The standard (likelihood ratio) test of the general linear hypothesis has been shown to possess numerous different optimum properties. A brief survey of these was included in a recent paper by Kiefer [2]. In the present note it is shown that all of these, and in fact a wide class of optimum properties of which the above are special cases, are consequences of the fact that the test is uniformly most powerful invariant.

We assume familiarity with the concepts defined in [1] and [2], where optimum $\beta$-expectation tolerance regions and their power functions were found for $k$-variate normal distributions. The method used is to reduce this problem to that of solving an equivalent hypothesis testing problem. It is the purpose of this paper to find optimum $\beta$-expectation tolerance regions for the single and double exponential distributions, and to exhibit the corresponding power functions. Let $X = (X_1, \cdots, X_n)$ be a random sample point in $n$ dimensions, where each $X_i$ is an independent observation, distributed by some continuous probability distribution function. It is often desirable to estimate on the basis of such a sample point a region, say $S(X_1, \cdots, X_n)$, which contains a given fraction $\beta$ of the parent distribution. We usually seek to estimate the center 100 $\beta$% of the distribution and/or one of the 100 $\beta$% tails of the parent distribution.

A distribution analogous to the canonical distribution used in testing the general linear hypothesis is developed for Model II analysis of variance for balanced classifications. As in the case of Model I analysis of variance, this standard distribution exhibits the sums of squares going into the analysis of variance table. By use of the standard form it is also shown that (i) all exact $F$-tests used in testing hypotheses based on balanced multiple classifications determine uniformly most powerful (u.m.p.) similar regions although they are not likelihood ratio (L.R.) tests, but (ii) in the balanced one-way classification, for all practical purposes, the test is an L.R. test, and is u.m.p. invariant. An exact $F$-test exists when we have a sum of squares, $S_1$ distributed as $(k + \sigma^2_0)$ times a chi-square variate, where $k > 0$, independently of $S_2$, which is distributed as $k$ times a chi-square variate. The test is then to reject the hypothesis that $\sigma^2_0 = 0$ whenever $S_1/S_2$ is greater than some suitably chosen number, $c$. As a corollary to property (i) it is shown that "of all invariant tests of $\sigma^2_0 = 0$ against $\sigma^2_0 > 0$ whose power is a function of $\sigma^2_0/(k + \sigma^2_0)$ only, the test $S_1/S_2 > c$ is most powerful, providing $S_1$ and $S_2$, as defined above can be found."

The purpose of this note is to present a lemma which will settle a question of completeness left open in Section 6 of the above mentioned paper [5]. We give two applications of the lemma, (i) by proving that, in addition to Herbach's results, also the standard $F$-test for $\sigma^2_{ab} = 0$ is a uniformly most powerful similar test, (ii) by pointing out that the standard form introduced in [5] together with our lemma provide convenient tools to prove that in a balanced model II design (with the usual normality assumptions) the standard estimates of variance components are minimum variance unbiased. This result is well known ([2], [3]) and it has in fact been pointed out by Graybill and Wortham [3] that a completeness argument may be used to demonstrate the minimum variance property of the usual estimators for the variance components. The present lemma shows that the estimators do indeed have the necessary completeness property. We will follow Herbach's notation throughout.

The use of sample quasi-ranges in estimating the standard deviation of normal, rectangular, and exponential populations is discussed. For the normal population, the expected value and the variance of the $r$th quasi-range for samples of size $n$ are tabulated for $r = 0(1) 8$ and $n = (2r + 2) (1) 100$. The efficiency of the unbiased estimate of population standard deviation based on one sample quasi-range is tabulated for the same values of $r$, with $n = (2r + 2) (2) 50 (5) 100$. Estimates based on a linear combination of two quasi-ranges are considered and a method is given for determining the weighting factor which maximizes the efficiency. The most efficient unbiased estimates based on one quasi-range for $n = 2 (1) 100$ and on linear combinations of two adjacent quasi-ranges and of two quasi-ranges among those with $r < r' \leqq 8$ for $n = 4 (1) 100$ are tabulated, along with their efficiencies. An example illustrates the use of these estimates. For the rectangular population, the efficient estimate of population standard deviation, which is based on the sample range, is tabulated for $n = 2(1) 100$. The bias, when estimates which assume normality are used, is tabulated for $n = 2(1) 100$ for rectangular and exponential populations.

The two-sample sign test is viewed as a test for the permutation symmetry of a bivariate distribution, and extensions to $k$-variate distributions are sought. Friedman's rank test, [1], although originally intended as a substitute for the $F$-test in a two-way classification, is such an extension. Study of the family of two-sample sign tests obtained by comparing the $k$ coordinates pairwise has yielded a statistic with an asymptotic Chi-square distribution from which a further test of symmetry can be constructed. The statistic is based on more degrees of freedom than Friedman's and is sensitive to a greater variety of alternatives. This extension is analogous to that obtained by Terpstra [2] from the Wilcoxon test. In this case, however, the limiting distribution turns out to be non-singular. The argument leading to the test is not restricted to the case of complete symmetry but may be carried through with any specified degree of asymmetry. The coordinates may also be compared $m$ at a time, $2 \leqq m \leqq k$. The argument can be extended and, with a slight modification, includes the derivation of Friedman's test. Thus a hierarchy of tests of permutation symmetry are available: Friedman's test corresponds to the case, $m = 1$; when $m = k$, the corresponding test turns out to be Pearson's Chi-square.

The one-sample problem is considered using techniques developed earlier [2], [3]. Let $Z = (Z_1, \cdots, Z_N)$ be a random vector with $Z_i = 1(0)$ if the $i$th smallest in absolute value in a sample of $N$ from the density $f(x)$ is positive (negative). Then $$P(Z = z) = N! \int_{\cdots_{0\leqq y_1\leqq\cdots\leqq yN\leqq\infty}}\int \prod_{i=1}^N \lbrack f^{1-z_i} (-y_i)f^{z_i}(y_i) dy_i\rbrack$$ Conditions are found implying $P(Z = z) > P(Z = z')$ where $z$ is derived from $z'$ by replacing a 0 by a 1, or interchanging a 0 and 1 in $z'$ by moving the 1 to the right. These conditions are met by the normal and other distributions. The results are useful in finding good tests of such null hypotheses as $X_1, \cdots, X_N$ are independently and identically distributed symmetrically about zero against such alternatives as slippage to the right. The Wilcoxon one sample signed rank test is a typical nonparametric procedure used under these conditions [4].

Let $F_n(x)$ be the empirical c.d.f. of $n$ independent random variables, each distributed according to the same continuous c.d.f. $F(x)$. The major object of this paper is to obtain in explicit form the probability law of the random variable $$D^+_n(\gamma) = \sup_{-\infty<x<\infty} \{F_n(x) - \gamma F(x)\}.$$ It is no loss of generality to suppose that $F(x)$ is the c.d.f. of the uniform distribution on [0, 1], so this assumption will be held throughout the paper. When $\gamma = 1$, then $D^+_n(1)$ is the usual one-sided goodness of fit statistic whose asymptotic distribution was first derived by Smirnov [6]. We obtain in several different forms (formulas 2.2 and 2.3) an expression for $$P(D^+_n(\gamma) < a) = P(F_n(x) \leqq a + \gamma^x, 0 \leqq x \leqq 1).$$ Formula (2.2) agrees with the one found by Birnbaum and Tingey [2] when $\gamma = 1$, which is the "classical" case. As a matter of fact, it seems to have been overlooked that this formula, for finite $n$, had already appeared in a paper by Smirnov [6]. The new formula (2.3) would seem to involve fewer computations for actual numerical evaluation. One rather remarkable fact which results from (2.3) is that \begin{equation*}P(F_n(x) \leqq \gamma x, 0 \leqq x \leqq 1) = \begin{cases}1 - \frac{1}{\gamma},\quad\gamma > 1\\ 0,\quad\gamma \leqq 1,\end{cases}\end{equation*} for any $n$. This was noted by Daniels [4] and was rediscovered by Robbins [5]. Using (2.3) it is easy to evaluate $\lim_{n\rightarrow\infty} P(F_n(x) \leqq a(n) + \gamma x)$ where $\gamma, (\gamma > 1)$ is fixed and $a(n) = d/n$, where $d$ is fixed. The limiting distribution when $\gamma > 1$ can be used to derive some facts about the Poisson Process which were recently discovered by Baxter and Donsker [1]. The methods used are elementary. To assist the reader, the results are all listed in Section 2 and Section 3 is devoted to giving proofs.

This note presents the solution to the problem of obtaining the full null distribution for the bivariate sign test proposed by J. L. Hodges, Jr., in 1955 [1]. The partial solution of [1] is completed, and the table of [1] is extended to give the full distribution up to a sample size $(n)$ of 30. In addition a partial table is included for sample size from 31 to 50.

A class of nonparametric procedures for testing the statistical identity of treatments in randomized block experiments is suggested and discussed. The suggested procedures are squarely based on experimental within-block randomizations, and they may be chosen so as to have special power against particular alternatives. The blocks are assumed to be statistically independent but no assumption is made concerning dependence within the various blocks. The basic idea is to obtain from each block a statistic that is, under the null hypothesis, symmetrically distributed about zero and then to apply to the set of these statistics a nonparametric test of symmetry about zero. The observational data can be of any quantitative type.

When a response (or dependent) variable $y$ can be observed for a continuous range of values of the independent variable $x$, which is at the control of the experimenter, the question arises as to how a given number of observations should be spaced. It will be assumed that $x$ is measurable without error and that $y$ differs from the true response function $f(x)$ by a random term $z$ with mean zero and constant variance $\sigma^2$. We suppose that the aim of the experimenter is to estimate $f(x)$, or possibly the mean response $\overline{f(x)}$, on the basis of $n$ observations $(x_i, y_i)$. Various aspects of this problem of optimal spacing have been studied for the case where $f(x)$ is known apart from some parameters (see e.g. Elfving [3], Chernoff [1], de la Garza [2], and Kiefer and Wolfowitz [8]). However, the functional form of $f(x)$ is often unknown or only approximately known. In the absence of a specific model to the contrary, polynomial approximations to $f(x)$ provide a convenient approach. Section 2 deals briefly with the non-statistical case $\sigma = 0$ when the problem of choosing $n$ abscissae in order to approximate to $f(x)$ by a polynomial of degree $n - 1$ reduces to one of optimum interpolation and that of integrating $f(x)$ reduces to Gaussian quadrature. For a fuller account of this part see Hildebrand [5] or Kopal [6]. If the response contains a random element, a polynomial of degree $n - 1$ or less may be fitted to the $n$ observations by least squares. The error of approximation will now be due, in general, both to random error and the use of an incorrect approximating function. We confine ourselves to the case of fitting a straight line when the true response, while roughly linear, may contain a quadratic component. Two criteria are considered in arriving at the two abscissae resulting in an optimal fit. The first of these criteria ((3.2) below) has also been discussed in a recent paper by Box and Draper [7] who have extended its use to the case of several independent variables. It is shown in Section 6 that for $x$-values symmetrically spaced about the centre of the region of interest nothing is gained in fitting a straight line by the use of more than two such abscissae. These optimal abscissae are determined in Sections 3 and 4. The emphasis of the present approach is on attaining an optimal straight line fit with a small number of observations, rather than on detecting departures from linearity. For the latter purpose more than two abscissae would, of course, be needed, but the number of observations required may well be uneconomically large. In Section 7 comparisons with some other simple spacings are made. As an illustration, consider the calibration of a large number of instruments for a range of $x$ in which $f(x)$ is known to be approximately linear. In this case adequate accuracy may be attainable by the use of two observations only. If $\sigma$ is not negligibly small several observations may be taken at each of two appropriately selected settings, especially if it is much easier to repeat measurements at a given setting than to turn to a new one (compare de la Garza [2]). An example illustrating the methods proposed is given in Section 8.

The technique of fitting a response surface is one widely used (especially in the chemical industry) to aid in the statistical analysis of experimental work in which the "yield" of a product depends, in some unknown fashion, on one or more controllable variables. Before the details of such an analysis can be carried out, experiments must be performed at predetermined levels of the controllable factors, i.e., an experimental design must be selected prior to experimentation. Box and Hunter [3] suggested designs of a certain type, which they called rotatable, as being suitable for such experimentation. Very few of these designs were then known. Since that time the work of R. L. Carter [6] has provided many new second order rotatable designs in two factors. However, additional methods were needed which would provide both second and third order designs in three and more factors. The present work represents an attempt to meet, in part, this need. New construction methods for obtaining rotatable designs of second order in three dimensions are here presented. By use of these methods various infinite classes of designs are obtained, and it may be shown that all the rotatable designs previously known can be derived as special cases of these infinite classes. Also derived is an infinite class of second order rotatable designs which contain only 16 points; only two particular designs contain fewer points.

Let $X_1, X_2, \cdots$ be independent and identically distributed random variables with possible values that are integers whose differences have g.c.d. one. Assume the m.g.f. of $X_1$ exists in an interval about 0, let $a$ be any number such that $E(X_1) < a < \sup X_1$, and let $\phi(a, t) = Ee^{t(X_{1-a})}$. There exists a unique value $t^{\ast}(a)$ of $t$ which minimizes $\phi(a, t)$ with respect to $t$; write $m(a) = \phi\lbrack a, t^{\ast}(a)\rbrack$ and $z = e^{-t^{\ast}(a)}$. Let $Y_1, Y_2, \cdots$ be independent and identically distributed random variables such that $Y_1$ and $X_1$ have the same range and $\Pr(Y_1 = x) = \Pr(X_1 = x) \cdot e^{t^{\ast}(a)(x-a)}/m(a)$, and let $\mu_2 = \sigma^2, \mu_3, \mu_4$ be central moments of $Y_1$. We show that $\Pr \{X_1 + \cdots + X_n = na\} = \lbrack m(a) \rbrack^n \Pr \{Y_1 + \cdots + Y_n = na\}$, and use this to establish the approximation $\Pr \{X_1 + \cdots + X_n = na\} = \pi^{\ast\ast}_n\lbrack 1 + 0(n^{-2})\rbrack$, where $na$ is a possible value of $X_1 + \cdots + X_n$ and $$\pi^{\ast\ast}_n = \frac{\lbrack m(a)\rbrack^n}{\sigma\sqrt{2\pi n}} \big\lbrack 1 + \frac{1}{8n} \big(\frac{\mu_4}{\mu^2_2} - 3 - \frac{5}{3} \frac{\mu^3_2}{\mu^3_2}\big)\big\rbrack.$$ Similarly we find that $\Pr \{X_1 + \cdots + X_n \geqq na\} = \Pi^{\ast\ast}_n\lbrack 1 + 0(n^{-2})\rbrack$, where $$\Pi^{\ast\ast}_n = \pi^{\ast\ast}_n \cdot \frac{1}{1 - z}\Bigg\{1 - \frac{1}{2n}\Bigg\lbrack\frac{(z\mu_3/\mu_2) + z(1 + z)/(1 - z)}{(1 + z)\mu_2}\Bigg\rbrack\Bigg\}.$$ We provide some numerical illustrations of the accuracy of these approximations, and give a conjectured analog of the leading term of $\Pi^{\ast\ast}_n$ for nonlattice variables.

Formulas closely related to $$u(t) = \lbrack n \log (1 + t^2/n)\rbrack^{\frac{1}{2}}\\ w(\chi^2) = \lbrack\chi^2 - n - n \log (\chi^2/n)\rbrack^{\frac{1}{2}}$$ are considered for converting upper tail values of Student's $t$ or chi-square variates with $n$ degrees of freedom to normal deviates. The chief object of the paper is to construct bounds on the deviation from the exact normal deviates such that the absolute deviation is bounded by $cn^{-\frac{1}{2}}$ uniformly in the entire tail. Two approximations for Student's $t$ are suggested that are remarkably accurate and an improvement over other available approximations. The bounds and approximations for Student's $t$ are given in Section 3 and those for chi-square in Section 4. Some of the methods used in obtaining bounds may be of value in other investigations. These are given in Section 2. The development of the bounds was stimulated by the work of Teichroew [3]. He obtains expansions for the normal deviates corresponding to tail values of Student's $t$ and chi-square and achieves spectacular accuracy even for small $n$. The idea and the construction of the expansion is set forth, briefly, in [4], p. 647. The first terms of these expansions are the $u(t)$ and $w(\chi^2)$ used here. The bounds of Theorems 3.1 and 4.2 show that these first approximations are correct to $O(n^{-\frac{1}{2}})$ uniformly for all $t > 0$ or $\chi^2 > n$. This fact can be used to show that the Teichroew expansions are valid asymptotic expansions.

Approximate expressions of more or less simple analytical form are derived for the conditional mean and variance over small intervals of a distribution having a probability density of somewhat restricted nature. An alternative formula for the mean is derived. This result is applied to an extremal problem in stratified sampling.

The first four moments of the sum of $s$ non-null latent roots of a matrix occurring in multivariate analysis are studied. In particular, the first four moments of the sum of six roots are found, and are used to compare the upper percentage points obtained directly from the moment ratios with those from Pillai's approximate distribution.

Suppose that a distribution $A$ is a mixture of distributions similar to $B$ but with different scale parameters; or (almost equivalently) that a distribution $F$ is a convolution of a given distribution $G$ with some other distribution. We derive conditions on (i) the moments of $A$ and $F$ and (ii) on the derivatives of $A$ and $F$; these conditions are necessary, but are not sufficient in general. The conditions (ii) are appropriate when $B$ (or $G$) is of Polya type 3.

Extension theorems of Tietze and Hahn-Banach play an important part in functional analysis. It seems reasonable to deal with similar questions for random transforms. In the present paper some measurability problems arising in connection with this probabilistic generalization are solved.

Let $x$ and $y$ be two independent normal variates each distributed with zero mean and a common variance; it is then well-known that the quotient $x/y$ follows the Cauchy law distributed symmetrically about the origin. Now the question that naturally arises is whether we can obtain a characterization of the normal distribution by this property of the quotient. This converse problem can be more precisely formulated as follows: Let $x$ and $y$ be two independently and identically distributed random variables having a common distribution function $F(x)$. Let the quotient $w = x/y$ follow the Cauchy law distributed symmetrically about the origin $w = 0$. Then the question is whether $F(x)$ is normal. But this converse is not true in general. The author [1] has recently constructed a very simple example of a non-normal distribution where the quotient $x/y$ follows the Cauchy law. Steck [7] has also given some examples of non-normal distributions with this property of the quotient. In the present paper we shall first derive some interesting general properties possessed by the class of distribution laws $F(x)$ [Section 2]. In Section 3 we deduce a characterization of the normal distribution under some conditions on the distribution function $F(x)$. Finally in Section 4 we construct an example of a non-normal distribution function $F(x)$ having finite moments of all orders where the quotient $x/y$ follows the Cauchy law. The method of proof is essentially based on the applications of Fourier transforms of distribution functions. For the proof of Theorem 3.1 we require somewhat deeper results in the theory of analytic functions.

Several modifications of the Dodge CSP-1 procedure [1] are presented. Changes are made in the rule of action when a defective item is observed while on sampling. The Average Outgoing Quality Limit (AOQL) for these new procedures are derived without the assumption of control. These results are compared with the AOQL assuming control. A production process is said to be in statistical control if there is a constant probability $p$ that an item is defective, and if the states of all the items (defective or nondefective) are stochastically independent. Further, the AOQL for the CSP-1 procedure using probability sampling (looking at every item with probability $1/k$ when on sampling) is derived without the assumption of control.

Shannon's basic theorem on the capacity of a channel is generalized to the case of a class of memoryless channels. A generalized capacity is defined and is shown to be the supremum of attainable transmission rates when the coding and decoding procedure must be satisfactory for every channel in the class.

For a memory less channel with finite input alphabet $A$, finite output alphabet $B$, and probability law $p(b\mid a)$, the capacity $C$ is defined as the maximum over all probability distributions $q$ on $A$ of $$\sum_{ab} q(a)p(b\mid a)\log_2(p(b\mid a)/\sum_a q(a)p(b\mid a)).$$ Shannon [1] has obtained the following result. Exponential error bound. For any $C_0 < C$ there is a number $\rho < 1$ such that, for every positive integer $N$, there is a set $S \subset A^{(N)}$ with at least $2^{C_0N}$ elements and a function $g$ from $B^{(N)}$ to $S$, such that, for every $s = (a_1, \cdots, a_N) \varepsilon S$, $$\sum p(b_1|a_1) \cdots p(b_N|a_N) < 2\rho^N,$$ where the sum extends over all sequences $b_1, \cdots, b_N$ for which $g(b_1, \cdots, b_N) \neq s$. Thus if the sender selects any $s \varepsilon S$ and places its letters $a_1, \cdots, a_N$ successively into the channel, and the receiver, on observing the resulting output sequence $b_1, \cdots, b_N$, decides that the input was $g(b_1, \cdots, b_N)$, the probability that he makes an error is less than $2\rho^N$, no matter what $s \varepsilon S$ was chosen. This result may be described as follows: it is possible to transmit at any rate $C_0 < C$, with arbitrarily small probability of error, by using block codes of sufficient length. We wish to draw a slightly stronger conclusion, as follows. We imagine an infinite sequence $x = (x_1, x_2, \cdots)$ of 0's and 1's, which we are required to transmit across the channel. At time $N$, the sender will have observed the first $\lbrack C_0N\rbrack$ coordinates of $x$, and will place the $N$th input symbol in the channel. The receiver, having at this point observed the first $N$ channel outputs, will estimate the first $M(N)$ coordinates of $x$. If $M(N)/C_0N \rightarrow 1$ as $N \rightarrow \infty$ and if, for every $x$, all but a finite number of his estimates are correct (i.e., agree with $x$ in every coordinate estimated) with probability 1, we shall say that the channel is being used at rate $C_0$. Our result is that, in this sense, a (memoryless) channel can be used at any rate $C_0 < C$. The result stated below is exactly this result, for the special case $C_0 = 1$. The general case involves no new ideas, but only more notation, and we shall restrict attention to the case $C_0 = 1$. The function $f_n$ of a code, as defined below, specifies the $n$th channel input symbol, as a function of the first $n$ coordinates of $x$. The number $M(n)$ is the number of $x$ coordinates to be estimated by the receiver after observing the first $n$ output symbols, and the function $g_n$ specifies the estimate. We now state the result precisely. For any finite set $S$, we denote by $S^{(N)}$ the set of all sequences $(s_1, \cdots, s_N)$, where $s_n \varepsilon S$ for $n = 1, 2, \cdots, N$. For a memoryless channel with finite input alphabet $A$, finite output alphabet $B$, an infinite code (for transmitting at rate 1) is defined as consisting of (a) a sequence $\{f_n\}$ of functions, where $f_n$ maps $I^{(n)}$ into $A$, and $I$ consists of the two elements 0 and 1, (b) a nondecreasing sequence $\{M(n)\}$ of positive integers such that $M(n)/n \rightarrow 1$ as $n \rightarrow \infty$, and (c) a sequence $\{g_n\}$ of functions, where $\{g_n\}$ maps $B^{(n)}$ into $I^{(M(n))}$. An infinite sequence $x = (x_1, x_2, \cdots)$ of 0's and 1's, together with an infinite code, defines a sequence of independent output variables $y_1, y_2, \cdots$, with $$\Pr \{y_n = b\} = p(b \mid f_n(x_1, \cdots, x_n)),$$ where $p(b \mid a)$ is the probability that the output symbol of the channel is $b$, given that the corresponding input symbol is $a$, and defines a sequence of estimated messages $t_1, t_2, \cdots$, where $t_n = g_n(y_1, \cdots, y_n)$. We shall say that the code is effective at $x$ if, with probability 1, $$t_n = (x_1, \cdots, x_{M(n)})$$ for all sufficiently large $n$, and shall say that the code is effective if it is effective for every $x$. The result of this note is the THEOREM: For any memoryless channel with capacity $C > 1$, there is an effective code.

The results of this note yield the mutual independence of certain matrices, characteristic roots, Hotelling's $T^2$ or Mahalanobis' $D^2$ statistics, and C. R. Rao's $R$ statistic. The result concerning the mutual independence of certain of Hotelling's $T^2$ statistics has been proved by K. S. Rao [1]. The results mentioned here occur (by implication and as a by-product) in [5], [6] and [7] in the course of investigations on some specific problems in statistical inference but are not explicitly stated. These results can be utilised in statistical inference and especially in simultaneous tests and simultaneous confidence interval estimation, and also in other problems.