We consider here the problem of minimizing and maximizing $\int^x_{-x\varphi}(x, F(x)) dx$ under the assumptions that $F(x)$ is a cumulative distribution function (cdf) on $\lbrack -X, X\rbrack$ with the first two moments given and that $\varphi$ is a certain known function having certain properties. The existence of the solution has been proved and a characterization of the maximizing and minimizing cdf's given. The minimizing cdf is unique when $\varphi(x, y)$ is strictly convex in $y$ and is completely characterized for some special forms of $\varphi$. The maximizing cdf is a discrete distribution and in the above case turns out to be a three-point distribution. Several statistical applications are discussed.

This paper is concerned with the spectral analysis of wide sense stationary time series which possess a spectral density function and whose fourth moment functions satisfy an integrability condition (which includes Gaussian processes). Consistent estimates are obtained for the spectral density function as well as for the spectral distribution function and a general class of spectral averages. Optimum consistent estimates are chosen on the basis of criteria involving the notions of order of consistency and asymptotic variance. The problem of interpolating the estimated spectral density, so that only a finite number of quantities need be computed to determine the entire graph, is also discussed. Both continuous and discrete time series are treated.

Group divisible incomplete block designs form an important class of incomplete block designs useful in a wide variety of experimental situations. Their properties, construction, and analysis have been thoroughly discussed in statistical literature, and we cite only several recent references [1], [2], and [3] to work of Bose and his co-workers dealing with partially balanced designs with two associate classes with which we shall be concerned. The utility of incomplete block designs would be increased with means of incorporating factorial treatment combinations in them. The use of factorials is widespread and stimulated by the concepts of confounding, partial confounding, and fractional replication. A mathematical summary on factorials is given by Kempthorne [4]. Kramer and Bradley [5] considered factorials in near-balance incomplete block designs, and here we generalize to the wider class of group divisible designs with two associate classes. Harshbarger [6] used a $2^3$ factorial in a Latinized rectangular lattice and this seems to be the first use of a factorial in a partially balanced incomplete block design. We obtain the intra-block analysis of variance for two-associate class group divisible designs with the adjusted treatment sum of squares in a modified form that more clearly indicates the structure of that quantity. Factorial treatment combinations are then identified with basic treatments through the association scheme of a design. This identification is effected in such a way that the factors are divided into two groups. For example, the design for 18 treatments (see [2], Design S60), divisible into six groups of three, in blocks of six, treatments replicated five times, can be adapted to a $6 \times 3$ factorial scheme; by regarding the six groups as made up of a $2 \times 3$ classification, the same design can be used for a $2 \times 3^2$ factorial scheme. Single-degree-of-freedom comparisons are obtainable in much the usual way and use of fractional replication, essentially within the groups of factors, is possible. The analyses for factorials depend on the estimators of basic treatment effects. We are not concerned with the construction of two-associate class group divisible designs and all known such designs for which $r \leqq 10, 3 \leqq k \leqq 10$, where $r$ is the number of replications and $k$ is the number of plots per block, are given in [7].

A report is presented on some statistical properties of the family of probability density functions $$\exp \lbrack -\lambda(x - \mu)^2/2\mu^2x\rbrack\lbrack\lambda/2\pi x^3\rbrack^{1/2}$$ for a variate $x$ and parameters $\mu$ and $\lambda$, with $x, \mu, \lambda$ each confined to $(0, \infty)$. The expectation of $x$ is $\mu$, while $\lambda$ is a measure of relative precision. The chief result is that the ml estimators of $\mu$ and $\lambda$ have stochastically independent distributions, and are of a nature which permits of the construction of an analogue of the analysis of variance for nested classifications. The ml estimator of $\mu$ is the sample mean, and for a fixed sample size $n$ its distribution is of the same family as $x$, with the same $\mu$ but with $\lambda$ replaced by $\lambda n$. The distribution of the ml estimator of the reciprocal of $\lambda$ is of the chi-square type. The probability distribution of $1/x$, and the estimation of certain functions of the parameters in heterogeneous data, are also considered.

The methods used in earlier papers of this series [2], [3] are extended from variances to third moments, and applied to the third moment about the mean (= third cumulant) of the usual estimate of the between variance component in a balanced single classification. The result is moderately complex, but manageable.

Consider a finite population consisting of $N$ elements $y_1, y_2, \cdots, y_N$. Throughout the paper we will assume that $N = nk$. A systematic sample of $n$ elements is drawn by choosing one element at random from the first $k$ elements $y_1, \cdots, y_k$, and then selecting every $k$th element thereafter. Let $y_{ij} = y_{i + (j - 1)k}(i = 1, \cdots, k; j = 1, \cdots, n)$; obviously systematic sampling is equivalent to selecting one of the $k$ "clusters" $$C_i = \{y_{ij}; j = 1, \cdots, n\}$$ at random. From this it follows that the sample mean $\bar y_i = 1/n \sum^n_{j = 1} y_{ij}$ is an unbiased estimate for the population mean $\bar y = 1/N \sum^k_{i = 1} \sum^n_{j = 1} y_{ij}$ and that $\operatorname{Var} \bar y_i = 1/k \sum^k_{i = 1} (\bar y_i - \bar y)^2$. We will denote this variance by $V^{(1)}_{sy}$ indicating by the superscript that only one cluster is selected at random. $V^{(1)}_{sy}$ can be written as \begin{equation*}\tag{1}V^{(1)}_{sy} = S^2 - \frac{1}{k} \sum^k_{i = 1} S^2_i, \text{where} S^2 = \frac{1}{N} \sum^k_{i = 1} \sum^n_{j = 1} (y_{ij} - \bar y)^2,\end{equation*} \\ \begin{equation*} S^2_i = \frac{1}{n} \sum^n_{j = 1} (y_{ij} - \bar y_i)^2.\end{equation*} It is natural to compare systematic sampling with stratified random sampling, where one element is chosen independently in each of the $n$ strata $\{y_1, \cdots, y_k\}, \{y_{k + 1}, \cdots, y_{2k}\}, \cdots$, and with simple random sampling using sample size $n$. The corresponding variances of the sample mean will be denoted by $V^{(1)}_{st} V^{(n)}_{ran}$ respectively. We consider now the following generalization of systematic sampling which appears to have been suggested by J. Tukey (see [3], p. 96, [4], [5]). Instead of choosing at first only one element at random we select a simple random sample of size $s$ (without replacement) from the first $k$ elements and then every $k$th element following those selected. In this way we obtain a sample of $ns$ elements and, if $i_1, i_2, \cdots, i_s$ are the serial numbers of the elements first chosen, the sample mean $1/s(\bar y_{i_1} + \cdots + \bar y_{i_s})$ can be used as an estimate for the population mean. This sampling procedure is clearly equivalent to drawing a simple random sample of size $s$ from the $k$ clusters $C_i(i = 1, \cdots, k)$. It therefore follows (see, for example, [2], Chapter 2.3 to 2.4) that the sample mean is an unbiased estimate for the population mean and that its variance, which we denote by $V^{(s)}_{sy}$, is given by begin{equation*} \tag{2}V^{(s)}_{sy} = \frac{k - s}{ks} \frac{1}{k - 1} \sum^k_{i = 1} (\bar y_i - \bar y)^2 = \frac{1}{s} \frac{k - s}{k - 1} V^{(1)}_{sy}.\end{equation*} Again, it is natural to compare this sampling procedure with stratified random sampling, where a simple random sample of size $s$ is drawn independently in each of the $n$ strata $\{y_1, \cdots, y_k\}, \{y_{k + 1}, \cdots, y_{2k}\}, \cdots$ or with simple random sampling employing sample size $ns$. We denote the corresponding variances of the sample mean (which in both cases is an unbiased estimate for the population mean) by $V^{(s)}_{st} ,V^{(ns)}_{ran}$ respectively. From well-known variance formulae (see, for example, [2], Chapters 2.4 and 5.3) it follows that \begin{equation*}\tag{3}V^{(s)}_{st} = \frac{1}{s} \frac{k - s} {k - 1} V^{(1)}_{st},\\ V^{(ns)}_{ran} = \frac{N - ns}{s(N - n)} V^{(n)}_{ran} = \frac{1}{s} \frac{k - s}{k - 1} V^{(n)}_{ran}. \end{equation*} Thus the relative magnitudes of the three variances $V^{(s)}_{sy}, V^{(s)}_{st}, V^{(ns)}_{ran}$ are the same as for $V^{(1)}_{sy}, V^{(1)}_{st}, V^{(n)}_{ran}$, of which comparisons were made for several types of populations by W. G. Madow and L. H. Madow [6] and W. G. Cochran [1]. Some of the results will be reviewed in Section 3. The object of this note is to compare systematic sampling with $s$ random starts, as described above, with systematic sampling employing only one random start but using a sample of the same size $ns$. To make this comparison we obviously have to assume that $k$ is an integral multiple of $s$, say $k = ls$. The latter procedure then consists in choosing one element at random from the first $l$ elements $\{y_1, \cdots, y_l\}$ and selecting every $l$th consecutive element. We denote the variances of the sample mean of the two procedures by $V^{(s)}_k, V^{(1)}_l$ respectively, indicating by the subscript the size of the initial "counting interval." (In our notation $V^{(s)}_{sy} \equiv V^{(s)}_k$.) We shall show in Section 4 that $V^{(1)}_l = V^{(s)}_k$ in the case of a population "in random order," but $V^{(1)}_l < V^{(s)_k$ for a population with a linear trend or with a positive correlation between the elements which is a decreasing convex function of their distance apart. Some numerical results on the relative precision of the two procedures will be given in Section 5 for the case of a large population with an exponential correlogram.

Orthogonal matrices having elements depending on certain random vectors provide a useful tool in various distribution problems in multivariate analysis. The method is applied to the derivation of the distributions of Hotelling's $T^2$ and Wilks' generalized variance, the Bartlett decomposition, and the Wishart distribution.

Let $\theta_p$ represent the unique 100$p$ per cent point of a continuous statistical population, while $x_r$ is the $r$th largest value of a sample of size $n$ from this population $(r = 1, \cdots, n)$. This paper considers estimation of $\theta_p$ on the basis of $x_{r(1)}, \cdots, x_{r(m)}$, where the $r(i)$ differ by $O(\sqrt{n + 1})$ and do not necessarily have values near $(n + 1)p$. Also considered is estimation of $x_R$ on the basis of $x_{r(1)}, \cdots, x_{r(m)}$, where the $r(i)$ differ by $O(\sqrt{n + 1})$ and do not necessarily have values near $R$. The results are of a nonparametric nature and based on expected value considerations. These estimation procedures may be useful for life-testing situations where time to failure is the variable and some of the items tested have not yet failed when observation is discontinued. Then $\theta_p$ and $x_R$ can be estimated for $p$ and $R$ values which extend a moderate way into the region where sample data is not available. Estimation of the $x_R$ value which would be obtained by continuing to observe the experiment represents a prediction of the future from the past. The results of this paper may be of value in the actuarial, population statistics, operations research, and other fields.

In this paper, we consider a disjoint decomposition, at three levels, of the total sample space for $n$-person, one-dimensional sociometric investigations. This results in a structure particularly suited to determination of the probability distributions of a large class of sociometric variables. Systematic methods for obtaining these distributions are presented and illustrated by two examples; while the first is trivial, the second produces a previously unknown result. It should be remarked that the methods developed here have application in the theory of communication networks and, indeed, in the study of any network situations which may be represented by either of the two models employed in the paper.

Suppose an experimenter $E$ wishes to compare the effectiveness of two treatments, $A$ and $B$, on a somewhat vaguely defined population. As individuals arrive, $E$ decides whether they are in the population, and if he decides that they are, he administers $A$ or $B$ and notes the result, until $nA$'s and $nB$'s have been administered. Plainly, if $E$ is aware, before deciding whether an individual is in the population, which treatment is to be administered next, he may, not necessarily deliberately, introduce a bias into the experiment. This bias we call selection bias. We propose to investigate the extent to which a statistician $S$, by determining the order in which treatments are administered, and not revealing to $E$ which treatment comes next until after the individual who is to receive it has been selected, can control this selection bias. Thus a design $d$ is a distribution over the set $T$ of the $\binom{2n}{n}$ sequences of length $2n$ containing $nA$'s and $nB's$. We shall measure the bias of a design by the maximum expected number of correct guesses which an experimenter can achieve, knowing $d$, attempting to guess the successive elements of a sequence $t \varepsilon T$ selected by $d$, and being told after each guess whether or not it is correct. The distribution of the number $G$ of correct guesses depends both on $d$ and on the prediction method $p$ used by the experimenter. We shall consider particularly two designs, the truncated binomial, in which the successive treatments are selected independently with probability $\frac{1}{2}$ each until $n$ treatments of one kind have occurred, and the sampling design, in which all $\binom{2n}{n}$ sequences are equally likely. We shall consider particularly two prediction methods, the convergent prediction, which predicts that treatment which has hitherto occurred less often, and the divergent prediction, which predicts that treatment which has hitherto occurred more often, except that after $n$ treatments of one kind have been administered, the divergent prediction agrees with the convergent predictions that the other treatment will follow; when both treatments have occurred equally often, either method predicts $A$ or $B$ by tossing a fair coin, independently for each case of equality. We find that among all designs, the truncated binomial minimizes the maximum expected number of correct guesses. For this design, the expected number of correct guesses is independent of the prediction method, and is $$n + n \binom{2n}{n} \big/ 2^{2n} \sim n + (n/\pi)^{1/2}$$ With the truncated binomial design, the variance in the number of correct guesses is largest for the divergence strategy and is $$3n/2 - D - D^2/4 \sim (3\pi - 2)n/2\pi - 2(n/\pi)^{1/2},$$ where $D = n \binom{2n}{n} \big/ 2^{2n - 1}$, and is smallest for the convergence strategy, and is $n/2 - D^2/4 \sim (\pi - 1)n/2\pi$. For the sampling design, convergent prediction maximizes the expected number of correct guesses; this maximum is $$n + 2^{2n - 1} \big/ \binom{2n}{n} - \frac{1}{2} \sim n + (\pi n/4)^{1/2}.$$ Finally we note that, if treatments are selected independently at random, bias of the kind we discuss disappears, but the treatment numbers can no longer be preassigned. Three such designs are considered: the fixed total design, in which the total number of treatments is a fixed number $s$, the fixed factor design, in which we continue until $1/X + 1/Y \leqq 2/n$, where $X$ is the number of $A$ treatments and $Y$ is the number of $B$ treatments administered, and the fixed minimum design, in which we continue until $\min (X, Y) = n$. For the fixed total design, we find that, for $s = 2n + 4, \mathrm{Pr}(1/X + 1/Y \leqq 2/n) \sim 0.955$ for large $n$; at the expense of 4 extra observations, we have a bias-free design whose variance factor will with probability 0.955 be smaller than that in which treatment numbers are preassigned. For the fixed factor design, the additional number of observations required to achieve the given precision has for large $n$ the distribution of the square of a normal deviate. For the fixed minimum design, in which we guarantee precision for the estimated effect of each treatment, the expected number of additional observations is roughly 1.13 $(n)^{1/2}$.

A normally distributed random vector $X$ is well known to be representable by $A \cdot Y$ (in the sense of having identical distributions), where $A$ is a matrix of constants and $Y$ is a random vector whose component random variables are independent. A necessary and sufficient condition for any infinitely divisible random vector to be so representable is given. The limiting case is discussed as are connections with the multivariate Poisson distribution and stochastic processes.

Some further methods are given for the construction of regular group divisible incomplete block designs, and designs derivable by these methods are tabulated. The methods are (i) designs containing complete and incomplete groups; (ii) designs with groups arranged in sets; and (iii) designs derivable by addition. In the first two of these methods, which are related, it may be possible to avoid having to take all the blocks that the full procedure would indicate.

The computations in the analysis of any equireplicate design can be carried out very easily if the number of treatments common to any two blocks is constant. A design with this property is called a Linked Block (LB) design and was introduced by Youden [9]. It is well known that for a Balanced Incomplete Block (BIB) design to have a constant number of treatments in common between any two blocks, it is necessary and sufficient that it is symmetric, that is, the number of blocks is equal to the number of treatments. In this paper, necessary and sufficient conditions are derived for any design with a given treatment-structure matrix to be of the LB type and the results applied to Partially Balanced Incomplete Block (PBIB) designs. Finally a list is prepared of all LB designs in the class of two-associate PBIB designs enumerated by Bose, Shrikhande and Clatworthy [2].

It is shown that the number of Borel Fields over a set $(S)$ of $n$ elements is equal to the number of equivalence relations within $S$. This number is asymptotically equal to $$(\beta + 1)^{-1/2} \exp \{n(\beta - 1 + \beta^{-1}) - 1\}\quad \text{where}\quad \beta \exp \beta = n$$.

We focus our attention herein on a Markov chain $x_0, x_1, \cdots$ with a countable number of states indexed by a subset I of the integers and with stationary transition probabilities $p_{ij}$, and explore the sets of states defined by: A transient set of states $C$ is said to be denumerably atomic if $P(x_n \varepsilon C i.o.) > 0$ and if for every infinite set $A \subset C$ we have $x_n \varepsilon C i.o.$ implies $x_n \varepsilon A i.o.$ with probability one (a.s.). Following Blackwell's basic paper [1] which introduced the systematic use of martingales into the study of Markov chains, we use the semi-martingale convergence theorem [2] to characterize denumerably atomic sets in terms of the bounded solutions of the inequality $$\phi(i) \leqq \sum_{j \varepsilon I} p_{ij}\phi(j),\quad i \varepsilon I.$$ For chains whose state space contains a denumerably atomic set a convergence criterion for certain sums $\sum^\infty_{n = 0}f(x_n)$ is then developed. The application of this criterion to a restricted class of continuous parameter Markov processes gives simple necessary and sufficient conditions for the existence of a unique process satisfying given infinitesimal conditions. This last result illuminates the connection between the necessary and sufficient conditions given by Feller [3] for uniqueness and the simpler conditions for birth and death processes given recently by Dobrusin [4], more recently by Karlin and McGregor [5], and by Reuter and Lederman [6] (see also [7]).

A straightforward expansion and integration of the frequency function for Fisher's $z$ produces a formula for the probability that $z$ is not exceeded, of which the successive terms decrease rapidly when $n_1$ and $n_2$ are large. It is given in terms of incomplete normal moment functions (or $\chi^2$ probabilities), and as a polynomial in $zN^{1/2}$, where $N$ is the harmonic mean of $n_1$ and $n_2$. This last form is identical with the inverted Cornish-Fisher expansion, originally deduced by quite different methods.

Let $\mathfrak{X}$ be the space of random variables defined on an abstract probability space $(\Omega, \mathcal{a}, P)$ where we consider any two elements of $\mathfrak{X}$ which are equal a.s. (almost surely) as the same. Frechet [2] exhibited a metric on $\mathfrak{X}$ (for example, $E\lbrack|X - Y|/(1 + |X - Y|)\rbrack)$ with the property that convergence in the metric is equivalent to convergence in probability, and he showed that for some probability spaces the same cannot be done for convergence a.s. Dugue [1] showed that it is not in general possible to define a norm on $\mathfrak{X}$ such that convergence in the norm is equivalent to convergence in probability. These results are contained in and completed by the following fact which was stated without proof by the author in [5] and which follows easily from the two theorems stated and proved in this note. There exists a metric (norm) on $\mathfrak{X}$ with convergence in the metric (norm) equivalent to convergence a.s. (in probability) if, and only if, $\Omega$ is the union of countable (finite) number of disjoint atoms. After these results were obtained it was found that the equivalence of parts (ii) and (iii) of Theorem 1 had been proved by Marczewski [4], p. 121. An atom of a probability space is a measurable set $A$ with $P(A) > 0$, such that any measurable subset has probability 0 or $P(A)$. It is easy to show that a random variable is a.s. constant on an atom. $f$ will always designate a real-valued function defined on $\mathfrak{X}$. Convergence in $f$ is said to be equivalent to convergence a.s. (in probability) if, for every sequence $\{X_n\}$ of elements from $\mathfrak{X}, f(X_n) \rightarrow 0$ if, and only if, $X_n \rightarrow 0$ a.s. (in probability). THEOREM 1. The following conditions on a probability space are equivalent. (i) There exists a function $f$, such that convergence in $f$ is equivalent to convergence a.s. (ii) For any sequence $\{X_n\}$ from $\mathfrak{X}$, if $X_n \rightarrow 0$ in probability, then $X_n \rightarrow 0$ a.s. (iii) $\Omega$ is a countable union of disjoint atoms. THEOREMS 2. The following conditions on a probability space are equivalent. (a) There exists a function $f$, such that convergence in $f$ is equivalent to convergence in probability and $f$ satisfies $|f(\alpha X)| = |\alpha|\cdot|f(X)|$ for any $X \varepsilon \mathfrak{X}$ and any real number $\alpha$. (b) $\Omega$ is a finite union of disjoint atoms.

The invariance method is applied to a regression problem for which the "errors" have a rectangular distribution. The invariance method can also be applied to produce good estimates for the regression problem when the "errors" form a sample from any fixed distribution.

If $\{Z_n\}, n = 1, 2, \ldots$ is a stationary stochastic process with $D$ states $0, 1, \cdots, D - 1$, and $X = \sum^\infty_1 Z_k/D^n$, Harris [1] has shown that the distribution of $X$ is absolutely continuous if and only if the $Z_n$ are independent and uniformly distributed over $0, 1, \cdots, D - 1$, i.e., if and only if the distribution of $X$ is uniform on the unit interval. In this note we show that if $\{Z_n\}, n = 1, 2, \cdots$ is any stochastic process with $D$ states $0, 1, \cdots, D - 1$ such that $X = \sum^\infty_1 Z_n/D^n$ has an absolutely continuous distribution, then the conditional distribution of $R_k = \sum^\infty_{n = 1} Z_{k + n}/D^n$ given $Z_1, \cdots, Z_k$ converges to the uniform distribution on the unit interval with probability 1 as $k \rightarrow \infty$. It follows that the unconditional distribution of $R_k$ converges to the uniform distribution as $k \rightarrow \infty$. Since if $\{Z_n\}$ is stationary the distribution of $R_k$ is independent of $k$, the result of Harris follows.