_n(X) = \sum_{V \neq \varnothing} e(S_v),\\ \end{align*} where $X = (X_1, \cdots, X_n), e$ is the indicator function of $(0, \infty)$, and $\varnothing$ denotes the empty set. Recently, Kraft and van Eeden [9] have pointed out that since \begin{align*}\tag{1.2a}P(M_n = k) &= 4^{-n}\binom{2k}{k}\binom{2n - 2k}{n - k},\quad k = 0, \cdots, n, \\ \tag{1.2b}P(N_n = k) &= 2^{-n},\quad k = 0, \cdots, 2^n - 1, \\ \end{align*} if $F$ is symmetric about zero (in the sense that $F(x) = 1 - F(-x), -\infty < x < \infty)$, both $M_n$ and $N_n$ may be used to test the hypothesis $H_0$ which specifies that $F$ is symmetric about zero. They also considered the consistency of such tests. The present paper gives some further sufficient conditions for the consistency of tests based on $M_n$ and $N_n$ and computes a measure of their asymptotic relative efficiency with respect to each other. The latter, of course, involves finding the asymptotic distributions of $M_n$ and $N_n$ under a sequence of local alternatives. In a final section the asymptotic properties of some confidence intervals and point-estimates based on $M_n$ and $N_n$ are considered. The alternatives of interest specify that $X_1$ is stochastically larger than a symmetric random variable in the sense that \begin{equation*}\tag{1.3}F(x) \leqq G(x),\quad -\infty < x < \infty, F \neq G,\end{equation*} for some $G$ which is symmetric about zero. The tests considered will be denoted by $\varphi_n$ and $\delta_n$ and reject for large values of $M_n$ and $N_n$ respectively. It should be noted that (1.2a) does not require the continuity of $F$; in fact, none of our results in Sections 2 and 3 which concern only $M_n$ or $\varphi_{nd}$ do.

In this paper is studied a technique based on samples of the form $N, X_1, X_2, \cdots, X_N$ where $N$ has a Poisson distribution, and each $X_i$ has the same continuous distribution function. Such samples, rather than fixed number samples, are appropriate for fixed time period observations where the number of occurrences is a Poisson variate, and are used in biology, insurance, and telephone engineering. We shall introduce a one-sided Kac statistic which is similar to the one-sided Kolmogorov statistic, derive forms for its finite dimensional and asymptotic distributions, find a lower bound for the power of the test, and prove that the test is "modified" consistent. Tabulations of the distributions will be given.

Let $\mathbf{A}_1$ and $\mathbf{A}_2$ be two symmetric matrices of order $p, \mathbf{A}_1$, positive definite and having a Wishart distribution [2], [18] with $f_1$ degrees of freedom and $\mathbf{A}_2$, at least positive semi-definite and having a (pseudo) non-central (linear) Wishart distribution [1], [3], [5], [18], [19] with $f_2$ degrees of freedom. Now let $\mathbf{A}_2 = \mathbf{CYY'C'}$ where $\mathbf{Y}$ is $p \times f_2$ and $\mathbf{C}$ is a lower triangular matrix such that $\mathbf{A}_1 + \mathbf{A}_2 = \mathbf{CC}'.$ Now consider the $s$(= minimum $(f_2, p)$) non-zero characteristic roots of the matrix $\mathbf{YY}'$. It can be shown that the density function of the characteristic roots of $\mathbf{Y'Y}$ for $f_2 \leqq p$ can be obtained from that of the characteristic roots of $\mathbf{YY}'$ for $f_2 \geqq p$ if in the latter case the following changes are made [6], [18]: \begin{equation*}\tag{1.1}(f_1, f_2, p) \rightarrow (f_1 + f_2 - p, p, f_2).\end{equation*} Now define $U^{(s)} = \mathrm{tr} (\mathbf{I}_p - \mathbf{YY}')^{-1} - p = \mathrm{tr} (\mathbf{I}_{f_2} - \mathbf{Y'Y}^{-1} - f_2$. In view of (1.1), we only consider $U^{(s)}$ when $s = p$, i.e. $U^{(p)}$, based on the density function [9] of $\mathbf{L} = \mathbf{YY}'$ for $f_2 \geqq p$. The first four moments of $U^{(s)}$ have been studied by Pillai in the central case [11], [12], [13], [14], [17] those for $U^{(2)}$ also by Pillai [15] in the non-central (linear) case and the first two moments of $U^{(p)}$ by the authors [7]. These results are extended in the present paper, obtaining the third and fourth moments of $U^{(p)}$ and further, two approximations to the distribution of $U^{(p)}$ are suggested in the linear case.

Robust estimates of contrasts in treatment effects for experiments with one observation per cell were proposed by Lehmann [6] for complete (randomized) blocks designs. The model for the observations $X_{i\alpha} (i = 1, \cdots, c; \alpha = 1, \cdots, n)$ is in this case \begin{equation*}\tag{1}X_{i\alpha} = \nu + \xi_i + \mu_\alpha + U_{i\alpha}\quad (\sum \xi_i = \sum \mu_\alpha = 0)\end{equation*} where the $\xi$'s are the treatment effects, the $\mu$'s are the block effects, and the $U$'s are independent with a common continuous distribution. Here we shall generalize these estimates to experiments in which the block size is smaller than the number of treatments to be compared, and we shall obtain their asymptotic efficiencies relative to the classical estimates. Since we are concerned with large sample theory, we shall be interested in designs in which the blocks are replicated a large number (at least 4) times. Such designs could be applied to situations in which only a few different treatment combinations are practicable but each could be replicated several times. For example, in an experiment to compare various diets for pigs, the natural block is the litter. One may wish to compare $c$ diets and have available a number of litters of size $b < c$. An incomplete blocks design using some $J$ litters could first be selected and then the whole design replicated several times using the remaining litters or (e.g. if some comparisons were of greater interest than others) some groups of $b$ diets could be given to more litters than others. Thus the situation to be considered is that in which $c$ treatments are to be compared and the blocks of experimental units are all of size $b < c$. An incomplete blocks design $D$ consisting of $J$ blocks of size $b$ is selected, the number $n_j$ of replications of the $j$th block is decided upon $(j = 1, \cdots, J; n_j = \rho_jn), \sum n_j$ blocks of experimental units are selected and numbered, and $\sum n_j$ sets of $b$ treatments are assigned to the selected blocks as specified by $D$ and the $n_j$. The set of blocks receiving the same treatments will be called a replication set. After the assignment of treatments to blocks, the order of application within the blocks is randomized. Assuming fixed effects and no interaction between treatment and block effects, the model for $D$ is \begin{equation*}\tag{2}X_{ij} = \nu' + \xi_i + \mu_j + U_{ij} (j = 1, \cdots, J; i \varepsilon S_j)\end{equation*} $\sum^c_{i = 1} \xi_i = \sum^J_{j = 1} \mu_j = 0$ where $S_j$ consists of the numbers of the $b$ treatments applied in the $j$th block, the $\xi$'s are the treatment effects, the $\mu$'s are the block effects, and the $U$'s are independent and identically distributed according to a continuous distribution $F$ with mean zero and variance $\sigma^2$ (not necessarily finite). Under the same assumptions, the model for the whole design is \begin{equation*}\tag{3}X_{ij\alpha} = \nu + \xi_i + \mu_j + \beta_{j\alpha} + U_{ij\alpha},\end{equation*} $(j = 1, \cdots, J; i \varepsilon S_j; \alpha = 1, \cdots, n_j)$ $\sum^c_{i = 1} \xi_i = \sum^J_{j = 1} \mu_j = 0; \sum^{n_j}_{\alpha = 1} \beta_{j\alpha} = 0\quad \text{for each} j$ where $S_j$ is defined as before (the treatments applied to the $j$th block of $D$ now being applied in the $n_j$ blocks of the $j$th replication set), $\xi_i$ is the effect of the $i$th treatment, $\mu_j$ is the effect of the $j$th replication set, $\beta_{j\alpha}$ is the effect of the $\alpha$th block in the $j$th replication set, and the $U$'s are distributed as in the model (2). (Although it might appear that the models (2) and (3) are valid only for a fixed order of application of the assigned treatments to the units within a block, the models remain valid under randomization. For, consider a model in which any of the $b$ treatments may be assigned to each unit within the block. If the corresponding $U$'s are independent, identically distributed random variables, then any selection according to specified probabilities will again be independent and identically distributed, which justifies the assumptions of the above models.)

Within the set of all linear parametric functions, $\lambda'\beta$, estimable from either or both of two uncorrelated sets of data, $y_1 = X_1\beta + e_1$ and $y_2 = X_2\beta + e_2$ with known non-singular variances, a general characterization is presented of those $\lambda'\beta$'s for which the best linear unbiased estimator (b.l.u.e.) is obtainable from one source of information alone or by simple weighting of respective b.l.u.e.'s from each of the two sources. It is shown that if the intersection of the row spaces of $X_1$ and $X_2$ has rank $r$ then in the intersection space there are exactly $r$ independent $\lambda'$ vectors for which the b.l.u.e. of $\lambda'\beta$ is obtainable by simple weighting. Some related statements are made for $k > 2$ uncorrelated sources of information. In the case of incomplete block designs the reduced intrablock normal equations and the interblock normal equations may be regarded as originating from two uncorrelated sources of information on the treatment parameter vector $\tau$. It is shown that an estimable treatment contrast, $\gamma'\tau$, is best estimated from one source alone or by simple weighting of b.l.u.e.'s from the respective sources if and only if $\gamma$ is an eigenvector of $\Lambda = (\lambda_{ij})$, where $\lambda_{ij}$ is the number of times treatments $i$ and $j$ occur together in a block. For symmetric factorial or quasifactorial designs, it is shown that any effect or interaction degree of freedom contrast is an eigenvector of $\Lambda$, and hence is best estimated by simple weighting of its interblock and intrablock estimates.

It was shown in [11] that one can construct orthogonal arrays $(\lambda 2^3, k + 1, 2, 3)$ from arrays $(\lambda 2^2, k, 2, 2)$ with the maximum number of constraints $k + 1$ provided that $k$ is the maximum number of constraints for the arrays of strength two. This result is generalized here to construction of arrays $(\lambda 2^{t + 1}, k + 1, 2, t + 1)$ from arrays $(\lambda 2^t, k, 2, t)$. The structure of arrays $(\lambda 2^t, t + 1, 2, t)$ is analyzed and for $\lambda = q2^n, q$ odd, a method of extending any array $(\lambda 2^t, t + 1, 2, t)$ to $t + n + 1$ constraints is described. Orthogonal arrays $(\lambda 2^4, k, 2, 4)$ are discussed in detail for $\lambda = 1$ through $\lambda = 5$. The maximum value of $k$ is established in each of these cases and arrays assuming these values are effectively constructed.

This paper attempts to add to the existing understanding of weighing designs (WD). Let us suppose that $n$ objects are to be weighed in $n$ weighings with a chemical balance having no bias. In the previous papers [14], [15], for odd $n$, optimum weighing designs are given for some particular cases subject to the conditions, viz., (i) variances of the estimated weights are equal and (ii) the estimated weights are equally correlated. Assuming these conditions we find some more optimum designs when $n$ is odd. Throughout this paper $n$ is assumed to be odd, except in Section 4, where $n$ may take any value. Let $x_{ij} = 1$ if the $j$th object is placed in the left pan in the $i$th weighing $= -1$ if the $j$th object is placed in the right pan in the $i$th weighing $= 0$ if the $j$th object is not weighed in the $i$th weighing. The $n$th order matrix $X = ((x_{ij}))$ is known as the design matrix. Also let $y_i$ be the result recorded in the $i$th weighing, $\epsilon_i$ is the error in the result, $w_j$ the true weight of the $j$th object, so that we have $n$ equations $x_{i1}w_1 + x_{i2}w_2 + \cdots + x_{in}w_n = y_i + \epsilon_i,\quad i = 1, 2, \cdots, n.$ We assume $X$ to be a non-singular matrix. The method of least squares or the theory of linear estimation gives the estimated weights $\hat{w}$ by the equation $\hat{w} = (X'X)^{-1}X'y$ where $\hat{w}$ and $y$ are the column vectors of the estimated weights and the observations respectively. If $\sigma^2$ is the variance of each weighing, then $V (\hat{w}) = (X'X)^{-1}\sigma^2 = ((c_{ij}))\sigma^2$ where $((c_{ij}))$ is the inverse of $X'X$. The design will be called optimum in the sense of Mood, if $\det ((c_{ij}))$ ($\det$ stands for determinant) is minimum and this is the case when $\det X'X$ is maximum. The efficiency of the weighing design can be measured, in the sense of Mood, by $\det X'X/\max \det X'X$ [12]. If $\lambda_{\min}$ be the minimum characteristic root of $X'X$, then the efficiency of the design can be measured, in the sense of Ehrenfeld, by $\lambda_{\min}/\max \lambda_{\min}$ [6]. According to Kishen, the efficiency can be measured by $1/\sum c_{ii}$ [11]. The first two definitions of efficiency were first introduced by Wald in [16]. Kiefer gave some optimality criteria for experimental designs in which the three criteria mentioned above were included and discussed them fully in [9] and [10]. He named these three definitions as $D$-optimum, $E$-optimum and $A$-optimum. We use Kiefer's notations in Section 6 for the definitions of efficiency of a weighing design given above.

Consider the problem of estimating unbiasedly a linear function, $\lambda?' \omega, \omega' = (\omega_1, \cdots, \omega_p)$, of $p$ weights $\omega_1, \cdots, \omega_p (0 < \omega_i < \infty$ for all $i = 1, \cdots, p)$ when the number of possible weighing operations, $n$, is smaller than $p$. Assume further that the weighing design is of the chemical type (each entry of the design matrix can assume one of the values $-1, 0, +1)$. A weighing design in which the number of weighing operations is smaller than the number of objects, $n < p$, will be called a fractional weighing design. A fractional weighing design is singular. As is well known, if a non-randomized weighing design is singular, one cannot estimate unbiasedly each of the $p$ weights. Some linear functions $\lambda?'\omega$ can be estimated, however, unbiasedly. For example, the sum of all $p$ weights can be estimated unbiasedly by a singular weighing design in which, at each weighing operation, all the objects are placed on one pan. Such a singular weighing design is in fact optimal. One can not attain the same precision in estimating the sum of all the $p$ weights by any non-singular design (see K. S. Banerjee [1]). Some linear functions of the weights can thus be estimated, with a sufficient degree of precision, on the basis of a number of weighing operations, $n$, smaller than the number of weights, $p$. A. M. Mood [4] has shown that if the design matrix of a non-singular weighing design is a Hadamard matrix (if it exists) then the design is optimal. We shall therefore study fractional weighing designs based on Hadamard matrices. If $p$ is such that no Hadamard matrix of order $p$ exists, we shall consider the smallest integer $p'$ larger than $p$, for which a Hadamard matrix exists and add $(p' - p)$ dummy weights. The linear functions $\lambda?'\omega$ can be extended in a way that assigns the dummy weights the coefficient zero. We shall then consider randomized fractional weighing designs which are constructed by choosing at random $n$ rows, independently and with replacement, from the given Hadamard matrix according to a probability vector $\xi$ of order $p$. Every such row specifies a weighing operation to be performed. Non-randomized fractional designs are special cases of the randomized designs under consideration. It is well known that if the weighing design is fractional $(n < p)$ then not all the linear functions $\lambda?'\omega$ are estimable under non-randomized designs. It is shown in the present paper that any linear function $\lambda?'\omega$ can be estimated unbiasedly under a proper randomization scheme. Optimal randomization procedures depend on the functional $\lambda?$. Complete class of unbiased estimation procedures is given for any linear function $\lambda?'\omega$. It is shown that every functional $\lambda?$ specifies a subset of, say $r (1 \leqq r \leqq p)$, admissible weighing operations (rows of the Hadamard matrix); in the sense that if other weighing operations are chosen then, either the estimation procedure is biased, or has a variance larger than the one that can be attained under the admissible weighing operations. It is also proven that if each of the $r$ admissible weighing operations $(1 \leqq r \leqq p)$ is chosen with probability equal to $1/r$, then the corresponding unbiased estimator has a uniformly (in $\omega$ and $\sigma^2$) minimum variance, relative to all the above specified randomization procedures. The formulae of the unbiased estimator of $\lambda?'\omega$ and its variance are extended in Section 4 to the case of random choice without replacement, of $n$ rows out of the $r$ admissible ones. This randomization procedure improves the results that can be obtained under the best randomization procedure with replacement. Finally, the relationship of the present study to the studies of S. Ehrenfeld and S. Zacks [2], [3] is discussed in Section 5.

Shah [5] obtained an upper bound for the number of disjoint blocks in certain PBIB designs. The results obtained there [5] were based on theorems given in References [1], [3], and [6]. It should be pointed out here that theorems on Semi-Regular Group Divisible designs [5] are also implicitly given by Saraf [4]. In this note, the similar results are given for Singular Group Divisible (SGD) designs and the main tool used to establish them is Theorem 3.1 given below.

Youden [8] introduced a new class of designs called Linked Block (LB) designs, where a LB design is defined as that in which any two blocks of the design have the same number of treatments in common. Later, the same author together with Connor [9] presented a set of designs called `chain block designs' useful in the field of Physical Sciences. These designs can be considered as `not fully linked block designs'. In this paper we shall extend the idea of LB designs in yet another way by defining Partially Linked Block (PLB) designs. The basic motive in introducing PLB designs is to generalise the notion of LB designs and also to see how far the existing designs yield new PBIB designs. Consider an arrangement of $v$ treatments in $b$ blocks of $k$ plots each $(k < v)$, such that each treatment occurs at most once in any block and altogether in $r$ blocks. This is called an incomplete block design and is denoted by $D(v, b, k, r)$. We have $bk = vr$. When $b = v$, the design becomes symmetric. The incidence matrix $N = (n_{ij}) (i = 1, 2, \cdots, v; j = 1, 2, \cdots, b)$, where $n_{ij} = 1$ or 0 according as the $i$th treatment occurs or does not occur in the $j$th block characterises such a design completely.