Let $\pi_1, \pi_2$ be normal populations with means $m_1, m_2$ respectively and a common variance $\sigma^2$, the parameter point $\omega = (m_1, m_2:\sigma)$ which characterizes the two populations being unknown, and let $\Omega$ be an arbitrary given set of possible points $\omega$. Random samples of fixed sizes $n_1, n_2$ are drawn from $\pi_1, \pi_2$ respectively, giving the combined sample point $\nu = (x_{11}, x_{12}, \cdots, x_{1n_1}; x_{21}, x_{22}, \cdots, x_{2n_2})$. For reasons which will be made clear later in connection with practical examples, any function $f(\nu)$ such that $0 \leq f(\nu) \leq 1$ is called a decision function, and for any such $f(\nu)$ the risk function is defined to be \begin{equation*}\tag{(1)} r(f\|\omega) = \max \lbrack m_1, m_2\rbrack - m_1E\lbrack f\mid \omega\rbrack - m_2E\lbrack 1 - f\|\omega\rbrack \geq 0\end{equation*} where $E$ denotes the expectation operator. A decision function $\bar{f}(\nu)$ is said to be (a) uniformly better than $f(\nu)$ if $r(\bar{f} \| \omega) \leq r(f\| \omega)$ for all $\omega$ in $\Omega$, the strict inequality holding for at least one $\omega$, (b) admissible if no decision function is uniformly better than $\bar{f}(\nu)$, and (c) minimax if $\sup_{\omega\epsilon\Omega} \lbrack r (\bar{f} \| \omega)\rbrack = \inf_f \sup_{\omega\epsilon\Omega} \lbrack r(f\| \omega)\rbrack$. The "problem of the greater mean" is, for any given $\Omega$, to determine the minimax decision functions, particularly those which are also admissible. Special interest attaches to the case in which there exists a unique minimax decision function $\bar{f}(\nu)$ (in the sense that if $f(\nu)$ is any minimax decision function then $f(\nu) = \bar{f}(\nu)$ for almost every $\nu$ in the sample space); such an $\bar{f}(\nu)$ is automatically admissible. The problem of the greater mean is, of course, a special problem in Wald's general theory of statistical decision functions [1]. Our results will, however, be derived by very simple direct methods which make no use of Wald's general theorems. We cite without proofs a few examples in order to show how strongly the solution of the problem of the greater mean depends on the structure of $\Omega$. In each case the minimax decision function is a function only of the two sample means $\bar{x}_1, \bar{x}_2$. (i) Let $\Omega'$ consist of the two points $(a, b: \sigma)$ and $(b, a: \sigma),$ with $a < b$. Then \begin{equation*}\tag{(2)} f^\ast(\nu) = \begin{cases}1 \text{if} n_1\bar{x}_1 - n_2\bar{x}_2 > (n_1 - n_2)(a + b)/2,\\0 \text{otherwise}\end{cases}\end{equation*} is the unique minimax decision function. (ii) Let $\Omega"$ consist of the two points $(c + h, c: \sigma)$ and $(c - h, c: \sigma)$, with $h > 0$. Then \begin{equation*}\tag{(3)} f^0_c(\nu) = \begin{cases}1\text{if} \bar{x}_1 > c\\0\text{otherwise},\end{cases}\end{equation*} is the unique minimax decision function. (iii) Let $\Omega"'$ consist of the three points $(\frac{1}{2}, - \frac{1}{2}:1), (\frac{1}{2}, \frac{3}{2}:1), (-\frac{3}{2}, -\frac{1}{2}:1),$ and let $n_1 = n_2 = n$. Then \begin{equation*}\tag{(4)}f^{\ast\ast}(\nu) = \begin{cases}1\text{if} e^{-2n\bar{x}_1} + e^{2n\bar{x}_2} < \lambda,\\0 \text{otherwise}\end{cases}\end{equation*}, where $\lambda$ is a certain definite constant, is the unique minimax decision function. The parameter spaces of two or three points specified in these examples are rather trivial, but in fact the corresponding decision functions (2), (3), (4) remain the unique minimax solutions of the decision problem with respect to much more general parameter spaces. Thus, for example, it is clear that $f^\ast(\nu)$ will remain the unique minimax decision function with respect to any $\Omega$ which contains $\Omega'$ and is such that $\sup_{\omega\epsilon\Omega} \lbrack r(f^\ast \| \omega)\rbrack = \sup_{\omega\epsilon\Omega'} \lbrack r(f^\ast \|\omega)\rbrack$. Corresponding remarks apply to $f^0_c(\nu)$ and $f^{\ast\ast}(\nu)$. When $n_1 = n_2$, (2) reduces to \begin{equation*}\tag{(5)}f^0(\nu) = \begin{cases}1 \text{if} \bar{x}_1 > \bar{x}_2,\\0 \text{otherwise}\end{cases}\end{equation*}. This decision function is of particular interest when both the means $m_1, m_2$ are unknown. It will be shown that whether or not $n_1 = n_2, f^0(\nu)$ is the unique minimax decision function under certain conditions on $\Omega$ which are likely to hold in practice, at least when both $n_1$ and $n_2$ are sufficiently large (Theorem 3). Likewise, $f^0_c(\nu)$, which is the analogue of $f^0(\nu)$ when one of the means $(m_2)$ is known exactly, is apt to be the unique minimax decision function in such cases, at least when $n_1$ is sufficiently large (Theorem 4). These results on $f^0(\nu)$ and $f^0_c(\nu)$ form the main results of the present paper. So much by way of a general summary. We shall now give a practical illustration (another is given in Section 3) to show how the problem of the greater mean arises in applications. Suppose that a consumer requires a certain number of manufactured articles which can be supplied at the same cost by each of two sources $\pi_1$ and $\pi_2$. The quality of an article is measured by a numerical characteristic $x$, and it is known that in the product of $\pi_i, x$ is normally distributed with mean $m_i$ and variance $\sigma^2,$ but the values of these parameters are unknown. The consumer has obtained a random sample of $n_1$, and $n_2$ articles from $\pi_1$ and $\pi_2$ respectively, and has found the values of $x$ to be $(x_{11}, x_{12}, \cdots, x_{1n_1}; x_{21}, x_{22}, \cdots, x_{2n_2}) = \nu$. What is the best way of ordering a total of $N$ articles from the two sources? The usual statistical theory, which confines itself to estimating the unknown parameters and to testing hypotheses of the form $H_0(m_1 = m_2),$ has at best an indirect bearing on the problem at hand. We therefore adopt Wald's point of view and investigate the consequences of any given course of action. If the consumer orders $fN$ articles from $\pi_1$ and $(1 - f)N$ from $\pi_2$, where $0 \leq f \leq 1,$ then the expectation of the sum of the $x$-values in the articles he obtains will be $N(m_1f + m_2(1 - f))$. The maximum possible value of this quantity is $N \max\lbrack m_1, m_2\rbrack$, and the "loss" per article which he sustains may therefore be taken as $W(\omega, f) = \max \lbrack m_1, m_2\rbrack - m_1f - m_2(1 - f) \geq 0$, where $\omega = (m_1, m_2: \sigma)$ is the true parameter point. The consumer wants to choose $f$ so as to make $W$ as small as possible. If he knew $m_1$ to be greater, or to be less, than $m_2$, then by choosing $f = 1$ or 0 respectively he could make $W = 0$. But since he does not know which $m_1$ is the greater he will presumably choose $f$ as some function of the sample point $\nu$. Suppose, therefore, that a "decision function" $f(\nu)$, such that $0 \leq f(\nu) leq 1$ but not necessarily taking on only the values 0 and 1, is defined for all points $\nu$ in the sample space and that the consumer sets $f = f(\nu).$ In repeated applications of this procedure, the "risk" or expected loss (a double expectation is involved: the expected loss for a given $f$ and the expected value of $f$ in using the decision function $f(\nu))$ per article is given by (1), and the consumer will try to find an $f(\nu)$ which minimizes this risk. Since the value of the risk depends on $\omega$ it is necessary to specify which values of $\omega$ are to be regarded as possible in the given problem; let the set of all such $\omega$ be denoted by $\Omega$. If the consumer agrees to adopt the "conservative" criterion of minimizing the maximum possible risk, then the statistician's problem is to find the minimax decision functions in the sense defined above. We have given the solutions of this problem for certain types of parameter spaces. The reader will observe that each of the minimax decision functions (2), (3), (4) was of the "all or nothing" type, with values 0 and 1 only. (Whether this remains true for every $\Omega$ we do not know.) By using one of these decision functions in a given instance one arrives at either the best possible decision or the worst. The attitudes of doubt sometimes associated with the non-rejection of the hypothesis $H_0(m_1 = m_2)$ are therefore irrelevant to the problem of the greater mean in the examples cited. (Cf. footnote 2; also Example 1 in Section 3.) The risk function (1) is but one of a general class $R$ of risk functions, to be defined in Section 2, which are associated with the problem of the greater mean. The most important members of $R$ are (1) and \begin{equation*}\tag{(6)}\bar{r}(f\| \omega) = P\text{(incorrect decision using} f(\nu)\|\omega),\end{equation*} where $"m_1 \leq m_2"$ and $"m_1 \geq m_2"$ are the two possible decisions. The risk function (6) is relevant to applications of a purely "scientific" nature in which the statistician is asked merely to give his opinion as to which population has the greater mean. Although the problem of constructing a suitable decision function for (6) is akin in spirit to the problems considered in the now classical Neyman-Pearson theory of statistical tests, no satisfactory solutions seem to be available. It is easy to see, however, that (1) and (6) are quite similar. Of course, in the case of (1) a decision function $f(\nu)$ may take on any value between 0 and 1 inclusive, while for (6) we allow only functions which take on only the values 0 and 1, corresponding respectively to the decisions $"m_1 \leq m_2"$ and $"m_1 \geq m_2"$. We then have for any such $f(\nu)$, \begin{equation*}\tag{(6')}\bar{r}(f\mid\omega) = \begin{cases}P(f(\nu) = 1 \| \omega) = E\lbrack f\| \omega\rbrack \text{if} m_1 < m_2,\\P(f(\nu) = 0 \| \omega) = E\lbrack 1 - f \|\omega\rbrack \text{if} m_1 > m_2,\\0\quad \text{if} m_1 = m_2,\end{cases}\end{equation*} and by comparison with (1) we see that $r(f\| \omega) = \| m_1 - m_2\| \bar{r}(f\| \omega)$ for all $\omega$. Now, in the three examples (i), (ii), (iii) cited above the unique minimax decision functions happen to take on only the values 0 and 1, and $\|m_1 - m_2\|$ is constant on each of the respective parameter sets. It follows that (2), (3), (4) are also the unique minimax decision functions relative to (6) and to $\Omega', \Omega", \Omega'''$ respectively. The remarks above following Example (iii) also remain valid for the risk function (6). We conclude this section with a remark on the methods of this paper. Any decision function relevant to (6) is equivalent to a test of the hypothesis $H_0(m_1 < m_2)$ against the alternative $H_1(m_1 > m_2)$, the region $\{\nu: f(\nu) = 1\}$ being the "critical region." Hence the Neyman-Pearson probability ratio method can be used to obtain the unique minimax decision function with respect to (6) and an $\Omega$ consisting of two (or more) points, and the result carries over to more general types of $\Omega$ in the manner already indicated. It turns out, however, that the dominant properties of the probability ratio tests are not confined to the class of tests alone, but extend to the class of all functions $f(\nu)$ such that $0 \leq f(\nu) \leq 1$. This result (Theorem 1) enables us to solve the problem of the greater mean for the risk function (1) as well as for (6). The reader who is interested in applications may turn to Section 3.

We consider here the relative merits of different statistics available for testing two means or two regression coefficients in relation to one-sided (asymmetric) and two-sided (symmetric) alternatives in case of unequal population variances. In so far as the Behrens-Fisher statistic is concerned we confine ourselves to the consideration of the behavior of its probability of Type I error in repeated sampling from populations with a fixed value of the unknown ratio of variances. In connection with the tests between two means, the present study takes its point of departure from the existing tests and investigates the question of utilizing an approximately determinate knowledge about the unknown ratio of variances. In connection with the comparison of two regression coefficients and also of two linear regression functions, we consider the effect of two concomitant sources of variation, viz., the unknown ratio of residual variances and the ratio of the sums of squares of the fixed variates, on the probability of Type I and Type II errors of certain well known statistics.

The extremal quotient is defined as the ratio of the largest to the absolute value of the smallest observation. Its analytical properties for symmetrical, continuous and unlimited distributions are obtained from a study of the auto-quotient defined as the ratio of two non-negative variates with identical distributions. The relation of the two statistics is established by proving that, for sufficiently large samples from an initial distribution with median zero, the largest (or smallest) value may be assumed to be positive (or negative) and that the extremes are independent. It follows that the distribution and the probability of the extremal quotient possess certain symmetries, and that its median is unity. As many moments exist for the extremal quotient as moments and reciprocal moments exist simultaneously for the initial variate. The logarithm of the extremal quotient is symmetrically distributed. These properties hold for all continuous symmetrical unlimited variates which possess a monotonically increasing probability function. For the exponential type, the asymptotic distribution of the extremal quotient can only be expressed by an integral. In this case, no moments exist. For the Cauchy type, the asymptotic distribution is very simple, and the logarithm of the extremal quotient has the same distribution as the midrange for initial distributions of the exponential type. It is not necessary to consider asymmetrical distributions since, in this case, for sufficiently large samples, one of the extremes will outweigh the other, unless the distribution is nearly symmetrical or has rapidly varying tails.

The paper describes the consequences of performing a preliminary $F$-test in the analysis of variance. The use of the 5% or 25% significance level for the preliminary test results in disturbances that are frequently large enough to lead to incorrect inferences in the final test. A more stable procedure is recommended for performing the preliminary test in which the two mean squares are pooled only if their ratio is less than twice the 50% point.

This paper is concerned with the problem of estimating the mean and variance of normal populations from singly and doubly truncated samples having known truncation points. Maximum likelihood estimating equations are derived which, with the aid of standard tables of areas and ordinates of the normal frequency function, can be readily solved by simple iterative processes. Asymptotic variances and covariances of these estimates are obtained from the information matrices. Numerical examples are given which illustrate the practical application of these results. In Sections 3 to 8 inclusive, the following cases of doubly truncated samples are considered: I, number of unmeasured observations unknown; II, number of unmeasured observations in each `tail' known; and III, total number of unmeasured observations known, but not the number in each `tail'. In Section 9, singly truncated samples are treated as special cases of I and II above.

In a previous paper [2] the authors have given a method for estimating the coefficients of a single equation in a complete system of linear stochastic equations. In the present paper the consistency of the estimates and the asymptotic distributions of the estimates and the test criteria are studied under conditions more general than those used in the derivation of these estimates and criteria. The point estimates, which can be obtained as maximum likelihood estimates under certain assumptions including that of normality of disturbances, are consistent even if the disturbances are not normally distributed and (a) some predetermined variables are neglected (Theorem 1) or (b) the single equation is in a non-linear system with certain properties (Theorem 2). Under certain general conditions (normality of the disturbances not being required) the estimates are asymptotically normally distributed (Theorems 3 and 4). The asymptotic covariance matrix is given for several cases. The criteria derived in [2] for testing the hypothesis of over-identification have, asymptotically, $\chi^2$-distributions (Theorem 5). The exact confidence regions developed in [2] for the case that all predetermined variables are exogenous (that is, that the difference equations are of zero order) are shown to be consistent and to hold asymptotically even when this assumption is not true (Theorem 6).

Let us consider a large number $n$ of observations which are statistically independent and drawn from continuous symmetrical populations. This paper presents some nonparametric tests of whether the $r$ largest observations of the set are too large to be consistent with the hypothesis that these populations have a common median value. Tests of whether the $r$ largest observations are too small to be consistent with this hypothesis are also considered. Here $r$ is a given integer which is independent of $n$. Subject to some weak restrictions, it is shown that the significance level of a test of the type presented tends to a value $\alpha$ as $n$ increases. For no admissible value of $n$, however, does the significance level of this test exceed $2\alpha$. If whether the largest observations are too large is considered, tests with values of $\alpha$ suitable for significance levels can be obtained for $r \geq 4$. Values of $\alpha$ suitable for significance levels can be obtained for any value of $r$ if whether the largest observations are too small is investigated ($n$ large). Properties of the power functions of these tests are considered for the special case in which the $r$ largest observations are from populations with common median $\theta$, the remaining observations are from populations with common median $\phi$, and each population has the property that the distribution of the quantity (sample value) - (population median) is independent of the value of the population median. For tests of $\theta > \phi$, the power function tends to zero as $\theta - \phi \rightarrow - \infty$ and to unity as $\theta - \phi \rightarrow \infty$. For tests of $\phi > \theta$, the power function tends to unity as $\theta - \phi \rightarrow - \infty$ and to zero as $\theta - \phi \rightarrow \infty$. Analogous tests of whether the smallest observations of a set are too small or too large can be obtained from the tests of the largest observations by symmetry considerations. If there is strong reason to believe that the set of observations is a random sample from a continuous population, the tests presented in this paper can be used to decide whether the population is symmetrical. Tests of this nature are sensitive to symmetry in the tails of the population but not to symmetry in the central part.

The properties of a measure of dependence $q'$ between two random variables are studied. It is shown (Sections 3-5) that $q'$ under fairly general conditions has an asymptotically normal distribution and provides approximate confidence limits for the population analogue of $q'$. A test of independence based on $q'$ is non-parametric (Section 6), and its asymptotic efficiency in the normal case is about 41% (Section 7). The $q'$-distribution in the case of independence is tabulated for sample sizes up to 50.

Stein [4] has exhibited a double sampling procedure to test hypotheses concerning the mean of normal variables with power independent of the unknown variances. This procedure is here adapted to test hypotheses concerning the ratio of means of two normal populations, also with power independent of the unknown variances. The use of a two sample procedure in a regression problem is also considered. Let $\{X_{ij}\} (i = 1, 2) (j = 1, 2, 3, \cdots)$ be independent random variables distributed according to $N(m_i, \sigma_i):$ all parameters are assumed to be unknown. Defining $k$ by the equation \begin{equation*}\tag{(1)} m_1 = km_2\end{equation*} we wish to test the hypothesis $H$ that $k$ has a specified value $k_0$. If $k_0 = 1$ the hypothesis $H$ reduces to a classical problem, often referred to in the literature as the Behrens-Fisher-problem (cf. Scheffe [3] for a bibliography). At the present time it is still an open question whether it is possible (or desirable) to find a non-trivial single sample test for $H$ with the size of the critical region independent of $\sigma_1$ and $\sigma_2$. In any case it is a simple extension of the result of Dantzig [1] (cf. also Stein [4]) to show that no non-trivial single sample test exists whose power is independent of $\sigma_1$ and $\sigma_2$. On the other hand the case $k_0 \neq 1$ may be expected to occur frequently in fields of application where a choice must be made between different products, methods of experimentation etc. which involve different costs. The statistician must make a choice on the basis of results relative to the ratio of costs involved. Nevertheless this problem appears to have received little attention in the literature. In general tests based on a two-sample procedure may not be as "efficient" in the sense of Wald [5] as a strict sequential procedure. On the other hand the two sample procedure reduces the number of decisions to be made by the experimenter and it will, in certain fields, simplify the experimental procedure.

The use of transformations to stabilize the variance of binomial or Poisson data is familiar(Anscombe [1], Bartlett [2, 3], Curtiss [4], Eisenhart [5]). The comparison of transformed binomial or Poisson data with percentage points of the normal distribution to make approximate significance tests or to set approximate confidence intervals is less familiar. Mosteller and Tukey [6] have recently made a graphical application of a transformation related to the square-root transformation for such purposes, where the use of "binomial probability paper" avoids all computation. We report here on an empirical study of a number of approximations, some intended for significance and confidence work and others for variance stabilization. For significance testing and the setting of confidence limits, we should like to use the normal deviate $K$ exceeded with the same probability as the number of successes $x$ from $n$ in a binomial distribution with expectation $np$, which is defined by $\frac{1}{2\pi} \int^K_{-\infty} e^{-\frac{1}{2}t^2} dt = \operatorname{Prob} \{x \leq k |mid \operatorname{binomial}, n, p\}.$ The most useful approximations to $K$ that we can propose here are $N$ (very simple), $N^+$ (accurate near the usual percentage points), and $N^{\ast\ast}$ (quite accurate generally), where $N = 2 (\sqrt{(k + 1)q} - \sqrt{(n - k)p)}.$ (This is the approximation used with binomial probability paper.) $N^+ = N + \frac{N + 2p - 1}{12\sqrt{E}},\quad E = \text{lesser of} np \text{and} nq, N^\ast = N + \frac{(N - 2)(N + 2)}{12} \big(\frac{1}{\sqrt{np + 1}} - \frac{1}{\sqrt{nq + 1}}\big), N^{\ast\ast} = N^\ast + \frac{N^\ast + 2p - 1}{12 \sqrt{E}}\cdot\quad E = \text{lesser of} np \text{and} nq.$ For variance stabilization, the averaged angular transformation $\sin^{-1}\sqrt{\frac{x}{n + 1}} + \sin^{-1} \sqrt{\frac{x + 1}{n+1}}$ has variance within $\pm 6%$ of $\frac{1}{n + \frac{1}{2}} \text{(angles in radians)}, \frac{821}{n + \frac{1}{2}} \text{(angles in degrees)},$ for almost all cases where $np \geq 1$. In the Poisson case, this simplifies to using $\sqrt{x} + \sqrt{x + 1}$ as having variance 1.