Let $W(t)$ denote a standard Wiener process for $0 \leqq t < \infty$. We compute the probability that $W(t) \geqq g(t)$ for some $t \geqq \tau > 0$ (or for some $t > 0$) for a certain class of functions $g(t)$, including functions which are $\sim (2t\log\log t)^{\frac{1}{2}}$ as $t\rightarrow\infty$. We also prove an invariance theorem which states that this probability is the limit as $m \rightarrow \infty$ of the probability that $S_n \geqq m^{\frac{1}{2}}g(n/m)$ for some $n \geqq \tau m$ (or for some $n \geqq 1)$, where $S_n$ is the $n$th partial sum of any sequence $x_1, x_2,\cdots$ of independent and identically distributed (i.i.d.) random variables with mean 0 and variance 1. The main results were announced in [19]. Some aspects of the invariance theorem were considered independently by Muller [14], who also studied the rate of convergence to the limiting distribution. Statistical applications of these ideas are indicated in [3] and [18]. In Section 2 we state the general theorems and give several examples. Sections 3-5 are devoted to the proof of these results. In Section 6 we indicate the applicability of our methods to stochastic processes other than the Wiener process. Of particular interest in this regard is the analogue of Theorem 1 for Bessel diffusion processes. Section 7 raises questions which will be treated in a subsequent paper.

An experimental design is called a multidimensional (MD) design, if it involves more than one factor; see e.g. Potthoff (1964a, b). A number of two-, three-, and four-dimensional designs are now in common use. For example, the ordinary balanced and partially balanced incomplete block designs are two-dimensional. The Latin squares, Youden squares, and the designs of Shrikhande (1951) are three-dimensional. Finally, the Graeco-Latin square designs are four-dimensional. The construction of multidimensional designs involving three or more factors has been discussed by several authors both when additivity is assumed and when interactions are present. To mention a few, we cite Plackett and Burman (1946), Plackett (1946), Potthoff (1964a, b), Agarwal (1966), Anderson (1968), and Causey (1968). A general class of multidimensional designs, which are partially balanced (PB), has been introduced in Srivastava (1961) and Bose and Srivastava (1964). These designs are called multidimensional partially balanced (MDPB) designs. The (MDPB) designs are useful for economizing on the number of observations to be taken, retaining at the same time a relative ease in analysis. MDPB designs for models containing interaction terms have been considered by Anderson (1968). The purpose of this paper is to obtain a class of necessary combinatorial conditions satisfied by the parameters of MDPB designs, and to provide a relatively easy method of determining whether a given design is "completely connected". This latter concept, also of a combinatorial nature, is a generalization of the concept of "connected" block-treatment designs. It signifies that for every factor included under the design, the "true" difference between any two factor levels possesses a best linear unbiased estimate. In a succeeding communication, Srivastava and Anderson (1968), general methods of construction of MDPB designs are considered.

An object is hidden in one of the cells $1,2,\cdots, R$ with probability distribution $s = (s(1), s(2),\cdots s(R))$ and remains in that cell while search is conducted. A searcher is informed of $s$ and continues search until the object is found. He is informed also of $p = (p(1), p(2),\cdots p(R))$ where $p(i)$ is the probability of finding the object if $i$ is searched and the object is in $i$. If the object is found on trial $n + 1$ its worth is discounted by the factor $\prod^R_{i=1}\beta^{N(i)}_i$, where $N(i)$ is the number of inspections of cell $i$ during the first $n$ trials and $0 \leqq \beta_i < 1, i = 1,2,\cdots, R$ is known by the searcher. For each $n$, if $s(\bar{f}(n)) = (s(1|\bar{f}(n)), s(2\mid\bar{f}(n)),\cdots,s(R|\bar{f}(n)))$ denotes the conditional location distribution, given the history $\bar{f}(n)$ of failures for $n$ trials, then it is shown that an optimal procedure selects an $i$ achieving $\overset{\max}{i}\frac{p(i)s(i\mid\bar{f}(n))}{1 - \beta_i}.$ The limiting behavior of the value is investigated as each element in any collection of components of $(\beta_1, \beta_2,\cdots, \beta_R)$ tends to one.

If $f$ is a monotone function subject to certain restrictions and $\varphi$ its inverse, then one can associate with any $x$, a real number between zero and one, a sequence $\{ a_n\}$ of integers such that $x = f(a_1 + f(a_2 + f(a_3 + f(a_4 + \cdots.$ If $T$ is the transformation $\langle\varphi(x)\rangle$ where $\langle\rangle$ stands for the fractional part, it has been shown that there is a unique measure $\mu$ invariant under $T$ which is absolutely continuous with respect to Lebesgue measure. Examples are $f(x) = x/10$ which gives rise to the decimal expansion with invariant measure Lebesgue measure, or $f(x) = 1/x$ which gives rise to the continued fraction, with measure $dx/\ln 2(1 + x)$. This induces a measure $P$ on the sequences $\{ a_n\}$ which is stationary ergodic and has other interesting properties. However, a large class of pairs $\{f, \mu\}$ gives rise to the pair $\{\{ a_n\}, P\}$. The paper is concerned with the problem of how, given a measure $\mu$ to find, when possible, and $f$, which corresponds to a pair $\{\{a_n\}, P\}$, or given an $\{ f, \mu\}$ pair, to reduce it to a canonical form. Interesting observations about the "memory" of the process arise from the "canonical form".

It is shown that, for sequences of median unbiased estimates, the asymptotic efficiency never exceeds Fisher's bound. In other words: assumptions on the convergence of the distributions of the estimates are superfluous in this case.

This paper demonstrates the direct relationship which exists between $O(p^m, p^m - 1)$ sets and a balanced set of $\ell$-restrictional lattice designs for $p^m$ treatments. For instance, we will show that \begin{equation*}A = \begin{bmatrix}0 & 1 \\ 1 & 1\end{bmatrix}\end{equation*} which, as has been shown by Raktoe [5] completely specifies a balanced set of 2-restrictional lattice designs for $7^2 = 49$ treatments, will also completely characterize a set of 48 mutually orthogonal latin squares of order 49, i.e. an $O(49, 48)$ set. In other words, if our interest is to exhibit an $O$(49, 48) set, the above $2 \times 2$ matrix will do the job. Strangely enough, as will be shown, $A$ also completely characterizes an $O$(4, 3) set. Note that, since a balanced set of 1-restrictional lattice designs is simply a BIB design, this paper shows in particular a different proof for the known equivalence of the $O(p^m, p^m - 1)$ sets with a class of resolvable BIB designs. Consequently, the content of this paper will be useful for those who are concerned with tabulating the designs or writing an efficient program for generating designs on a computer.

The distribution of the characteristic (ch.) roots of a sample covariance matrix $\mathbf{S}$ (one-sample case) or the matrix $\mathbf{S}_1\mathbf{S}_2^{-1}$ (two-sample case, see Section 3) depends on a definite integral over the group of orthogonal (in the complex case replaced by unitary) matrices. This integral, either in the one-sample case or the two-sample case, involves the ch. roots of both the population and sample matrices. Usually the integral in either case is expressed as a hypergeometric series involving zonal polynomials [4], [6]. Unfortunately, these series converge slowly unless the ch. roots of the argument matrices lie in very limited ranges. Furthermore, the computations of these series are not so easy and not convenient for further development. In the one-sample real case, Anderson [1] has obtained an asymptotic expansion for the distribution of the ch. roots of the sample covariance matrix. In the two-sample case, however, the situation is more complicated. Chang [2] has obtained an asymptotic expansion for the first term. In Section 3 and Section 4 of this paper, we extend Chang's results obtaining the second term and also derive a general formula which includes the formulae of Anderson [1], James [7], Chang [2] and Roy [12] as limiting or special cases. In Section 5 we are dealing with the asymptotic expansions in the two-sample case in the complex Gaussian population, from which the one-sample results are obtained as limiting cases. Finally, Section 6 gives a comparison of the four asymptotic expansions.

A recent theorem of Orey [12] (see also [1], [6], [7], [13]) asserts that if $T$ is an $L_1$ operator induced on a discrete measure space by an irreducible recurrent aperiodic Markov matrix, then the condition (C) holds: $f \epsilon L_1, \int f = 0$ implies that $T^n f$ converges to zero in $L_1$. In an attempt to determine when (C) holds for more general operators, we at first prove the following (Theorem 1.1): Let $T$ be a positive linear contraction operator on $L_1$; if $T^nf$ and $T^{n+1}f$ intersect slightly, but uniformly in $f$ in the unit sphere of $L_1$, then $T^nf - T^{n+1}f$ converges to zero in norm. (C) follows if $T$ is conservative and ergodic (Corollary 1.3). In Section 2 we derive from this a simple proof of Orey's theorem. The main result of the paper is in Section 3 and could be called a "zero-two" theorem: Let $P(x, A)$ be a Markov kernel, and assume that there is a $\sigma$-finite measure $m$ such that for each $A, m(A) = 0$ implies $P(x, A) = 0$ a.e. and $m(A) > 0$ implies $\sum^\infty_{n=0} P^{(n)}(x, A) = \infty$ a.e. Then the total variation of the measure $P^{(n)}(x, \cdot) - P^{(n+1)}(x, \cdot)$ is either a.e. 2 for all $n$ or it converges a.e. to 0 as $n \rightarrow \infty$. In Section 4 it is shown that a version of the zero-two theorem essentially contains the Jamison-Orey generalization of Orey's theorem to Harris processes. Section 1 and Section 2 of this paper do not assume any knowledge of either operator ergodic theory or probability. Some known results in ergodic theory are applied in Section 3, but the proof of the main theorem does not depend on them.

Let $X_{it} (t = 1,\cdots, n; i = 1,\cdots, k)$ be independent observations from $k$ populations with respective distribution functions $F(x - \theta_i)$, where the translation parameters $\theta_i$ are unknown. Consider the problem of selecting one population, the objective being to select the population with largest translation parameter. Procedures based on the joint ranking of all $nk$ observations have been considered by Lehmann [5], Bartlett and Govindarajulu [1], and Puri and Puri [9]. Robust procedures for related problems have been considered by Sobel [11] and McDonald and Gupta [7], among others. Define the $i$th population to be good if $\theta_i > \theta_{\max} - \Delta$ where $\theta_{\max} = \max \{\theta_1,\cdots, \theta_k\}$ and where $\Delta$ is a specified positive constant. The asymptotic relative efficiency (A.R.E.) of two procedures is then the limiting ratio of the sample sizes required to achieve a preassigned minimum probability of selecting a good population. It was hoped that procedures based on ranks would be more robust in terms of A.R.E. than corresponding parametric procedures. However, it has recently been shown that the slippage configuration used to find the A.R.E. in references [5] and [1] was not least favorable for the selection of a good population (See reference [10].). Puri and Puri [9] avoided this difficulty by restricting consideration to parameter points $\mathbf{\theta}^{(n)} = (\theta_1^{(n)},\cdots, \theta^{(n)}_k)$ for which $\theta^{(n)}_{\max} - \theta^{(n)}_i = b_i/n^{\frac{1}{2}} + 0(1/n ^{\frac{1}{2}})$ for $i = 1,\cdots, k$, where the $b_i$ are nonnegative constants. In Section 2, selection procedures are defined which are based on two-sample estimates of shift. It is shown in Section 3 that if the underlying distribution $F(x)$ is absolutely continuous then the procedures defined in Section 2 will select a unique population. Conditions are given under which the slippage configuration is the least favorable parameter point for the selection of a good population. This result does not require restrictions on the set of translation parameters comprising the parameter space. The A.R.E. of these procedures is defined in Section 4. If we consider the procedure based on the Hodges-Lehmann estimates of shift corresponding to the two-sample $F_0$-scores test, it is shown that the A.R.E. of this procedure relative to the normal theory procedure of Bechhofer [2] is simply the Pitman efficiency of the two-sample $F_0$-scores test relative to the $t$-test. Hence this approach yields efficiency results which are similar to those in references [5], [1], and [9]. However, the use of estimates in the definition of the selection procedure has the advantage of eliminating the difficulties concerning the least favorable parameter point.

Let $P_0$ and $P_1$ be two probability distributions on a measurable space $(\mathscr{H}, \mathscr{B})$. We consider the problem of testing the simple hypothesis $H: P = P_0$ against the simple alternative $K: P = P_1$ on the basis of $n$ independent random variables $X_1, X_2, \cdots, X_n$ with common distribution $P$. The method that is widely used in the literature for investigating large sample properties of optimal tests is the following: according to the Neyman-Pearson lemma an optimal test can be described by means of sums of independent random variables. Theorems from the theory of probabilities of large deviations or ergodic theorems then allow one to obtain results on the asymtotic behavior of the error probabilities of optimal tests. In this paper we use a representation of the power of an optimal test which has its origin in the duality theory of infinite linear programming, in order to derive upper and lower bounds for the power. The bounds holds for any sample size $n$. This is done in Section 1 for two different types of tests, namely for most powerful tests at leve $\alpha_n$ and for tests which minimize a weighted sum of the error probabilities. It turns out that those bounds allow one to derive the well-known asymptotic properties under slightly more general conditions, i.e., $\alpha_n$ is permitted to tend to zero faster than any negative power of $n$ (but not exponentially fast) and the weight $\lambda_n$ to tend exponentially fast to zero. This and another application are discussed in Section 2.

It is possible to obtain convergence to a $D$-optimum measure, as defined by Kiefer and Wolfowitz, by successively adding points to a given initial experimental design. The points added correspond to points of maximum variance of the usual least squares estimate of the response mean for the particular regression model at each stage. A new bound is given for the generalized variances involved and an example is worked out.

Let $x_1, x_2, cdots, x_n$ be $n$ independent and identically distributed random variables whose distribution depends on a parameter $\theta, \theta \in \Theta$. Let $\Theta_0$ be a subset of $\Theta$ and consider the test of the hypothesis that $\theta \in \Theta_0. L_n (x_1, \cdots, x_n)$ is the level attained by a test statistic $T_n(x_1, \cdots, x_n)$ in the sense that it is the maximum probability under the hypothesis of obtaining a value large or larger than $T_n$ where large values of $T_n$ are significant for the hypothesis. Under some assumptions Bahadur [3] showed that where a non-null $\theta$ obtains $L_n$ cannot tend to zero at a rate faster than $\lbrack\rho(\theta)\rbrack^n$ where $\rho$ is a function defined in terms Kullback-Liebler information numbers. In this paper this result has been shown to be true without any assumptions whatsoever (Theorem 1). Some aspects of the relationship between the rate of convergence of $L_n$ and rate of convergence of the size of the tests are also studied and an equivalence property is shown (Theorem 2).

The noncentral Wishart density depends on a definite integral over the group of orthogonal matrices. This integral defines a function of the latent roots of a matrix involving the parent normal vectors and their means and covariances. An expansion for the integral in increasing powers of the reciprocals of these roots is developed using two distinct methods--an integration procedure and a substitution into a set of differential equations.

A number of articles concerned with the problem of finding minimum variance unbiased estimates (MVUE) of certain "non-standard" parametric functions have appeared in the statistical literature. When a complete sufficient statistic (c.s.s)exists, the Rao--Blackwell Theorem gives a method of constructing the MVUE of an estimable parametric function based on any unbiased estimate. A parametric function $\varphi(\theta)$ is called estimable if there exists an unbiased estimate. As an example, let $X_1, \cdots, X_n$ be independent and identically distributed (i.i.d.), $X_i$ being distributed according to a $p$-dimensional multivariate normal distribution with mean $\mu \in R^p (R^p = p$-dimensional Euclidean space) and covariance matrix $\Sigma: p \times p$. This is denoted by $\mathscr{L}(X_i = \mathscr{N} _p(\mu, \Sigma), \mathscr{L}(X)$ reading "the law of $X$." All vectors in this paper are column vectors. For $p = 1$, Kolmogorov (1950) solved the problem of finding the MVUE for the parametric function $\varphi_t(\mu, \sigma^2) = P_{\mu,\sigma^2}(X_1 \geqq t),$ $\mu$ and $\sigma^2$ both unknown. Kolmogorov's method, which was to calculate the conditional distribution of $X_1$ given the c.s.s. and apply the Rao--Blackwell Theorem to the indicator function $I_{\lbrack t,\infty)}(X_1)$, was sufficiently general to be applicable to other estimable parametric functions for this family of distributions. Later Lieberman and Resnikoff (1955) gave an independent solution to this problem using the same method. Barton (1961) and A. P. Basu (1964) solved this and other problems using the same approach. An alternative method for finding MVUE's when the distribution of the c.s.s. is the given, is the transform (LaPlace, Mellin, etc.) method. When applicable, this method does not require having an initial unbiased estimate. The transform method was used by Tate (1959) for distributions involving location and scale parameters and by Washio, Morimoto and Ikeda (1956) for the one-parameter Koopman-Darmois family. Olkin and Pratt (1958) also determined MVUE's of certain correlation coefficients using the LaPlace Transform. Neyman and Scott (1960) developed a third method, which they term "the expansion method," for producing the MVUE of certain parametric functions. Their applications were restricted to univariate normal distributions. An obvious alternative to the above methods is to exhibit a function of the c.s.s. and verify that it is unbiased. This method was used by Ghurye and Olkin (1969) to estimate density functions of the multivariate normal and the Wishart distribution with various parametric assumptions. As it is not constructive, this method may be hard to apply. E. Lehmann (unpublished lecture notes prior to 1964) and Sathe and Varde (1969) used a theorem due to D. Basu 91955) to greatly simplify the application of the Rao-Blackwell Theorem. Our Theorem 2.1, which permits other (including multi-variate) applications, is a generalization of their approach. Theorem 2.1 is applicable when a given unbiased estimate may be written as a function of the c.s.s. and an ancillary statistic. An ancillary statistic is one whose distribution does not depend on the parameter. The preponderance of applications of all the above references have been to families of distributions which are invariant under a group of transformations. These applications are in the domain of our Theorem 2.2 which gives conditions under which a group of transformations may be used to construct the data as a function of a c.s.s. and an ancillary statistic determined by the group structure and the c.s.s. In most cases, this ancillary statistic will be a maximal invariant statistic. The group structure essentially provides an easy method for representing the conditional distribution of the data given the c.s.s. in terms of the marginal distribution of the ancillary statistic. This greatly simplifies the application of the Rao--Blackwell Theorem to a given unbiased estimate. Section 3 illustrates the application of Theorem 2.2. In Example 1, the $\mathscr{N} _p(\mu, \Sigma)$ distribution is considered, $\mu$ and $\Sigma$ both unknown. The distribution of a maximal invariant is characterized and used to represent in integral form the MVUE of any estimable parametric function. Application is given to set probabilities and to estimating the $\mathscr{N} _p(\mu, \Sigma)$ density. Example 2 is concerned with an application to $U$-statistics.

We shall assume that the motion of a target in a given region is a diffusion process and that a searcher tries to find the target. If the searcher is able to remember the track he has travelled, then the probability density--from the searcher's point of view--of the target will be changing also because of the search, since the searcher knows that the target was not in the immediate proximity of his track. In the case of a stationary target the search already carried out would be taken into account by a straightforward application of Bayes' formula [4]. We shall assume the customary form for the instantaneous probability density of detection [3] and then modify the diffusion equation of the target to include the above Bayesian effect. The properties of this equation are discussed. An optimal statistical control problem of finding the best way of search is formulated.

Exemplification of the theory developed in [9] using a linear space of random variables other than linear combinations of the components of a random vector, and unbiased estimation for the parameters of a mixed linear model using quadratic estimators are the primary reasons for the considerations in this paper. For a random vector $Y$ with expectation $X\beta$ and covariance matrix $\sum_i\nu_iV_i$ ($\nu_1, \cdots, \nu_m$, and $\beta$ denote the parameters), interest centers upon quadratic estimability for parametric functions of the form $\sum_{i\leqq j}\gamma_{ij}\beta_i\beta_j + \sum_k\gamma_k\nu k$ and procedures for obtaining quadratic estimators for such parametric functions. Special emphasis is given to parametric functions of the form $\sum_k\gamma_k\nu_k$. Unbiased estimation of variance components is the main reason for quadratic estimability considerations regarding parametric functions of the form $\sum_k\gamma_k\nu_k$. Concerning variance component models, Airy, in 1861 (Scheffe [6]), appears to have been the first to introduce a model with more than one source of variation. Such a model is also implied (Scheffe [6]) by Chauvenet in 1863. Fisher [1], [2] reintroduced variance component models and discussed, apparently for the first time, unbiased estimation in such models. Since Fisher's introduction and discussion of unbiased estimation in models with more than one source of variation, there has been considerable literature published on the subject. One of these papers is a description by Henderson [5] which popularized three methods (now known as Henderson's Methods I, II, and III) for obtaining unbiased estimates of variance components. We mention these methods since they seem to be commonly used in the estimation of variance components. For a review as well as a matrix formulation of the methods see Searle [7]. Among the several pieces of work which have dealt with Henderson's methods, only that of Harville [4] seems to have been concerned with consistency of the equations leading to the estimators and to the existence of unbiased (quadratic) estimators under various conditions. Harville, however, only treats a completely random two-way classification model with interaction. One other result which deals with existence of unbiased quadratic estimators in a completely random model is given by Graybill and Hultquist [3]. In Section 2 the form we assume for a mixed linear model is introduced and the pertinent quantiles needed for the application of the results in [9] are obtained. Definitions, terminology, and notation are consistent with the usage in [9]. Section 3 considers parametric functions of the form $\sum_{i\leqq j}\gamma_{ij}\beta_i\beta_j + \sum_k\gamma_k\nu_k$ and Section 4 concerns parametric functions of the form $\sum_k\gamma_k\nu_k$. One particular method for obtaining unbiased estimators for linear combinations of variance components is given in Section 4 that is computationally simpler than the Henderson Method III procedure which is the most widely used general approach applicable to any mixed linear model. The method described in Section 4 has the added advantage of giving necessary and sufficient conditions for the existence of unbiased quadratic estimators which is not always the case with the Henderson Method III. In the last section an example is given which illustrates the Henderson Method III procedure from the viewpoint of this paper.

The multiplication of $k$-statistics is treated by Fisher [4], Tukey [8], [9], Wishart [10], Kendall [5], Abdel-Aty [1], Dwyer and Tracy [3], and Tracy [7]. Various rules, procedures, functions and tables have been devised to aid in this task. The rules and procedures derived from the combinatorial properties of the $k$-statistics and other symmetric functions linearly related to them. When dealing with generalized polykas is was found convenient to ignore communtativity of multiplication in denoting the various symmetric functions and use ordered partitions to represent them rather than the usual partitions [2]. The simplicity of the relationships among the symmetric functions, which results from the use of ordered partitions, may be used to obtain simple multiplication procedures.

In a recent paper Raktoe [1969] has presented a new approach and also a generalized technique to combining elements from distinct finite fields. The results however were related only to distinct prime fields i.e. each of the Galois fields in question consisted of residue classes modulo a prime. This paper solves the problem of combining elements of the most general case, i.e. the fields are not necessarily based on distinct primes and they can be prime powered.

In this paper two convergence theorems are proved. One gives a strong law of large numbers for a class of linear rank statistics and the other gives weak convergence of a weighted empirical cumulative process to Gaussian process, concentrated on continuous sample functions. Of course, both of these results are true under some regularity condition on the quantities involved.