Two recent papers by Bose and Draper [1] and Draper [3] showed how it was possible, by combining certain sets of points, to construct infinite classes of second order rotatable designs in three and more dimensions. In this paper, third order rotatable designs in three dimensions are discussed. First, a general theorem is proved that provides the conditions under which a third order rotatable arrangement of points in $k$ dimensions is non-singular. The four previously known third order designs in three dimensions are stated; it is then shown how some of the second order design classes constructed earlier [1] may be combined in pairs to give infinite classes of sequential third order rotatable designs in three dimensions. One example of such a combination is worked out in full and it is shown that two of the four known designs are special cases of this class. A summary of further third order rotatable design classes that have been shown to exist, and that have been tabulated by the author, concludes the paper.

In a previous paper [8] the author proved that the $P_N$ and $S_N$ matrices are the most efficient weighing designs obtainable under Kishen's definition of efficiency [5], when $N$ is odd and $N \equiv 2 (\operatorname{mod} 4)$ respectively, subject to the conditions (i) The variances of the estimated weights are equal; (ii) The estimated weights are equally correlated. In this paper, assuming the above conditions, it is proved that the $P_N$ matrices are the best weighing designs under the definitions of Mood [6] and Ehrenfeld [2] when $N$ is odd, while the $S_N$ matrices are the best weighing designs under the definition of Ehrenfeld when $N \equiv 2 (\operatorname{mod} 4)$. Under Mood's definition of efficiency, the best weighing design $X,$ when $N \equiv 2 (\operatorname{mod} 4),$ is shown to be that for which $X'X = (N - 2)I_N + 2E_{NN},$ where $I_N$ is the $N$th order identity matrix and $E_{NN}$ is the $N$th order square matrix with positive unit elements everywhere. By applying the Hasse-Minkowski invariant, a necessary condition for the existence of the $S_N$ matrices is obtained, and the impossibilities of the $S_N$ matrices of orders 22, 34, 58 and 78 are shown.

Certain linear estimation procedures for randomized experimental designs are evaluated relative to the criteria of bias, variance and mean square error. For the designs considered, treatment combinations are randomly allocated to experimental units, the randomness being subject only to a wide symmetry condition. Statistical properties refer to the discrete probabilities induced by the randomization hypothesis. Section 2 defines the basic statistical model and discusses the question of conditional inference relative to this model. Certain vectorial notation and terminology is introduced in Section 3. Although the theory of the paper applies directly to $k$-factor designs with general $k,$ the notation is set up in Section 3 for a three factor design, and the three factor notation is used throughout, except for Section 5 which discusses an even simpler example. Two general classes of linear unbiased estimators are defined in Section 4 and illustrated in Section 5. In Section 6 it is shown that estimators of the types defined in Section 4 have optimum properties in a wide class of linear estimators. Finally, the theory for the basic model is generalized in Section 7 to cover the case of observations with error. Formal proofs of stated theorems are to be found at the ends of Sections 4.2, 4.3 and 6.

In the present paper the Mixed Model developed by Scheffe [10] for the complete two-way layout is extended to the complete three-way layout with two random-effects factors. The model involves three basic covariance matrices of unknown parameters in addition to the error variance and fixed effects. Assuming normality, tests of the usual statistical hypotheses, except that of no fixed main effects, are derived from the analysis of variance table. Those of no interaction between the fixed-effects and a random-effects factor are applicable only under a simplifying assumption. A reduced form of the model is derived which involves sets of independent identically distributed random vectors. These are used to obtain unbiased estimators of the basic covariance matrices and to construct a $T^2$ test of the hypothesis of no fixed main effects. This test involves nonoptimum estimators of the effects, but this is shown to result in general only in a small loss of power. Individual and simultaneous confidence intervals for the fixed main effects are obtained in terms of these nonoptimum estimators.

In a previous paper [6] the covariances of least-squares estimates of regression coefficients and the expected value of the estimate of residual variance were investigated when the errors are assumed to be correlated. In this paper we will investigate the distribution of the usual test statistics for regression coefficients under the same assumptions. Applications of the theory to the cases of testing a single sample mean, the difference between the means of two samples, the coefficients in a linear trend and in regression on trigonometric functions will be discussed in some detail under an assumed covariance matrix for errors.

The mixed model with one factor represented by fixed effects, one factor by random effects, and a normal error, has often stipulated that these random effects be a sample drawn from a normally distributed population. In the case of a single response (or univariate) experiment, the variance of this normal distribution is a natural measure of the dispersion of these random effects, and confidence bounds on the ratio of this variance to the error variance [13], [14], [9] and simultaneous confidence bounds on both variances (in the latter case with a confidence coefficient $\geqq$ a specified value) [9] have already been found for certain classes of experimental designs. But when a distribution is not normal--or not assumed at the outset to be normal--the variance may not reveal as much about the distribution as some other measure such as interquartile range. In the present paper we seek confidence bounds on what, in a sense to be explained presently, might be called representations of the interquartile range and of analogous differences between higher order quantiles of the population from which the random effects are drawn. The method of obtaining these bounds involves an element of approximation comparable to grouping continuous data into $k$ classes, since it replaces the actual random-effects variate by a "substitute variate" having $k$ equally probable discrete values. The main idea is this. Let us assume, for simplicity of discussion, that we have a real valued stochastic variate. One comment here might be helpful. If the stochastic variate is observable, it seems natural to attempt to approximate its unknown distribution by introducing unknown probabilities over a finite set of preassigned class intervals, then trying to estimate these probabilities and then (especially for a continuous distribution) increasing the number of class intervals. On the other hand, if the variate is unobservable, as in the present set-up, it seems natural to try to approximate the distribution by replacing the stochastic variate by a "substitute" variate which is supposed to take, as a first approximation, two (unknown) values with equal probabilities, or as a second approximation, three (unknown) values with equal probabilities, or in general $k$ (unknown) values with equal probabilities. We then try to estimate, in terms of our observations, these unknown values, which may be regarded as approximations to the 1st, 3rd, $\cdots, (2k - 1)$th quantiles of the unknown distribution. The random effects variate postulated in our model may have either a continuous or a discrete distribution, provided in the latter case there are enough distinct values to make these $k$ quantiles meaningful parameters of the distribution. From now on, for brevity, we shall refer to these unknown values as the quantiles of the unknown distribution. It will be seen later that it is the differences between these unknown values rather than the unknown values themselves which we can estimate or make inferences about, and the number, $k - 1,$ of such differences which can be estimated is restricted by the experiment. Turning now to the confidence bounds, we observe that in the derivation of these bounds use is made of the same kind of sums of squares as in the normal variance components analysis. Unlike the more familiar confidence statements where the confidence coefficient may be specified at will, here except for the case of two blocks, only a lower bound on the confidence coefficient is specifiable, and this includes as a factor a decreasing function of $k$, the number of discrete values of the substitute variable. For $k = 2, 3, 4, 5$ the geometric shape of a $(k - 1)$-dimensional confidence region has been found. It is also shown how the usual inference about the fixed effects can be made from this model and then how the above type of confidence bounds can be found for each of several random-effects factors in an experiment with orthogonal design. In the case of a multiresponse--usually called multivariate--experiment, the model frequently stipulates that the random effects be samples from a multivariate normal population. Although the variance matrix of this distribution has a readily available estimator, confidence bounds have presented many difficulties. For the extremely restricted model in which the variance matrix of each random-effects factor is proportional to the variance matrix of the error, Roy and Gnanadesikan [10] obtained simultaneous confidence bounds on the characteristic roots of this latter matrix and on the proportionality constants. In Part II of this paper the authors present (with a confidence coefficient greater than or equal to a preassigned value) confidence bounds on the characteristic roots of the variance-covariance matrix of a random-effects variate without assuming any such relation to the error matrix. The second part will also consider the $p$-variate extension of the univariate substitute variate and the associated confidence bounds for the case where the $p$-dimensional distribution of the random effects is not necessarily normal. This development will be only indicated in principle for $p > 2$ but will be discussed in some detail for the case $p = 2$.

The present paper is a continuation of another with similar title and the same overall objectives--confidence bounds on appropriate measures of the dispersion of the distribution from which random (block) effects are drawn in an experiment where fixed (treatment) effects are also under investigation. Specifically, confidence bounds are obtained on the maximum and minimum characteristic roots of the variance matrix of the block effects when the latter are assumed to come from a $p$-variate normal distribution (without the assumption made in [1], [5] that this variance matrix is proportional to that of the error). When the random block effects are not assumed to be normal, consideration is given to the approximation of an unknown multivariate distribution by means of marginal and conditional quantiles. Then for a rather restricted bivariate case, simultaneous confidence bounds are found for the two interquartile ranges. Since the ideas and notation of the first paper are presupposed by this one, much duplication is avoided by reference to appropriate sections or steps in the previous article. To facilitate such reference, the numbering is consecutive through both parts.

Tabular reconstruction of differences from a curtailed set of moments is ordinarily impossible because of a gap on the integral side of the difference table. This gap can be closed by substituting an associated polynomial for the one that is being operated upon. The associated polynomial has, at the end of a contracted range, terminal differences and moments individually proportional to corresponding differences and moments of the polynomial from which it was derived. The function is applied to the problem of matching moments to form least squares approximations.

Simple hypotheses $H_P$ and $H_Q$, specifying two distinct positive transition densities $p(x \mid y)$ and $q \mid y)$ and initial densities $p(x)$ and $q(x)$ with respect to a finite Lebesque-Stieltjes measure, are assumed for a discrete time parameter Markov process. Let $R_n$ be the likelihood ratio static based on the first $n + 1$ observations of the process, and consider the class of sequences of likelihood ratio tests $T(a, \alpha) = \{\lbrack R_n > n^\alpha a\rbrack: n = 0, 1, 2, \cdots\}$ generated by letting $a$ and $\alpha$ vary over the real numbers. Under certain regularity assumptions on $K_t(x, y) = p^{1-t} (x \mid y)q^t (x \mid y)$ and the initial densities $p$ and $q$, the subclass of consistent sequences is determined, and the limiting rates at which the error probabilities tend to zero for tests in this subclass are found. A definition of the best asymptotic rate for distinguishing between $H_P$ and $H_Q$ is made for the class of consistent tests. This "asymptotic rate of discrimination" is evaluated and is shown to be attained by a certain subclass of these tests. Some applications and extensions of the theory to infinite Lebesgue-Stieltjes measures are given.

If $X$ is a random variable with $EX^2 = \sigma^2$, then by Chebyshev's inequality, \begin{equation*}\tag{1.1}P\{|X| \geqq \epsilon\} \leqq \sigma^2/\epsilon^2.\end{equation*} If in addition $EX = 0$, one obtains a corresponding one-sided inequality \begin{equation*}\tag{1.2}\quad P\{X \geqq \epsilon\} \leqq \sigma^2/ (\epsilon^2 + \sigma^2)\end{equation*} (see, e.g., [8] p. 198). In each case a distribution for $X$ is known that results in equality, so that the bounds are sharp. By a change of variable we can take $\epsilon = 1$. There are many possible multivariate extensions of (1.1) and (1.2). Those providing bounds for $P\{\max_{1 \leqq j \leqq k} |X_j| \geqq 1\}$ and $P\{|\max_{1 \leqq j \leqq k} X_j \geqq 1\}$ have been investigated in [3, 5, 9] and [4], respectively. We consider here various inequalities involving (i) the minimum component or (ii) the product of the components of a random vector. Derivations and proofs of sharpness for these two classes of extensions show remarkable similarities. Some of each type occur as special cases of a general theorem in Section 3. Bounds are given under various assumptions concerning variances, covariances and independence. Notation. We denote the vector $(1, \cdots, 1)$ by $e$ and $(0, \cdots, 0)$ by 0; the dimensionality will be clear from the context. If $x = (x_1, \cdots, x_k)$ and $y = (y_1, \cdots, y_k)$, we write $x \geqq y(x > y)$ to mean $x_j \geqq y_j(x_j > y_j), j = 1, 2, \cdots, k$. If $\Sigma = (\sigma_{ij}): k \times k$ is a moment matrix, for convenience we write $\sigma_{jj} = \sigma^2_j, j = 1, \cdots, k$. Unless otherwise stated, we assume that $\Sigma$ is positive definite.

A Random Mapping Space $(X, \mathcal{J}, P)$ is a triplet, where $X$ is a finite set of elements $x$ of cardinality $n, \mathcal{J}$ is a set of transformations $T$ of $X$ into $X$, and $P$ is a probability measure over $\mathcal{J}$. In this paper, four choices of $\mathcal{J}$ are considered (I) $\mathcal{J}$ is the set of all transformations of $X$ into $X$. (II) $\mathcal{J}$ is the set of all transformations of $X$ into $X$ such that for each $x \varepsilon X Tx \neq x$. (III) $\mathcal{J}$ is the set of one-to-one mappings of $X$ onto $X$. (IV) $\mathcal{J}$ is the set of one-to-one mappings of $X$ onto $X$, such that for each $x \varepsilon X, Tx \neq x$. In each case $P$ is taken as the uniform probability distribution over $\mathcal{J}$. If $x \varepsilon X$ and $T \varepsilon \mathcal{J}$, we will define $T^kx$ as the $k$th iteration of $T$ on $x$, where $k$ is an integer, i.e. $T^kx = T(T^{k-1}x)$, and $T^0x = x$ for all $x$. The reader should note that, in general, $T^kx, k < 0$, may not exist or may not be uniquely determined. If for some $k \geqq 0, T^kx = y$, then $y$ is said to be a $k$th image of $x$ in $T$. The set of successors of $x$ in $T, S_T(x)$ is the set of all images of $x$ in $T$, i.e., $S_T(x) = \{x, Tx, T^2x, \cdots, T^{n-1}x\},$ which need not be all distinct elements. If for some $k \leqq 0, T^kx = y, y$ is said to be a $k$th inverse of $x$ in $T$. The set of all $k$th inverses of $x$ in $T$ is $T^{(k)} (x)$ and $P_T(x) = \mathbf{\bigcup}^0_{k=-n} T^{(k)} (x)$ is the set of predecessors of $x$ in $T$. If there exists an $m > 0$, such that $T^mx = x$, then $x$ is a cyclical element of $T$ and the set of elements $x, Tx, T^2x, \cdots, T^{m-1}x$ is the cycle containing $x, C_T(x)$. If $m$ is the smallest positive integer for which $T^m x = x$, then $C_T(x)$ has cardinality $m$. We note further an interesting equivalence relation induced by $T$. If there exists a pair of integers $k_1, k_2$ such that $T^{k_1}x = T^{k_2}y,$ then $x \sim y$ under $T$. It is readily seen that this is in fact an equivalence, and hence decomposes $X$ into equivalence classes, which we shall call the components of $X$ in $T$; and designate by $K_T(x)$ the component containing $x$. We define $s_T(x)$ to be the number of elements in $S_T(x), p_T(x)$ to be the number of elements in $P_T(x)$, and $l_T(x)$ to be the number of elements in the cycle contained in $K_T(x)$ (i.e. $l(x) =$ the number of elements in $C_T(x)$ if $x$ is cyclical). We designate by $q_T$ the number of elements of $X$ cyclical in $T$, and by $r_T$ the number of components of $X$ in $T$. Rubin and Sitgreaves [9] in a Stanford Technical Report have obtained the distributions of $s, p, l, q,$ and have given a generating function for the distribution of $r$ in case I. Folkert [3], in an unpublished doctoral dissertation has obtained the distribution of $r$ in cases I and II. The distribution of $r$ in case III is classical and may be found in Feller [2], Gontcharoff [4], and Riordan [8]. In the present paper, a number of these earlier results are rederived and extended. Specifically, for cases I and II, we compute the probability distributions of $s, p, l, q$ and $r$. In cases III and IV the distributions of $l$ and $r$ are given. In addition some asymptotic distributions and low order moments are obtained. For the convenience of the reader, an index of notations having a fixed meaning is provided in the appendix to the paper.

The distribution of the operating life of a series system of like elements supplied with a set of spare components has been obtained for the situation when failed elements are in turn replaced by spares. This distribution has been evaluated for some common types of component life densities, and tables of expected total system life have been constructed. These expectations have been compared with those of systems with no spares as a measure of the efficacy of the additional spare components. The reliability of systems with spares has also been studied.

Let samples of size $n$ be drawn from each of $k$ univariate continuous cumulative distribution functions on the same real line, and consider the intersection of the $k$ intervals between the $r$th and $(r + 1)$st order statistics in the several samples. Then, to maximize the probability that that intersection be nonempty the distributions should be identical. Furthermore, for each sample, consider two intervals--that between the $r$th and $(r + 1)$st and that between the $s$th and $(s + 1)$st order statistics-then to maximize the probability that both the intersection of the "$r$" intervals and the intersection of the "$s$" intervals be nonempty, the distributions again should be identical and the value of the maximum probability is $\frac{\binom{n}{r}^k \binom{n - r}{s - r}^k}{\binom{kn}{kr}\binom{k\lbrack n - r\rbrack}{k\lbrack s - r\rbrack}}, \qquad r \leqq s.$ Some possible directions for generalization are discussed. The problem arose in connection with a sociological study of interaction behavior in small groups. The results make it possible to provide a test of the hypothesis that several samples of the same size are randomly drawn from possibly different populations, against the alternative that the samples are not independently and randomly drawn from distributions. For example, suppose we observe the frequency of a particular sort of interaction for each member of five groups of size six. Suppose the five men with the highest frequencies each belong to different group. Then we can say (ignoring discreteness) that an event has occurred whose probability under random sampling is at most 144/2639 or about 0.055. (The statistic would have but two values, either the five highest belong to different groups, or they do not. Such a test would be especially appropriate if group structure were thought to develop automatically certain specialized functions in members.) However, the main interest in this paper is in the problem in probability.

Let $X$ be a random variable governed by one of a family of distributions which is conveniently parameterized by $\mu$, the expectation of $X$, so that, in particular, the variance of $X, \sigma^2$, is a function of $\mu$, which we denote by $\sigma^2(\mu)$. A transformation, $\psi(X)$, is sometimes sought so that the variance of $\psi(X)$, as $\mu$ sweeps over its domain, is independent of $\mu$ (or much more nearly constant than $\sigma^2(\mu)$). A standard method of obtaining such a transformation for stabilization of the variance is to consider $X$ as one of a sequence of random variables, the sequence converging asymptotically in distribution, usually to a normal distribution. One form of the basic theorem is stated and proved by C. R. Rao [8], pp. 207-8, as follows. THEOREM (Rao). If $X$ is asymptotically normally distributed about $\mu$, with asymptotic variance $\sigma^2(\mu)$, then any function $\psi = \psi(X)$, with continuous first derivative in some neighborhood of $\mu$, is asymptotically normally distributed with mean $\psi(\mu)$ and variance $\sigma^2(\mu)(d\psi/d\mu)^2$, where $(d\psi/d\mu)$ denotes the derivative of $\psi(X)$ with respect to $X$, evaluated at the point $\mu$. From this we immediately have the following well-known COROLLARY. The random variable \begin{equation*}\tag{1}\psi(X) = c \int^X_K \frac{d\mu}{\sigma(\mu)}\end{equation*}, where $0 < x < \infty$, and where $K$ is an arbitrary constant, has a variance which is stabilized asymptotically at $c^2$ It is assumed, of course, that the integrand in (1) is integrable. If $\psi(X)$ is not a real-valued function on the domain of $X$, then the mapping is meaningless. Transformations such as (1), perhaps slightly modified, not only often work well for stabilizing non-asymptotic variances, but also often serve as well to normalize non-normal distributions. In general, however, nothing is known about the relative closeness to normality of the distribution of a random variable before and after a variance-stabilizing transformation is applied. Nor can anything general be said about the relative rapidity of approach to asymptotic normality. The study of concrete examples, however, suggests some connection between variance stabilization and normalization of non-normal distributions. A theoretical connection that may be relevant in certain cases has been put forward by N. L. Johnson [3], pp. 150-1. Johnson shows that, when the random variable of interest has a certain structure, then the differential equation for the normalizing transformation is similar to the differential equation for the variance-stabilizing transformation. The specified structure is that $X_n = Y_1 + Y_2G(X_1) + \cdots + Y_nG(X_{n-1})$, where the $Y'$s are independent and small, and $G(\cdot)$ is some function. In what follows, we obtain the variance-stabilizing transformation for the noncentral $t$ distributions and consider its normalizing properties. We repeat the same procedure for the topside noncentral $F$ distributions, although the variance-stabilizing transformation in this case is not well-defined. We then derive two other (well-defined) transformations for the approximate normalization of the topside noncentral F. Numerical comparisons of these approximations and the exact values are given.

Numerous investigations, both theoretical and numerical, have been made of the distribution of the range in normal samples. One of the first investigators was Student [1] who examined the distribution on an empirical basis. Somewhat earlier, Tippett [2] presented tables and charts for the mean, standard deviation, and of the measures of skewness and kurtosis $(\beta_1 \text{and} \beta_2)$ for the range; further studies of the moments were made by E. S. Pearson [3], Hartley and Pearson [6] and by Ruben [7]. Tables of the moment constants are available in [6] and in Pearson and Hartley's book of statistical tables [10] (Tables 20 and 27), while, more recently, tables of the moment constants were provided by Harter and Clemm [11]. Approximations to the probability integral and percentage points of the distribution were suggested by E. S. Pearson [4], Cox [12], Patnaik [14] and Tukey [15]. Pearson's approximations were based on Pearsonian distributions of Type I and VI, while Cox used the Gamma Function (i.e., a multiple of $\chi^2$ with fractional degrees of freedom) and Patnaik a multiple of $\chi$ with fractional degrees of freedom as the basis for their approximations. These last two approximations have been compared by Pearson [5]. An approximation of a different type was derived by Johnson [16] who gave a series expansion for the probability integral suitable for low sample sizes and low values of the range. The behavior of the distribution for large sample size has been studied by Gumbel [17], [18], [19], Elfving [20], Cox [13] and Harley and Pearson [21]. For theoretical studies of the exact distribution reference is made to McKay and Pearson [22], Pillai [23], [24] and Cadwell [25] (see also Hartley [26]). Finally, tables of the probability integral and percentage points of the distribution have been provided by Hartley and Pearson [27], [10] (Tables 23 and 22), as well as by Harter and Clemm [11]. In the present paper, the following new results relating to the range distribution will be obtained: (i) The latter function may be expressed as the product of the sample size and the probability content of a certain parallelotope relative to a hyperspherical normal distribution, and (ii) the function can be evaluated as an infinite series involving the even moments of a sum of independent truncated normal variables. The distribution function may also be directly related to the moment generating function of the square of the sum of these truncated variables.

A description is given of the computation of tables of percentage points of the range, moments of the range, and percentage points of the studentized range for samples from a normal population. Percentage points of the (standardized) range $W = w/\sigma$ corresponding to cumulative probability $P = 0.0001, 0.0005, 0.001, 0.005, 0.01, 0.025, 0.05, 0.1 (0.1) 0.9, 0.95, 0.975, 0.99, 0.995, 0.999, 0.9995$ and $0.9999$ are given to six decimal places for samples of size $n = 2 (1) 20 (2) 40 (10) 100$. Moments (mean, variance, skewness, and elongation) of the range $W$ are given to eight or more significant figures for samples of size $n = 2 (1) 100$. Percentage points of the studentized range $Q = w/s$ corresponding to cumulative probability $P = 0.9, 0.95, 0.975, 0.99, 0.995$, and $0.999$ are given to four significant figures or four decimal places, whichever is less accurate, for samples of size $n = 2 (1) 20 (2) 40 (10) 100$, with degrees of freedom $\nu = 1 (1) 20, 24, 30, 40, 60, 120$, and $\infty$ for the independent estimate $s^2$ of the population variance. All tabular values are accurate to within a unit in the last place.

In this paper an asymptotic formula for the cumulative distribution function (c.d.f.) of Hotelling's generalized $T_0^2$ statistic is derived for general values of the number of dimensions in the non-null case. This result includes as a special case the previous result of the author for the null distribution of $T_0^2$ [5], and shows certain properties of $T_0^2$ which help determine the power of the test based on the statistic for moderately large samples. Both of the author's results provide an approximate complete analysis of the $T_0^2$ test, although the exact null and non-null distributions of $T_0^2$ are not available at present.

Let $\pi_1, \pi_2$ be two normal populations with common mean $\mu$ and variances $\sigma^2_1, \sigma^2_2$, where the parameter values are unknown. Suppose that it is desired to estimate $\mu$, and that the experimental procedure is to take $m$ observations from each population, compute variance estimates, and then take $n - 2m$ observations from that population with the smaller observed variance, where $n$ has been fixed beforehand. Let $R_n(\theta, m) = V_0^{-1} E(\hat\mu - \mu)^2$ be the risk of the estimator $\hat\mu$, where $V_0 = n^{-1} \min (\sigma^2_1, \sigma^2_2)$ and where $\theta = \sigma^2_2/\sigma^2_1$. For a class of "best" estimators, it is shown in this paper that $\sup_\theta R_n(\theta, m) \rightarrow 1$ as $n \rightarrow \infty$ if and only if $m/n \rightarrow 0$ and $m \rightarrow \infty$ as $n \rightarrow \infty$; that $\min_m \sup_\theta R_n(\theta, m) \sim 1 + Cn^{\frac{1}{3}}$ as $n \rightarrow \infty$; and that the minimax sample size is $m \sim Cn^{\frac{2}{3}}$ as $n \rightarrow \infty$.

This paper deals with non-parametric two-sample tests on dispersions. Two samples, $X$- and $Y$-samples of $m$ and $n$ independent observations from populations with continuous cumulative distribution functions $F(u)$ and $G(u)$ respectively, are considered. It is required for the basic test that the difference in locations (medians) of the two populations be known and, when this is so, the two samples may be adjusted to have equal locations. Taking these location parameters to be zero without loss of generality, we test the hypothesis that $G(u) \equiv F(u)$ against alternatives of the form $G(u) \equiv F(\theta u), \theta \neq 1$. The two samples are ordered in a single joint array and ranks are assigned from each end of the joint array towards the middle. The statistic used is $W$, the sum of ranks for the $X$-sample. The distribution of $W$ is studied and tables of significant values of $W$ are provided for $m + n \leqq 20$ and both upper- and lower-tail significance levels .005, .01, .025 and .05. The first four moments of $W$ are developed and a normal approximation to the null distribution of $W$ is devised. Large-sample properties of the $W$-test are considered. A proof of limiting normality is based on a theorem of Chernoff and Savage. Consistency of the $W$-test is indicated and its relative efficiency in comparison with the variance-ratio $F$-test is obtained as $6/\pi^2$ when $F(u)$ is the normal distribution function. Other non-parametric tests of dispersions are reviewed. The $W$-test is less efficient asymptotically than some of these other tests but is easier to apply, particularly with the tables provided. A modified test is suggested for the case where the difference in population locations is not known. This involves replacing the two original samples by two corresponding samples of deviations from sample medians. The procedure of the $W$-test is applied to the two samples of deviations. The properties of the modified test have not been investigated except for a sampling study of rather limited scope. That study indicates that the moments of $W$ for the modified test are not greatly different from those under the basic procedure.

Let $X(t), t \geqq 0$, be a Wiener process with unknown mean $\mu$ per unit time and unit variance per unit time. Thus, $X(0) = 0$ and for any $t_2 > t_1 \geqq 0, X(t_2) - X(t_1)$ is normally distributed with mean $(t_2 - t_1)\mu$ and variance $t_2 - t_1$. Furthermore, for any sequence $0 \leqq t_{11} < t_{12} \leqq t_{21} < t_{22} \leqq \cdots \leqq t_{k1} < t_{k2},$ the random variables $X(t_j2) - X(t)j1), j = 1, \cdots, k,$ are independent. The process may be observed continuously beginning at $t = 0$ and the problem is to decide between the hypotheses that $\mu \leqq \mu_0$ and $\mu > \mu_0$, where $\mu_0$ is a given number, which without loss of generality is taken as 0. Thus the hypotheses are \begin{align*}\tag{1.1}H_0 &: \mu \leqq 0 \\ H_1 &: \mu > 0.\end{align*} It is assumed that the cost of observing the process for a time $t$ is $bt$, where $b > 0$, and that $W_i(\mu)$, the cost of accepting $H_i(i = 0, 1)$ when $\mu$ is the true mean, is of the form \begin{align*}W_0(\mu) &= \begin{cases}0 & \text{for} \mu\leqq 0\\c\mu^r & \text{for} \mu > 0\\ \tag{1.2}W_1(\mu) = \begin{cases}c |\mu |^r & \text{for} \mu \leqq 0\\0 \text{for} \mu > 0\end{cases}\end{cases}\end{align*} where $c > 0$ and $0 < r \leqq 2$. The main result of this paper is that under these conditions the minimax decision procedure is a certain sequential probability ratio test (SPRT). The reason for restricting $r$ to the interval $0 < r \leqq 2$ will be brought out in the derivation given in Section 3. In Section 6, the analogous problem of testing the hypotheses (1.1) about the mean of a normal distribution is considered. The minimax procedure found for the Wiener process provides, in an obvious fashion, an approximation to the minimax procedure for this problem. Approximations of this type have been discussed in the literature. For $r = 1$, Moriguti [10] and Maurice [9] have found the approximate minimax procedure in a certain class of symmetric SPRT's. The same procedure is mentioned by Johnson in the discussion following [8]. Breakwell [2], [3], [4], has treated similar problems for the binomial and Poisson distributions. The work to be presented here not only puts all of this on a rigorous basis for the Wiener process but shows that, for the Wiener process, the minimax SPRT is in fact minimax among all decision procedures. Finally, it is shown in Section 6 that if the cost per observation is large, the true minimax procedure for the normal decision problem is to take exactly one observation and then accept one of the hypotheses.