In this paper we obtain some multivariate generalizations of Chebyshev's inequality, two of which are extended to continuous parameter stochastic processes. The extensions are obtained in a natural way by taking into account separability and letting the number of variables approach infinity. Particular attention is paid to the question of sharpness. To show that the bound of the inequality cannot be improved, examples are given in a number of cases that attain equality.

This paper concerns the following problem posed by R. Pyke (1958). What is the cardinality, $M_k$, of the maximal family of stochastically independent random variables defined on a given space $\Omega$, of cardinality ${\bar{\bar{\Omega}}} = k$? Since maximality is sought, the investigation is limited to two-valued, nontrivial (tvnt) random variables; and the $\sigma$-algebra of measurable subsets of $\Omega$ is taken to be that generated by the family of random variables. With these restrictions the problem is essentially one of cardinality. The results are summarized in the table below. Theorem 1 follows from the fact that stochastic independence entails the non-vanishing of certain finite intersections of elementary sets. Theorem 2 is a result of Kakutani, Kodaira and Oxtoby [4, 5, 6]. Theorem 3 is a consequence of a set theoretic result of Tarski [11], and a theorem of Banach [1, 2, 10, 11], which results were used in proofs of Theorem 2. Theorem 4 follows from a construction and a lemma of Marczewski [9]. The paper is divided into five sections. Section 1 introduces the notation and terminology. Section 2 discusses two types of independence. Sections 3, 4, and 5 treat, respectively, the finite, non-countable and countable cases.

If a finite Markov chain (discrete time, discrete states) has a number of absorbing states, one of these will eventually be reached. In this paper are given theoretical formulae for the probability distribution, its generating function and moments of the time taken to first reach an absorbing state, and these formulae are applied to an example taken from genetics. While first passage time problems and their solutions are known for a wide variety of Markov chain processes (e.g., [2], [7], [4]), the theory seems not to have been used in population genetics. Suppose a genetic population consists of a constant number of individuals and the state of the population is defined by the numbers of the various genotypes existing at a given time. Then if mutation is absent, all individuals will eventually become of the same genotype because of random influences such as births, deaths, mating, selection, chromosome breakages and recombinations. The population behavior may in some circumstances be approximated by a Markov chain with absorbing states. In Section 2, two alternative approaches are given for the theoretical determination of absorption time properties, using well known techniques. In Section 3, the consequences of the theoretical results are investigated for a particular population model introduced by Moran [9], [10], and explicit expressions for the distribution of the gene fixation time are obtained in terms of Chebyshev's orthogonal polynomials. The derivation requires finding the pre- and post-eigenvectors of the matrix of transition probabilities, and an incidental by-product is the proof of certain identities for the orthogonal polynomials. The material presented in Section 2 and Section 3 is obtained by exact methods. In Section 4, the Fokker-Planck diffusion equation is used to obtain approximate results, and these are compared with those of the exact theory to ascertain the accuracy of the diffusion approximation.

This paper extends some results of Grenander [1] relating to discrete real stationary normal processes with absolutely continuous spectrum to the case in which the spectrum also contains a step function with a finite number of salt uses. It is shown by Grenander [1] that the periodogram is an asymptotically unbiased estimate of the spectral density $f(\lambda)$ and that its variance is $\lbrack f(\lambda)\rbrack^2$ or $2\lbrack f(\lambda)\rbrack^2$, according as $\lambda \neq 0$ or $\lambda = 0$. In the present paper the same results are established at a point of continuity. The consistency of a suitably weighted periodogram for estimating $f(\lambda)$ is established by Grenander [1]. In this paper a weighted periodogram estimate similar to that of Grenander (except that the weight function is more restricted) is constructed which consistently estimates the spectral density at a point of continuity. It appears that this extended result leads to a direct approach to the location of a single periodicity irrespective of the presence of others in the time series.

The theory of collective risk deals with an insurance business, for which, during a time interval $(0, t)$ (1) the total claim $X(t)$ has a compound Poisson distribution, and (2) the gross risk premium received is $\lambda t$. The risk reserve $Z(t) = u + \lambda t - X(t)$, with the initial value $Z(0) = u$, is a temporally homogeneous Markov process. Starting with the initial value $u$, let $T$ be the first subsequent time at which the risk reserve becomes negative, i.e., the business is "ruined". The problem of ruin in collective risk theory is concerned with the distribution of the random variable $T$; this distribution has not so far been obtained explicitly except in a few particular cases. In this paper, the whole problem is re-examined, and explicit results are obtained in the cases of negative and positive processes. These results are then extended to the case where the total claim $X(t)$ is a general additive process.

In [2] and [3], H. Chernoff discussed the Sequential Design of Experiments. In [2], a procedure was exhibited and was proved to be asymptotically optimal for the hypothesis testing problem when there are finitely many states of nature. This paper extends Chernoff's results to infinitely many states of nature.

The use of 2-associate PBIB design is fairly common in experimental work. However, PBIB designs with more than two associate classes are not widely used because of the complicated nature of the analysis and construction involved. Recently, in an interesting paper [2], Shah constructed a number of 3-associate PBIB designs by what may be called the matrix substitution method. In this method, the incidence matrix of the 3-associate PBIB design is constructed by replacing the integers of a balanced matrix in $S$ integers (for example, the matrix might be the incidence matrix of a BIB design, that is, a balanced matrix in two integers) by the incidence matrices of $S$ associable BIB designs. The present author [1] and Shah [3] have shown that the above method may be used to construct a PBIB design with $2m + 1$ associate classes by replacing the integers of a PBIB design with $m$ associate classes by the incidence matrices of two associable BIB designs. Shah [3] has also given a simple method of analysis for PBIB designs with $2^n - 1$ associate classes constructed by the matrix substitution method. Thus, Shah's method of analysis may be used to analyze PBIB designs with 3 and 7 associate classes (corresponding to $n = 2$ and 3) which are of practical interest. In the present paper, simple methods of analysis for a class of PBIB designs with 3 and 5 associate classes are given. The method given here for PBIB designs with three associate classes provides an alternate method, and it is hoped that this method is more simple and direct than that given by Shah.

By definition $(X, Y)$ is a bivariate Poisson vector if $(X, Y) = (X^\ast + U, Y^\ast + U)$ where $X^\ast, Y^\ast$ and $U$ are three independent Poisson variables with, say, respective expectations $a, b$ and $d$. Let $(X_n, Y_n)n = 1, 2, \cdots, N$ be independent observations on a bivariate Poisson vector $(X, Y)$. It is shown that no test for the independence of $X$ and $Y$ can be both boundary randomized similar in $a$ and $b$ and also uniformly most powerful. However, a test of the form \begin{equation*} \varphi_0(Z\mid s, t) = \begin{cases} 1 \\ \gamma 0 \\ 0 \end{cases} \text{according as} Z = \sum^N_{n = 1} X_nY_n \gtreqqless k_0(N, s, t)\end{equation*} given $S = s$ and $T = t$, where, $\sum^N_{n = 1} X_n = S,\quad\sum^N_{n = 1} Y_n = T$, is boundary randomized similar and locally most powerful. Using a lemma on the convergence to a Normal probability distribution function of the conditional probability distribution function of $\sum^N_{n = 1} (X_n - sN^{-1}) (Y_n - tN^{-1})$ given $S = s$ and $T = t$, asymptotic formulae for the values of the $k_0(N, s, t)$ corresponding to a given level of significance are derived. In addition, it is shown that the asymptotic power of the test can be obtained from an approximation to a Normal probability function and that, in case instead of $k_0(N, s, t)$ its value calculated from the asymptotic formulae is used, the modified test is asymptotically locally most powerful in the sense of Definition 2. To extend the domain of application of this test, we replace $U$ in the definition of the bivariate Poisson vector by another random variable $W$ also taking non-negative integral values according to the probability function $P\{W = w \mid \sigma \geqq 0\} = \int^\infty_0 e^{-\sigma t}\frac{{(\sigma t)}^w} {w!} f(t) dt$ where $f(t)$ is any continuous probability function satisfying $\frac{\partial}{\partial\sigma} \int^\infty_0 e^{-\sigma t} \frac{{(\sigma t)}^w}{w!} f(t) dt \bigg |_{\sigma = 0} = \int^\infty_0 \frac{\partial}{\partial\sigma} \big\{e^{-\sigma t} \frac{{(\sigma t)}^w}{w!}\big\}|_{\sigma = 0} f(t) dt$ $\sigma \geqq 0, w = 0, 1, 2, \cdots$ and $\int^\infty_0 tf(t) dt < \infty$. In this way a class of bivariate probability functions is obtained such that for every member of this class also, regarded as a probability function of the random vector $(X, Y)$, the locally most powerful boundary randomized similar test for the independence of the two random variables $X$ and $Y$ is the same as the one given in the Poisson case.

Suppose a random sample of size $n$ is drawn from a normal population with mean $\mu$ and standard deviation $\sigma$ and that the sample has been censored either to the right or the left. Suppose the censoring is at a fixed point of the distribution or at a pre-specified sample percentage point, or is a combination of these two types of censoring. In this paper we present small sample bounded confidence intervals for $\mu$ and $\sigma$, based on a joint bounded confidence region at a confidence level with fixed bound. The limits for $\mu$ and $\sigma$ so obtained converge in probability, as $n \rightarrow \infty$, to the parameter values. The procedure of the paper allows similar results for some other scale-translation families of distributions. One such case, which is briefly discussed, is that of the exponential distribution with unknown initial point. The somewhat general applicability of the procedure mitigates the fact that it is not based on sufficient statistics.

Under the nonparametric approach, various methods have been suggested to avoid the assumption of normality and homoscedasticity implicit in the analysis of variance. For the one-way classification, i.e., to decide whether $c$ samples come from the same population, Kruskal and Wallis [9] have proposed the $H$-test based on ranks; Mood and Brown [10] have proposed the $M$-test, utilizing the numbers of observations above the median of the combined sample; while the present author [2] has offered the $V$-test based on the number of $c$-plets that can be formed by choosing one observation from each sample such that the observation from the $k$th sample is the least, $k = 1, 2, \cdots, c$. For the two-way classification, Friedman [8] has made use of ranks. His statistic, to test the hypothesis that the rankings by $m$ "observers" of $n$ "objects" are independent, essentially offers a test for the two-way classification with one observation per cell. Durbin [7] has given a generalization for the balanced incomplete block design. Benard and Van Elteren [1] have generalized it still further. Mood and Brown [6, 10] also have considered the two-way classification with one observation per cell or the same number of observations per cell. In the first part of the present paper, their test has been extended to cover incomplete block situations. Mood and Brown [6, 10] have also considered some simple regression problems. In the present paper their methods are extended to discuss some additional regression problems. Next some bivariate analysis of variance problems are considered. The "step-down procedure" [11, 12] is used to reduce the problem to the univariate case with the other variate as a concomitant variate. The regression methods developed earlier are used here in these bivariate problems. The method seems to be perfectly general and could be extended to three or more variates, that is, to the multivariate situation. Most of the tests are offered on heuristic considerations. They are expected to be good for large samples.

Under certain conditions, to be specified in Theorems 2 and 4, the latent roots of the symmetric random matrix $F$ with $\varepsilon F = \Phi$ are biased estimators of the latent roots of $\Phi$; the smallest (largest) root is negatively (positively) biased. Here bias includes both expectation-bias and median-bias. Further properties of the distribution of the latent roots are given, among them some relations between covariances of the latent roots, covariances of elements of $F$, and the amounts of expectation-bias of the latent roots. Also, a sufficient condition is given for a certain type of symmetry in the joint distribution of the latent roots. For applications of the theory presented in this paper to the theory of response surface estimation see van der Vaart [9].

The noncentral means distribution for infinite error degrees of freedom and the noncentral Wishart distribution are derived as expansions in zonal polynomials. A method of calculating the zonal polynomials is outlined, and orthogonality properties of their coefficients stated.

In this paper we generalize the result of James Pachares [1] on the distribution of a definite quadratic form to the case of non-central normal variates.

This paper deals with the order statistics from the normal distribution. Equations are obtained for the percentage points and the modal values of the $k$th order statistic in a sample of size $n$. Table I gives some percentage points for selected values of $k$ and $n$. In Table II the modal values of the largest order statistic are given. Appropriate symmetry relations which enable one to obtain certain missing values in Tables I and II are mentioned.

This note suggests that expressing a distribution function as a mixture of suitably chosen distribution functions leads to improved methods for generating random variables in a computer. The idea is to choose a distribution function which is close to the original and use it most of the time, applying the correction only infrequently. Mixtures allow this to be done in probability terms rather than in the more elaborate ways of conventional numerical analysis, which must be applied every time.

For a resolvable balanced incomplete block design, R. C. Bose [1] obtained the inequality $b \geqq v + r - 1$, and P. M. Roy [2] and W. F. Mikhail [3] proved this inequality without the assumption of resolvability, but with the weaker assumption that $v$ is a multiple of $k$. In this note an alternative and simpler proof of Roy's theorem is given.

A recent paper [3] described a method for constructing infinite classes of third order rotatable designs in three dimensions. This note gives nine specific designs selected from some of the infinite classes that have been shown to exist.