It is shown that there exist strictly unbiased and consistent tests for the univariate and multivariate two- and $k$-sample problem, for the hypothesis of independence, and for the hypothesis of symmetry with respect to a given point. Certain new tests for the univariate two-sample problem are discussed. The large sample power of these tests and of the Mann-Whitney test are obtained by means of a theorem of Hoeffding. There is a discussion of the problem of tied observations.

A general method of testing the significance of nonlinear regression, suggested by Hotelling, is adapted to the regression equations $Y = be^{px}$ and $Y = a + be^{px}$. The values of $x$ are taken to be in arithmetic progression, and the standard deviation of the observed $y$ is supposed constant for all $x$. This is in contrast to the assumption, implicit in the usual procedure of fitting a straight line to $\log y$, that the standard deviation of $\log y$ is constant. It will be observed that the distribution of $y_1, y_2, \cdots, y_n$ must be such that the joint probability density for $y_1, y_2, \cdots, y_n$ is a function of $x^2_1 + x^2_2 + \cdots + x^2_n$, and this condition implies the assumption of normality. The null hypothesis is that $be^{px} = 0$ for all $x$, while the alternative hypotheses are specified by $b \neq 0, p \neq - \infty$. The method involves the calculation of the volume of a "tube" on a hypersphere in $n$-dimensional space. An asymptotic expression for the length of the tube is developed, and it is shown that the curvature of the axis is everywhere finite. From this expression, for values of the correlation coefficient $R$ between observed and fitted values of $y$ at least as great as 0.894, a function of $R$ is obtained giving the probability that a random sample would yield at least as great a value of $R$. A short table giving $R$ for various significance levels and various sizes of sample is calculated for each of the equations mentioned, and the application to certain experimental data is discussed.

In a recent paper [1] the author investigated the mean number of steps in random walks in $n$-dimensional domains. The purpose of the present article is to generalize those results by applying similar methods to the study of the moment generating function for the number of steps and of its distribution function. As an application explicit asymptotic expressions for the variance in special cases and estimates for the likelihood of very long walks are obtained. The author wishes to express his thanks to Professor R. Fortet for many helpful discussions. The walks take place in an open bounded domain $B$ of $n$-dimensional Euclidean space $E$ with boundary $C$. A point moves in $E$ according to a given transition probability law $F(y, x)$. Here $x$ and $y$ are points of $E$ with coordinates $x_i, i = 1, 2, \cdots, n$, and $y_i, i = 1, 2, \cdots, n$, and $F(y, x)$ is the probability that a jump known to start at $x$ end at a point all of whose coordinates are less than the corresponding ones of $y$. The function $F(y, x)$ is a distribution function with respect to $y$, and it is assumed to be Borel measurable with respect to all variables. Let $N = N_x$ be the number of steps in a random walk that begins at a point $x$ of $B$ and ends with the step on which the moving point leaves $B$ for the first time. If the probability of the moving point eventually leaving $B$ is equal to one, then $N$ is a random variable. It is called the duration of the walk. It is useful to extend the definition of $N$ by setting $N_x = 0,\quad x \varepsilon E - B$.

Unbiased critical regions of type D for testing simple hypotheses specifying the values of several parameters are defined and their properties studied. These regions constitute a natural generalization of the Neyman-Pearson regions of type A for testing simple hypotheses specifying the value of one parameter. A theorem is obtained which plays the role of the Neyman-Pearson fundamental lemma in the type A case. Illustrative examples of type D regions are given.

Sometimes in a design the position within the block is important as a source of variation, and the experiment gains in efficiency by eliminating the positional effect. The classical example is due to Youden in his studies on the tobacco mosaic virus [1]. He found that the response to treatments also depends on the position of the leaf on the plant. If the number of leaves is sufficient so that every treatment can be applied to one leaf of a tree, then we get an ordinary Latin square, in which the trees are columns and the leaves belonging to the same position constitute the rows. But if the number of treatments is larger than the number of leaf positions available, then we must have incomplete columns. Youden used a design in which the columns constituted a balanced incomplete block design, whereas the rows were complete. These designs are known as Youden's squares, and can be used when two-way elimination of heterogeneity is desired. In Fisher and Yates statistical tables [2] balanced incomplete block designs in which the number of blocks $b$ is equal to the number of treatments $v$ have been used to obtain Youden's squares, and the authors state that "in all cases of practical importance" it has been found possible to convert balanced incomplete blocks of the above kind to a Youden's square by so ordering the varieties in the blocks that each variety occurs once in each position. F. W. Levi noted ([3], p. 6) that this reordering can always be done, in virtue of a theorem given by Konig [4] which states that an even regular graph of degree $m$ is the product of $m$ regular graphs of degree 1. Smith and Hartley [5] give a practical procedure for converting balanced incomplete blocks with $b = v$ into Youden's squares. In this paper I have considered some general classes of designs for two-way elimination of heterogeneity. In Section 3 balanced incomplete block designs for which $b = mv$ have been used to obtain two-way designs in which each treatment occurs in a given position $m$ times. The case $m = 1$ gives Youden's squares. In Section 4 it has been shown that balanced incomplete block designs for which $b$ is not an integral multiple of $v$ can be used to obtain designs for two-way elimination of heterogeneity in which there are two accuracies (i.e., some pairs of treatments are compared with one accuracy, while other pairs are compared with a different accuracy) as in the case of lattice designs for one-way elimination of heterogeneity. In Sections 5 and 6 partially balanced designs have been used to obtain two-way designs with two accuracies. In every case the method of analysis and tables of actual designs have been given.

A general discrete distribution is obtained whose random variable is the number of "hits" on a target. The target is $k$-dimensional and Gaussian diffuse, that is, the probability of a hit is given to within a constant factor by a Gaussian probability density function of the position of the "trajectory" in $k$ dimensions. For a series of $n$ rounds, the $n$ positions of the trajectory have a multivariate Gaussian distribution. An expression is given, using Theorems 1 and 2 or 1 and 3, for the probability of $r$ hits as a linear combination of probabilities of all hits on each possible set of rounds. Theorems 4, 5, and 6, with Theorem 1, give three limiting distributions as $n$, the number of rounds, tends to infinity. Theorems 7, 8, and 9, with Theorem 1, present three other limiting cases, and Theorems 10 and 1 give a time average result.

A method based on higher moments is presented in this paper by which the type of a univariate Pearson frequency distribution (population) can be determined and its parameters estimated from truncated samples with known points of truncation and an unknown number of missing observations. Estimating equations applicable to the four-parameter distributions involve the first six moments of a doubly truncated sample or the first five moments of a singly truncated sample. When the number of parameters to be estimated is reduced, there is a corresponding reduction in the order of the sample moments required. A sample is described as singly or doubly truncated according to whether one or both "tails" are missing. Estimates obtained by the method of this paper enjoy the property of being consistent and they are relatively simple to calculate in practice. They should be satisfactory for (a) rough estimation, (b) graduation, and (c) first approximations on which to base iterations to maximum likelihood estimates. Previous investigations of truncated univariate distributions include studies of truncated normal distributions by Pearson and Lee [1], [2], Fisher [3], Stevens [4], Cochran [5], Ipsen [6], Hald [7], and this writer [8], [9]. In addition, the truncated binomial distribution has been studied by Finney [10], and the truncated Type III distribution by this writer [11].

Distributions of characteristic roots have been obtained by Girshick [1], Fisher [2], Hsu [3], and Roy [4]. The present paper outlines an alternative derivation of these distributions which is somewhat more elementary than those that have been published and which may have some pedagogical utility. The primary object of the paper, however, is to obtain the normalizing constants for these distributions; though the correct values of the constants have been published in the references cited above, no convincing derivation seems to have been recorded.

Let $x, y$, and $z$ be three random variables with continuous cumulative distribution functions $f, g$, and $h$. In order to test the hypothesis $f = g = h$ under certain alternatives two statistics $U, V$ based on ranks are proposed. Recurrence relations are given for determining the probability of a given $(U, V)$ in a sample of $l x$'s, $m y$'s, $n z$'s and the different moments of the joint distribution of $U$ and $V$. The means, second, and fourth moments of the joint distribution are given explicitly and the limit distribution is shown to be normal. As an illustration the joint distribution of $U, V$ is given for the case $(l, m, n) = (6, 3, 3)$ together with the values obtained by using the bivariate normal approximation. Tables of the joint cumulative distribution of $U, V$ have been prepared for all cases where $l + m + n \leqq 15$.

From an algebraic point of view the analysis of variance tests of effects and interactions can be based on the minimum values of a certain quadratic expression in which the "$h$-matrix" (defined in Section 3) is fundamental. The arbitrariness in the choice of this matrix reflects the arbitrariness in the definition of effects and interactions. The paper considers the dependence of the result of these tests on the $h$-matrix used and expresses the answer by the two theorems of Section 4, which are proved in the subsequent sections.

In the three papers, [1], [2], [3], entitled "Nonparametric estimation", Scheffe and Tukey generalized previous results on tolerance regions and extended them to cover all continuous and discontinuous distribution functions. This note contains four comments arising from these papers: first, on a method for giving bounds to the confidence level in the discontinuous case which can lower the probability that the end points need to have part, a random variable, of their probability neglected to maintain the given confidence level; second, on a correction of a statement of results in [2]; third, on a proof in [2] requiring a further statement; fourth, a necessary restatement of theorems in [3].

A probability density function, $f(x; \theta)$, is considered for which there exists a sufficient statistic. It is shown, under certain regularity conditions on the family of distributions and on the class of estimates, that if there exists an unbiased sufficient estimate of $\theta$, it will be unique. This result is used to show that when the regularity conditions are satisfied, the method of Blackwell for improving an unbiased estimate of $\theta$ merely yields a natural estimate.

Mood ([1], p. 394) discusses a test for the hypothesis that two samples come from populations having the same continuous cumulative frequency distributions. It consists of arranging the observations from the two samples in a single group in order of size and then comparing the numbers in the two samples above the median. This technique is extended to using several order statistics from the combined samples, and to the case of several samples. The test is non-parametric and might be a good substitute for the single variable of classification analysis of variance in cases of doubtful normality. The application of the test would be the same as in Mood ([1], p. 398) except that there would be more than two rows in the table.

Moments of order statistics multiplied by appropriate factors are called normalized moments. These normalized moments are shown to be successive differences of normalized moments of largest order statistics.