A class of multiple decision procedures is described and its members are shown to possess uniformly minimum risk among all procedures that are unbiased with respect to a certain loss function. This provides a justification for a number of procedures considered by Tukey, Duncan, and others, for certain classes of point estimates, and for some nonparametric decision procedures based on sample cumulative distribution functions and related to tests of the Kolmogoroff-Smirnoff type.

The variance of the usual estimate of between variance components in an unbalanced single classification has been found for arbitrary infinite populations by Hammersley [1], who found it necessary to use rather heavy algebra. The methods of polykays are here applied to a family of weighted estimates to obtain the variances and covariances of the estimates of between and within variance components. These apply to arbitrary finite populations. Weighting column means equally seems to give a better estimate than the classical proportional weighting for the between variance component as soon as (i) the between component exceeds $\frac{1}{2}$ of the within component in a moderately unbalanced design, or (ii) the between component exceeds the within component in a substantially unbalanced design. Slight further gains come from intermediate weighting. Numerical examples are given. While pooling mean squares instead of sums of squares across columns loses accuracy, notably for the within variance component, doing the same in calculating the between variance component seems to have a minor effect. If the within contributions are sufficiently non-normal, this effect will be favorable.

Generalized sequential probability ratio tests (hereafter abbreviated GSPRT's) for testing between two simple hypotheses have been defined in [1]. The present paper, divided into four sections, discusses certain properties of GSPRT's. In Section 1 it is shown that under certain conditions the distributions of the sample size under the two hypotheses uniquely determine a GSPRT. In the second section, the admissibility of GSPRT's is discussed, admissibility being defined in terms of the probabilities of the two types of error and the distributions of the sample size required to come to a decision; in particular, notwithstanding the result of Section 1, many GSPRT's are inadmissible. In Section 3 it is shown that, under certain monotonicity assumptions on the probability ratios, the GSPRT's are a complete class with respect to the probabilities of the two types of error and the average distribution of the sample size over a finite set of other distributions. In Section 4, finer characterizations are given of GSPRT's which minimize the expected sample size under a third distribution satisfying certain monotonicity properties relative to the other two distributions; these characterizations give monotonicity properties of the decision bounds.

The present paper gives the formal statements and proofs of the results illustrated in [1]. In a series of papers ([2], [3], [4]) the present author has been developing the minimum distance method for obtaining strongly consistent estimators (i.e., estimators which converge with probability one). The method of the present paper is much superior, in simplicity and generality of application, to the methods used in the papers [2] and [4] cited above. Roughly speaking, the present paper can be summarized by saying that, in many stochastic structures where the distribution function (d.f.) depends continuously upon the parameters and d.f.'s of the chance variables in the structure, those parameters and d.f.'s which are identified (uniquely determined by the d.f. of the structure) can be strongly consistently estimated by the minimum distance method of the present paper. Since identification is obviously a necessary condition for estimation by any method, it follows that, in many actual statistical problems, identification implies estimatability by the method of the present paper. Thus problems of long standing like that of Section 5 below are easily solved. For this problem the whole canonical complex (Section 6 below; see [1]) has never, to the author's knowledge, been estimated by any other method. The directional parameter of the structure of Section 4 seems to be here estimated for the first time. As the identification problem is solved for additional structures it will be possible to apply the minimum distance method. The proofs in the present paper are of the simplest and most elementary sort. In Section 8 we treat a problem in estimation for nonparametric stochastic difference equations. Here the observed chance variables are not independent, but the minimum distance method is still applicable. The treatment is incomparably simpler than that of [4], where this and several other such problems are treated. The present method can be applied to the other problems as well. Application of the present method is routine in each problem as soon as the identification question is disposed of. In this respect it compares favorably with the method of [4], whose application was far from routine. As we have emphasized in [1], the present method can be applied with very many definitions of distance (this is also true of the earlier versions of the minimum distance method). The definition used in the present paper has the convenience of making a certain space conditionally compact and thus eliminating the need for certain circumlocutions. Since no reason is known at present for preferring one definition of distance to another we have adopted a convenient definition. It is a problem of great interest to decide which, if any, definition of distance yields estimators preferable in some sense. The definition of distance used in this paper was employed in [9]. As the problem is formulated in Section 2 below (see especially equation (2.1), the "observed" chance variables $\{X_i\}$ are known functions (right members of (2.1)) of the "unobservable" chance variables $\{Y_i\}$ and of the unknown constants $\{\theta_i\}$. In the problems treated in [3], [9], and [11], it is the distribution of the observed chance variables which is a known function of unobservable chance variables and of unknown constants, and not the observed chance variables themselves. However, the latter problems can easily be put in the same form as the former problems. Moreover, in the method described below the values of the observed chance variables are used only to estimate the distribution function of the observed chance variables (by means of the empiric distribution function). Consequently there is no difference whatever in the treatment of the problems by the minimum distance method, no matter how the problems are formulated. The unobservable chance variables $\{Y_i\}$ correspond to what in [11] are called "incidental parameters"; the unknown constants $\{\theta_i\}$ are called in [11] "structural parameters". In [9] there is a discussion of the fact that in some problems treated in the literature the incidental parameters are considered as constants and in other problems as chance variables. In contradistinction to the present paper [3] (in particular its Section 5) treats the incidental parameters as unknown constants. The fundamental idea of both papers is the same: The estimator is chosen to be such a function of the observed chance variables that the d.f. of the observed chance variables (when the estimator is put in place of the parameters and distributions being estimated) is "closest" to the empiric d.f. of the observed chance variables. The details of application are perhaps easier in the present paper; the problems are different and of interest per se.

Maximum likelihood estimates and their asymptotic distribution are obtained for the transition probabilities in a Markov chain of arbitrary order when there are repeated observations of the chain. Likelihood ratio tests and $\chi^2$-tests of the form used in contingency tables are obtained for testing the following hypotheses: (a) that the transition probabilities of a first order chain are constant, (b) that in case the transition probabilities are constant, they are specified numbers, and (c) that the process is a $u$th order Markov chain against the alternative it is $r$th but not $u$th order. In case $u = 0$ and $r = 1$, case (c) results in tests of the null hypothesis that observations at successive time points are statistically independent against the alternate hypothesis that observations are from a first order Markov chain. Tests of several other hypotheses are also considered. The statistical analysis in the case of a single observation of a long chain is also discussed. There is some discussion of the relation between likelihood ratio criteria and $\chi^2$-tests of the form used in contingency tables.

In recent years, problems related to the characterization of the normal distribution from the property of stochastic independence of linear functions of independent random variables have been investigated by various authors. The most general result in this direction is that obtained independently by Darmois [4] and Skitovich [14], who proved that if there exist two linear functions $$X = \sum^n_{j = 1} a_jx_j \text{and} Y = \sum^n_{j = 1} b_jx_j\quad\text{with} a_ib_j \neq 0$$ $(j = 1, 2, \cdots n)$, such that they are stochastically independent, where $x_1, x_2, \cdots, x_n$ are $n$ independently (but not necessarily identically) distributed proper random variables, then each $x_j$ is normally distributed. Their methods of proof are similar in nature, both being based on the use of characteristic functions, without any assumption about the existence of moments. The same theorem has been proved independently by Basu [1], under the assumption that the random variables are identically and independently distributed and have finite moments of all orders. This result is also obtained independently by Lukacs and King [11], under the assumption that the random variables are independently (but not necessarily identically) distributed, each having finite moments up to order $n$, and by Linnik [10], under the assumption that the random variables have only finite variances. The special case of this theorem for $n = 2$ was proved earlier independently by Kac [6], Gnedenko [5], and Darmois [3], without any assumption about the existence of moments. Thus we see that the problems on the consequences of stochastic independence of linear statistics have been exhaustively studied. Hence the question that naturally arises is whether similar investigations into the distribution laws of the random variables are possible under the assumption of a suitable type of stochastic dependence of the linear statistics. In this direction, the author [8] has derived a characterization of the stable law with finite expectation from the property of linearity of regression of one linear statistic on the other for the case $n = 2$. The author [7] has also obtained some characterizations of the normal distribution from the consequence of the linearity of the multiple regression of one random variable on several others, when the variables have the linear structural setup as in the case of the bi-factor theory of Spearman. For the formulation of the problem investigated in the present paper, we require a precise definition of the terms conditional distribution, linearity of regression, and homoscedasticity. Let $F(x, y)$ and $F_0(x)$ denote respectively the distribution function of the two-dimensional random variable $(x, y)$ and the marginal distribution function of $x$. Then we define the conditional distribution function of $y$ for fixed $x$ by $F_x(y)$ so that it satisfies the relation \begin{equation*}\tag{1.1}\int^x_{-\infty} F_\xi(y) dF_0(\xi) = F(x, y).\end{equation*} In the present investigation, the following assumptions on the distributions of the random variables will be made: Assumption 1. The conditional distribution of $y$ for fixed $x$ as defined in (1.1) is assumed to exist, wherever needed. Assumption 2. Each of the random variables concerned has a finite second moment. This assumption allows us to take derivatives of the characteristic functions of the corresponding random variables up to and including the second order. Assumption 3. Without any loss of generality in the proof, it is assumed that the expectation of each of the random variables concerned is equal to zero. The role of these assumptions is to ensure the existence of the expectation and the variance of the conditional distribution of $y$ for fixed $x$, which may be defined respectively as \begin{align*}\tag{1.2}E_x(y) &= \int^\infty_{-\infty} y dF_x(y), \\ \tag{1.3}V_x(y) &= \int^\infty_{-\infty} \{y - E_x(y)\}^2 dF_x(y) \\ &= S_x(y) - \{E_x(y)\}^2, \end{align*} where $$S_x(y) = \int^\infty_{-infty} y^2 dF_x(y).$$ In this case, the regression of $y$ on $x$ is said to be linear, if the relation \begin{equation*}\tag{1.4}E_x(y) = \beta x\end{equation*} is satisfied for all $x$, except for a set of probability measure zero, as the expectations of both the random variables $x$ and $y$ are already assumed to be zero. The constant $\beta$ in equation (1.4) is called the coefficient of regression of $y$ on $x$. Similarly the conditional distribution of $y$ for fixed $x$ is said to be homoscedastic, if the conditional variance $V_x(y)$ as defined in (1.3) is a constant $\sigma^2_0$ not involving $x$. Thus if the regression of $y$ and $x$ is linear and given by $\beta x$ and the conditional variance of $y$ for fixed $x$ is a constant $\sigma^2_0$, we have the relation \begin{equation*}\tag{1.5}S_x(y) = \sigma^2_0 + \beta^2x^2\end{equation*} to be satisfied for all $x$, in addition to the condition (1.4). For simplicity in notation, throughout the present paper we shall use the term the conditional distribution of $y$ for fixed $x$ is L.R.H. ($\beta, \sigma^2_0)$ meaning thereby that the regression of $y$ on $x$ is linear and given by $\beta x$ and that the conditional variance of $y$ for fixed $x$ is $\sigma^2_0$, being free of $x$, i.e., equivalent to the statement that both the relations (1.4) and (1.5) are simultaneously satisfied. In the following sections we shall derive some characterizations of the normal distribution from the property of linearity of regression and homoscedasticity of suitable linear statistics. The main theorem (Theorem 3.2) is given in Section 3. The proof of this theorem uses as a starting point a very simple set of necessary and sufficient conditions for the linearity of regression and homoscedasticity (Lemma 2.1) and finally a theorem of Linnik (Lemma 2.3) on an analytical extension of Cramer's theorem [2] on the normal law. Several important corollaries are deduced in the subsequent section.

Proposed is a method for approximating the distribution of the ranks, which is the basis for evaluating the power of an arbitrary rank test (see definition of "rank test" in Section 2 below). The method involves, in essence, a transformation of the original distributions, by means of interpolating polynomials, into distributions defined on the unit interval (0, 1). A somewhat detailed discussion is given to the problem of testing the hypothesis that two populations are identical against the alternative hypothesis that they have two specified (non-identical) distributions. Explicit formulas for approximating the distributions of the ranks under the alternative hypothesis are given. A few tables are computed for the case where both distributions are normal with the same variance but different means. The last section is devoted to the investigation of the asymptotic power efficiency of certain rank tests.

Confidence intervals for, and tests of hypotheses about, the interquantile distance are obtained, using one or two properly chosen quasi-ranges. Consistency (of the estimates and tests) is proved. Applications are also given to making inferences about the standard deviations of distributions whose cdf's are of the form $F((x - \mu)/\lambda)$.

Suppose $X_1, \cdots, X_m, Y_1, \cdots, Y_n$ are $m + n = N$ independent random variables, the $X$'s identically distributed and the $Y$'s identically distributed, each with a continuous cdf. Let $$z = (z_1, \cdots, z_m, z_{m + 1}, \cdots, z_N) = (x_1, \cdots, x_m, y_1, \cdots, y_n)$$ represent an observation on the $N$ random variables and let $$u(z) = (1/m) \sum^m_{i = 1} z_i - (1/n) \sum^N_{i = m + 1} z_i = \bar x - \bar y$$. Consider the $r = N! N$-tuples obtained from $(z_1, \cdots, z_N)$ by making all permutations of the indices $(1, \cdots, N)$. Since we assume continuous cdf's, then with probability one, these $r N$-tuples will be distinct. Denote them by $z^{(1)}, \cdots, z^{(r)}$, and suppose that they have been ordered so that $$u(z^{(1)} \geqq \cdots \geqq u(z^{(r)})$$. Notice that since $$\bar x - \bar y = (1/m) \sum^N_{i = 1} z_i - (N/m)\bar y = (N/n)\bar x - (1/n) \sum^N_{i = 1} z_i,$$ the same ordering can be induced by choosing $u(z) = c\bar x$ or $u(z) = - c\bar y$ for any $c > 0$. Assuming that the cdf's of $X_1, Y_1$ are of the form $F(x), F(x - \Delta)$ respectively, Pitman [2] suggested essentially the following test of the hypothesis $H'$ that $\Delta = 0$. Select a set of $k (k > 0)$ integers $i_1, \cdots, i_k, (1 \leqq i_1 < \cdots < i_k \leqq r)$. If the observed $z$ is one of the points $z^{(i_1)}, \cdots, z^{(i_k)}$, reject $H'$, otherwise accept. When $H'$ is true, the type one error does not depend on the specific form of the distribution of the $X$'s and the $Y$'s and is in fact equal to $k/r$. The choice of the rejection set $i_1, \cdots, i_k$ should depend on the alternative hypothesis. For instance, if the experimenter wants protection against the alternative that the "$X$'s tend to be larger than the $Y$'s," then the labels $1, \cdots, k$ might be reasonable. For the alternative that the "$X$'s tend to be smaller than the $Y$'s" the analogous procedure is to use the other tail, $r - k + 1, \cdots, r$. Against both alternatives, a two-tail procedure could be used. Lehmann and Stein have shown in [1] that in the class of all tests (of size $\alpha = k/r$) of the hypothesis $$H: \text{the distribution of} X_1 \cdots, X_m, Y_1, \cdots, Y_n \text{is invariant under all permutations},$$ the single-tail test based on $1, \cdots, k$ is uniformly most powerful against the alternatives that $F_1$ is an $N(\theta, \sigma)$ cdf, $F_2$ is an $N(\theta + \Delta, \sigma)$ cdf, $\Delta < 0$; the test based on $r - k + 1, \cdots, r$ is uniformly most powerful for $\Delta > 0$. A practical shortcoming of this procedure is the great difficulty in enumerating the points $z^{(i)}$ and the evaluation of $u(z^{(i)})$ for each of them. For instance, even after eliminating those permutations which always give the same value of $u$, then for sample sizes $m = n = 5$, there are $\binom{10}{5} = 252$ permutations to examine, and for sample sizes $m = n = 10$, there are $\binom{20}{10} = 184,765$ permutations to examine. In the following section, we propose the almost obvious procedure of examining a "random sample" of permutations and making the decision to accept or reject $H$ on the basis of those permutations only. Bounds are determined for the ratio of the power of the original procedure to the modified one. Some numerical values of these bounds are given in Table 1. The bounds there listed correspond to tests which in both original and modified form have size $\alpha$, and for which the modified test is based on a random sample of $s$ permutations drawn with replacement. These have been computed for a certain class of alternatives which is described below. For simplicity, we have restricted the main exposition to the two-sample problem. In Section 5, we point out extensions to the more general hypotheses of invariance studied in [1].

Suppose that a relationship $\eta = \varphi(\xi_1, \xi_2, \cdots, \xi_k)$ exists between a response $\eta$ and the levels $\xi_1, \xi_2, \cdots, \xi_k$ of $k$ quantitative variables or factors, and that nothing is assumed about the function $\varphi$ except that, within a limited region of immediate interest in the space of the variables, it can be adequately represented by a polynomial of degree $d$. A $k$-dimensional experimental design of order $d$ is a set of $N$ points in the $k$-dimensional space of the variables so chosen that, using the data generated by making one observation at each of the points, all the coefficients in the $d$th degree polynomial can be estimated. The problem of selecting practically useful designs is discussed, and in this connection the concept of the variance function for an experimental design is introduced. Reasons are advanced for preferring designs having a "spherical" or nearly "spherical" variance function. Such designs insure that the estimated response has a constant variance at all points which are the same distance from the center of the design. Designs having this property are called rotatable designs. When such arrangements are submitted to rotation about the fixed center, the variances and covariances of the estimated coefficients in the fitted series remain constant. Rotatable designs having satisfactory variance functions are given for $d = 1, 2$; and $k = 2, 3, \cdots, \infty$. Blocking arrangements are derived. The simplification in the form of the confidence region for a stationary point resulting from the use of a second order rotatable design is discussed.

Let $X_1, \cdots, X_m; Y_1, \cdots, Y_n$ be independently distributed on the unit interval. Assume that the $X$'s are uniformly distributed and that the $Y$'s have an absolutely continuous distribution whose density $g(y)$ is bounded and has at most finitely many discontinuities. Let $Z_0 = 0, Z_{n + 1} = 1$, and let $Z_1 < \cdots < Z_n$ be the values of the $Y$'s arranged in increasing order. For each $i = 1, \cdots, n + 1$ let $S_i$ be the number of $X$'s which lie in the interval $\lbrack Z_{i - 1}, Z_i\rbrack$. For each nonnegative integer $r$, let $Q_n(r)$ be the proportion of values among $S_1, \cdots, S_{n + 1}$ which are equal to $r$. Suppose $m$ and $n$ approach infinity in the ratio $(m/n) = \alpha > 0$. In Section 2 it is shown that $$\operatornamewithlimits{\lim}{n \rightarrow \infty} \operatornamewithlimits{\sup}{r \geqq 0} |Q_n(r) - Q(r)| = 0$$ with probability one, where $$Q(r) = \alpha^r \int^1_0 \frac{g^2(y)}{\lbrack\alpha + g(y)\rbrack^{r + 1}}dy.$$ This result may be used to prove consistency of certain tests of the hypothesis that the two samples have the same continuous distribution. Several such examples are given in Section 3. A further property of one of these tests is briefly discussed in Section 4.

Generalizing earlier observations by Fisher and Hotelling, Tukey [1] showed that if a family of distributions admits a set of sufficient statistics, then the family obtained by truncation to a fixed set, or by a fixed selection, also admits the same set of sufficient statistics (this wording is Tukey's; we give a precise mathematical statement later). Tukey's proof assumed the relevant family of probability measures to be dominated by a fixed measure function and made use of the factorization theorem concerning sufficient statistics in this case. In the present short note we shall first re-prove Tukey's result without assuming domination (and, hence, without appealing to the factorization theorem). Then we shall show that, under general conditions, if a sufficient statistic has one or more of the properties of completeness, bounded completeness, or minimality, before truncation, then it preserves such after truncation. The treatment is on the lines of the abstract discussion of sufficient statistics given by Halmos and Savage [2]. We shall assume familiarity with the results given in this latter paper. For definitions of completeness, bounded completeness, and minimality, and for a discussion of the significance of these concepts we refer to Lehmann and Scheffe [3].