It is shown that, under usual regularity conditions, the maximum likelihood estimator of a structural parameter is strongly consistent, when the (infinitely many) incidental parameters are independently distributed chance variables with a common unknown distribution function. The latter is also consistently estimated although it is not assumed to belong to a parametric class. Application is made to several problems, in particular to the problem of estimating a straight line with both variables subject to error, which thus after all has a maximum likelihood solution.

The assumptions appropriate to the application of analysis of variance to specific examples, and the effects of these assumptions on the resulting interpretations, are today a matter of very active discussion. Formulas for average values of mean squares play a central role in this problem, as do assumptions about interactions. This paper presents formulas for crossed (and, incidentally, for nested and for non-interacting completely randomized) classifications, based on a model of sufficient generality and flexibility that the necessary assumptions concern only the selection of the levels of the factors and not the behavior of what is being experimented upon. (This means, in particular, that the average response is an arbitrary function of the factors.) These formulas are not very complex, and specialize to the classical results for crossed and nested classifications, when appropriate restrictions are made. Complete randomization is only discussed for the elementary case of "no interactions with experimental units" and randomized blocks are not discussed. In discussion and proof, we give most space to the two-way classification with replication, basing our direct proof more closely on the proof independently obtained by Cornfield [17], than on the earlier proof by Tukey [20]. We also treat the three-way classification in detail. Results for the general factorial are also stated and proved. The relation of this paper to other recent work, published and unpublished, is discussed in Section 4 (average values of mean squares) and in Section 11 (various types of linear models).

A measure is introduced of the information provided by an experiment. The measure is derived from the work of Shannon [10] and involves the knowledge prior to performing the experiment, expressed through a prior probability distribution over the parameter space. The measure is used to compare some pairs of experiments without reference to prior distributions; this method of comparison is contrasted with the methods discussed by Blackwell. Finally, the measure is applied to provide a solution to some problems of experimental design, where the object of experimentation is not to reach decisions but rather to gain knowledge about the world.

Let $F$ and $G$ be continuous univariate cdf's. For testing the hypothesis $F = G$ against general alternatives, E. Lehmann [4] has proposed and found certain properties of a test based on the particular measure of discrepancy $\int (F - G)^2 d\lbrack (F + G) / 2\rbrack.$ In this note will be given some additional properties of Lehmann's test (cf. also [8]) and a closely related test proposed by Mood [2].

We are concerned with the asymptotic behavior of stochastic approximation processes of the Robbins-Monro type [9], the Kiefer-Wolfowitz type [7], and related types. Our main interest is establishing the asymptotic normality, under appropriate conditions, of processes of certain kinds. However, a number of results on convergence with probability one are also obtained as immediate consequences of a theorem needed in the work on asymptotic normality. Our results contain and extend some of the results in this area reported by Blum [1], Chung [3], Hodges and Lehmann [5], and others. In addition, we establish a number of results for cases not previously investigated. For instance, we show that the Kiefer-Wolfowitz processes are asymptotically normal under quite general conditions. The rapidity of convergence depends on the amount that the function $M$ (of Corollary 3.2) departs from symmetry in the neighborhood of the location of the maximum. We give results on convergence with probability one and asymptotic normality of stochastic approximation processes useful in connection with the problem of finding the location of the point of inflection of a function. For all cases in which we establish asymptotic normality we also show how the unknown quantities in the variance of the limiting normal distribution can be estimated. These results make possible the construction of asymptotic confidence intervals free of unknowns. Other results, too detailed to be summarized here, are established which also might be of interest in practical applications. Our method of procedure is as follows. We define a class of stochastic approximation processes, denoted by $A_0,$ which contains the class of Robbins-Monro processes, the class of Kiefer-Wolfowitz processes, and some other related classes. We first study the class $A_0$ using methods similar to those used in [1], [3], and [5]. After various results at this level have been obtained, the results for the special cases follow with an economy of effort.

In this paper the asymptotic expansion of a percentage point of Hotelling's generalized $T^2_0$ distribution is derived in terms of the corresponding percentage point of a $\chi^2$ distribution. Our result generalizes Hotelling's and Frankel's asymptotic expansion for the generalized Student $T$ [3], [4]. The technique used in this paper for obtaining the asymptotic expansion of $T^2_0$ is an extension of the previous methods of Welch [8] and of James [5], [6], who used them to solve the distribution problem of various statistics in connection with the Behrens-Fisher problem. An asymptotic formula for the cumulative distribution function (c.d.f.) of $T^2_0$ is also given together with an upper bound for the error committed when all but the first few terms are omitted in the series. This formula is a sort of multivariate analogue of Hartley's formula of "Studentization" [2].

In this paper a few results (of a purely mathematical nature) are obtained, which are useful for studying certain distribution problems in multivariate analysis--e.g., those relating to the characteristic roots of a determinantal equation ([1], [2], [5]). In particular, the results are shown to be readily applicable to the moment problems of the sum of the roots and the distributions of the extreme roots. Most of the results given are in the form of certain recursion formulae for reducing special types of $k$-th order Vandermonde determinants in terms of those of orders $(k - 1)$ and $(k - 2).$ The applications of these results are given by S. N. Roy [6] and the present author [3].

It is shown that in simple statistical designs in which a covariance adjustment is made for concomitant variation, an unbiased between-treatment mean square can be produced by weighted randomization, i.e., by selecting an arrangement at random from a set of arrangements, giving different arrangements in the set unequal chances of selection.

Suppose $X_1, X_2, \cdots, X_n$ are known to be independently and identically distributed, each with the density function $f(x),$ with $\int^1_0f(x) dx = 1.$ Let $Y_1 \leqq Y_2 \leqq \cdots \leqq Y_n$ be the ordered values of $X_1, X_2, \cdots, X_n,$ and define $W_1 = Y_1, W_2 = Y_2 - Y_1, \cdots, W_n = Y_n - Y_{n -1},$ and $W_{n + 1} = 1 - Y_n,$ so that $W_1 + \cdots + W_{n + 1} = 1.$ Finally, define $Z_1, \cdots, Z_{n + 1}$ as the ordered values of $W_1, \cdots, W_{n + 1},$ so that $0 \leqq Z_1 \leqq Z_2 \leqq \cdots \leqq Z_{n + 1},$ with $Z_1 + \cdots + Z_{n+1} = 1.$ We are going to test the hypothesis that $f(x) = 1$ for $0 < x < 1,$ and we are going to consider only tests based on $Z_1, Z_2, \cdots, Z_n.$ The intutitive justification for this is that, roughly speaking, deviations from the hypothesis on any part of the unit interval are treated alike. Several authors have discussed tests based on $Z_1, \cdots, Z_n.$ (See references [1], [2], [3].) If $u$ is a number greater than unity, it is shown that the test of the form "reject the hypothesis if $Z^u_1 + \cdots + Z^u_{n+1} > K"$ is consistent against a very wide class of alternatives. When $u = 2,$ the resulting test has some desirable properties with respect to alternatives with linear density functions.

The following theorems are proved. THEOREM 1. If $x_1, x_2, \cdots$ satisfy $-1 \leqq x_n \leqq a, a \leqq 1$ and $E(x_n \mid x_1, \cdots, x_{n-1}) \leqq - u \max (| x_n | \mid x_1, \cdots, x_{n-1}), 0 < u < 1$, then for any positive $t$ $$\mathrm{Pr}\{x_1 + \cdots + x_n \geqq t \text{for some} n\} \leqq \theta^t,$$ where $\theta$ is the positive root (other than $\theta = 1$) of \begin{equation*}\tag{1} \frac{a + u}{a + 1} \theta^{a+1} - \theta^a + \frac{1 - u}{a + 1} = 0.\end{equation*} This choice of $\theta$ is the best possible. THEOREM 2. If $x_1, x_2, \cdots$ satisfy $| x_n | \leqq 1$ and $E(x_n \mid x_1, \cdots, x_{n - 1}) = 0,$ then for all $N > 0,$ $$\mathrm{Pr}\big\{\big|\frac{x_1 + \cdots + x_n}{n}\big| \geqq \epsilon \text{for some} n \geqq N\big\} \leqq 2_\varphi^N,$$ where $\varphi = (1 + \epsilon)^{-(1+\epsilon)/2}(1 - \epsilon)^{-(1 - \epsilon)/2}.$ This choice of $\varphi$ is, for every $\epsilon$ between 0 and 1, the best possible. Both results are improvements of results of Blackwell [1], and the methods of proof are somewhat similar.

The statistical literature combines two types of investigations concerning the consistency of maximum likelihood (M.L.) estimates. A few of these, such as the most excellent paper of A. Wald [1], do prove directly the consistency of M.L. estimates. However, most investigators seem to have concentrated their efforts on proving the existence and consistency of suitably selected roots of the successive likelihood equations. Some authors, see [2], for example, add the supplementary remark that such consistent roots will eventually be unique in suitably small neighborhoods of the true value and will achieve a local maximum. It is the purpose of the present note to point out by means of examples that this second mode of attack is not adequate. In the examples given below, the "usual regularity conditions" of Cramer [3] or Wald [4] are satisfied, but the M.L. estimates are not consistent. It should also be pointed out that the direct proofs of existence of roots, simple in the case of a unidimensional parameter, become unwieldy in more than one dimension. On the other hand, if one has proved the consistency of the M.L. estimates, the existence of roots follows trivially from the fact that when a differentiable function reaches its maximum in an open set, the derivatives vanish at that point.

Under certain mild restrictions on the distributions involved, it is shown that the probabilities of the two types of error uniquely determine the two bounds characterizing the Wald sequential probability ratio test.