The phrase "testing against trend" in the title refers to a situation in which observations are made with equal sample sizes on several populations belonging to a common univariate exponential family. Order relations among the parameters associated with the various populations are assumed known, and it is desired to test the null hypothesis that the parameters are all equal. The likelihood ratio test is described in Section 3. Slight extensions, developed in Section 1, of known theorems suffice to determine, in a certain sense, the asymptotic distribution of an appropriate function of the likelihood ratio. This asymptotic distribution is that of Bartholomew's combination of Chi-squares.

Independent identically distributed observations, $X_1, X_2, \cdots$, are taken sequentially. All that is known a priori about their common probability measure, $P$, is that it is a member of a given (at most countable) family, $\pi = \{P_n\}^\infty_{n=1}$, of such measures. At some time, depending only on the observed data and the tolerable probability of error, one wants to stop and decide which $P_k$ nature has chosen. Two sampling situations are considered, with and without error, as well as two stopping time requirements, uniformly (over $\pi$) bounded and $P_k$-dependent. Necessary and/or sufficient conditions for the distinguishability of the measures in $\pi$ in terms of a variety of measure metrics are obtained. The Levy-Prokhorov metric proves to be particularly relevant.

In this note we discuss the notions of biorthogonal and dual configurations and their relevance in certain statistical applications. The first application is to the distribution of a random matrix related to a multi-variate-normal sample matrix. As with the latter, the distribution is preserved by (certain) linear transformations. One consequence of this is the familiar result that if $\mathbf{Q}$ is a non-singular Wishart matrix, then for any non-zero vector $\alpha, 1/\alpha'\mathbf{Q}^{-1}\alpha$ is a multiple of a chi-square variable. Application is also made to the Gauss-Markov theorem and to certain estimates of mixing proportions due to Robbins.

"Best" estimators of variance components for the unbalanced cases of random-effects models are not known. In fact, even for the very simplest of the unbalanced "designs", the balanced incomplete block designs, the question of the existence of minimum variance unbiased estimators remains open (Kapadia and Weeks [5]). The traditional approach to the derivation of variance-component estimators for unbalanced cases has been to pick several quadratic functions of the data, set these functions equal to their expectations, and then solve the resulting system of equations for the variance components. Two of the estimators derived in this fashion for the variance components associated with the unbalanced two-way cross classification are those referred to as the Methods-1 and -3 estimators of Henderson [4]. Method-1 utilizes quadratics analogous to the sums of squares in a balanced analysis of variance. The quadratics employed in Method-3 represent differences between reductions in sums of squares due to fitting different models. Since in Method-3 more differences between reductions are available than one has variance components to estimate, the method is not uniquely defined. Here, the Method-3 estimators of the components associated with the two-way classification are taken to be those in Harville [3], which are the ones most commonly used. Searle [9] obtained algebraic expressions for the sampling variances of the Method-1 estimators of the "two-way" components. Low [6] gave similar expressions for the Method-3 estimators for the zero-interaction case. Their results were obtained by applying well-known formulas for the variances and covariances of quadratic functions of multivariate-normal random variables. These formulas state that if $\mathbf{y}$ is a random vector having the multivariate normal distribution with mean $\mathbf{u}$ and variance-covariance matrix $\mathbf{V}$ and if $\mathbf{A}$ and $\mathbf{B}$ are square symmetric matrices of appropriate dimension having fixed elements, then \begin{align}\tag{1}\operatorname{var} \lbrack\mathbf{y}'\mathbf{Ay}\rbrack &= 4\mathbf{u}'\mathbf{AV}\mathbf{Au} + 2\mathrm{tr}(\mathbf{VA})^2\end{align}and\begin{align}\tag{2} \operatorname{cov} \lbrack\mathbf{y}'\mathbf{Ay},\mathbf{y}' \mathbf{B}y\rbrack &= 4\mathbf{u}'\mathbf{AVBu} + 2 \mathrm{tr} (\mathbf{VAVB})\end{align} Searle [8], [10] and Mahamunulu [7] have also used these formulas to obtain algebraic expressions for the variances of commonly-used estimators of the components of variance associated with other unbalanced classifications. In the present paper, results (supplementary to those of Searle) are given which lead to expressions for the sampling variances of Method-3 estimators of the variance components associated with the unbalanced two-way cross classification with interaction. By using these results in combination with those of Searle, the variances of Method-1 and Method-3 estimators can be directly compared for a given set of subclass numbers. The results are shown to simplify when the "unbalancedness" is of the type associated with a balanced incomplete block design. Neither estimator of any component is uniformly better than the other for any such design. (Except for the estimators of the residual component which are identically equal.)

Consider the problem of Bayesian sequential estimation of a real parameter $\theta$ with quadratic loss and fixed cost $c$ per observation. It is well known (cf. [1], [2]) that, under simple regularity conditions, this problem reduces to the following one. If $Z_1, Z_2, \cdots, Z_n, \cdots$ are the observations (independent and identically distributed given $\theta$) let \begin{equation*}\tag{1.1}Y_n = \operatorname{Var} (\theta\mid Z_1, \cdots, Z_n),\end{equation*} the posterior variance of $\theta$, and, \begin{equation*}\tag{1.2}X_n(c) = Y_n + nc.\end{equation*} The problem is then to find a stopping time $s(c)$ such that $E(X_{s(c)}(c)) = \inf \{E(X_t(c)):t \varepsilon T\}$ where $T$ is the set of all stopping times. In general, although $s(c)$ can usually be shown to exist finding it in explicit form is difficult. In [2] we proposed the following stopping time $\tilde{t}(c)$ for this problem: "Stop as soon as $Y_n \leqq c(n + 1)$". We showed in [2] (generalized in [3]) that under some regularity conditions this rule is asymptotically pointwise optimal (A.P.O.) i.e., \begin{equation*}\tag{1.3}\lim_{c\rightarrow 0} X_{\tilde{t}(c)}(c)\lbrack X(c)\rbrack^{-1} = 1\operatorname{a.s.} \end{equation*} where, \begin{equation*}\tag{1.4}X(c) = \inf_nX_n(c).\end{equation*} In fact, we proved that, \begin{equation*}\tag{1.5}X_{\tilde{t}(c)}(c) = 2c^{\frac{1}{2}}V^{\frac{1}{2}}(\theta) + o(c^{\frac{1}{2}}) \operatorname{a.s.}\end{equation*} and, \begin{equation*}\tag{1.6}X_{\tilde{t}(c)}(c) - X(c) = o(c^{\frac{1}{2}}) \operatorname{a.s.}\end{equation*} where $V(\theta)$ is the reciprocal of the Fisher information number. Later, (in [3]) we showed, under some additional conditions that $\tilde{t}(c)$ is asymptotically optimal i.e., that, \begin{equation*}\tag{1.7}\lim_{c\rightarrow 0}\lbrack E(X_{s(c)}(c))\rbrack\lbrack E(X_{\tilde{t}(c)}(c))\rbrack^{-1} = 1,\end{equation*} and in fact, that \begin{equation*}\tag{1.8}E(X(c)) = 2c^{\frac{1}{2}}E(V(\theta)) + o(c^{\frac{1}{2}})\end{equation*} and \begin{equation*}\tag{1.9}E(X_{\tilde{t}(c)}(c)) - E(X(c)) = o(c^{\frac{1}{2}}).\end{equation*} In this paper we seek to refine the term $o(c^{\frac{1}{2}})$ in (1.5)-(1.6) and (1.8)-(1.9). Our analysis, as in our previous work, is based on looking at the asymptotic properties of $Y_n$. We showed in [2] and [4] that, \begin{equation*}\tag{1.10}Y_n = V(\theta) n^{-1} + R_n\end{equation*} where $R_n = o(n^{-1})$ a.s. In [4] we further showed that, under suitable conditions, \begin{equation*}\tag{1.11}Y_n = V(\theta)n^{-1} + S_n(\theta)n^{-2} + R'_n\end{equation*} a.s. where $R'_n = o(n^{-3/2})$ and \begin{equation*}\tag{1.12}S_n(\theta) = \sum^n_{i=1} W_i(\theta)\end{equation*} where the $W _i$ are independent and identically distributed with mean $0$ given $\theta$. If $W_1(\theta)$ has a second moment and is non degenerate the law of the iterated logarithm enables us to conclude that, \begin{equation*}\tag{1.13}R_n = O(n^{-3/2}\lbrack\log\log n\rbrack^{\frac{1}{2}}) \operatorname{a.s.}\end{equation*} This suggests Theorem 2.1 which asserts that if (1.13) holds then, \begin{equation*}\tag{1.14} X_{\tilde{t}(c)}(c) - X(c) = 0(c^{3/4-\epsilon})\end{equation*} a.s. for all $\epsilon > 0$. The analogues of (1.8) and (1.9) pose greater difficulty. In Section 3 we show that (Theorems 3.1, 3.2), \begin{equation*}\tag{1.15}E(X(c) - 2\lbrack V(\theta)c\rbrack^{\frac{1}{2}}) = \max (o(c^{\frac{1}{2}+\delta(\lambda, b)-\epsilon}), O(c)),\end{equation*} for every $\epsilon > 0$ where, \begin{equation*}\tag{1.16}\delta(\lambda, b) = \frac{1}{2}(\lambda - 1)b(b + (\lambda - 1))^{-1}\end{equation*} and $b$ and $\lambda$ depend on the problem. (Typically $\lambda = \frac{3}{2}$.) On the other hand, in Section 4 we establish, (Theorem 4.1), \begin{equation*}\tag{1.17}E(X_{\tilde{t}(c)}(c) - 2\lbrack V(\theta)c\rbrack^{\frac{1}{2}})^+ = \max (O(c^{\lambda/2}), O(c)),\end{equation*} for every $\epsilon > 0$ where again typically $\lambda = \frac{3}{2}$. Finally in Section 5 we apply our general results to two special situations. (i) Estimating the mean of a normal distribution with a normal prior. (ii) Estimating $p$ on the basis of binomial trials with a beta prior. In case (i) our conditions yields $O(c)$ in both (1.15) and (1.17) and this is best possible. In (ii) when for instance we have a uniform prior the best $\lambda = \frac{3}{2}$ and the best $b = 1$ and we therefore get $o(c^{-3/4\epsilon})$ for every $\epsilon > 0$ in (1.15) and $o(c^{2/3-\epsilon})$ for every $\epsilon > 0$ in (1.17). We do not believe these are best possible. A further analysis of (1.11) would seem to be required for anything better.

Let $X_1, X_2, \cdots$ be independent random variables with distribution functions $V_1, V_2, \cdots$, zero means and finite non-zero variances $\sigma_1^2,\sigma_2^2, \cdots$. Set $s_n^2 = \sum^n_1 \sigma_i^2$ and $\Phi(x) = (2\pi)^{-\frac{1}{2}} \int^x_{-\infty}e^{-t{}^2/2} \operatorname{dt}$. Define \begin{equation*}\tag{1.1}\psi_n(c) = \sum^n_1 \int_{|x| > c}x^2 dV _i(x).\end{equation*} According to the well-known Lindeberg-Feller Theorem [1] the condition $s_n^{-2}\psi_n(\xi s_n) \rightarrow 0 \text{as} n \rightarrow \infty\quad\text{for all} \xi > 0$ is both necessary and sufficient in order that $P\lbrack(X_1 + \cdots + X_n)s_n^{-1} \leqq x\rbrack \rightarrow \Phi(x)$ uniformly in $x$ as $n \rightarrow \infty$ and that $\max_{1 \leqq j \leqq n}\sigma_js_n^{-1} \rightarrow 0 \text{as} n \rightarrow \infty.$ Using the method of [3] and [4], it is shown that there exists an absolute constant $K$, independent of $n$ and of the particular sequence $V_1, V_2, \cdots$ such that \begin{equation*}\tag{1.2}\sup_{-\infty<x<\infty}|P\lbrack(X_1 + \cdots + X_n)s_n^{-1} \leqq x\rbrack - \Phi(x)| \leqq Ks_n^{-3} \int^{s_n}_0 \psi_n(u) du.\end{equation*} Some corollaries are deduced and the accuracy of this bound is investigated. Using a truncation scheme, an absolute upper bound is also derived for $\sup_{-\infty<x<\infty}|P\lbrack(X_1 + \cdots + X_n)B_n^{-1} \leqq x\rbrack - \Phi(x)|$, where the assumption of finite variances is now dropped and $B_n$ is a norming constant defined in (4.1).

Let $f(\mathbf{x} \mid P_1)$ be the $\operatorname{pdf}$ of a $(k - 1)$-dimensional normal distribution with zero means, unit variances, and correlation matrix $P_1$. Consider the integral, for $\delta > 0$, \begin{equation*}\tag{1}\int^\infty_{-\delta} \cdots \int^\infty_{-\delta} f(\mathbf{x} \mid P_1)dx \cdots dx_{k-1} = \alpha(\delta), \text{say}.\end{equation*} Assume that no element of $P_1$ is a function of $\delta$. Note that $\alpha(\delta)$ is an increasing function of $\delta$ and $\alpha(\delta) \rightarrow 1$ as $\delta \rightarrow \infty$. The problem is to obtain an approximation to $\delta$, for a large specified value, $\alpha$, of $\alpha(\delta)$. This is given by the theorem of Section 1. This result is used to obtain approximations to the sample size in a selection procedure of Bechhofer and in a problem of selection from a multivariate normal population. The closeness of the approximation is illustrated for the procedure of Bechhofer (Table 1).

This paper extends and unifies the use of Galois or finite fields in mixed factorials utilizing concepts of ring and ideal theory. A method of combining elements from $k$ distinct prime fields is presented and its use indicated in formulating properties of treatment combinations and effects in a mixed factorial.

The theory of sequential analysis was originally developed by Wald [12] in the context of testing a simple hypothesis against a specific alternative. Wald and Stein [11] subsequently considered the problem of finding a sequential confidence interval of prescribed width $(w)$ and confidence coefficient $(1 - \alpha)$ for the mean $(m)$ of a normal distribution with known variance $(\sigma^2)$. Their basic result, to the effect that no sequential procedure existed for this problem which had an average sample size less than the number of measurements required by the classical single-sample procedure, implies a substantial limitation as to what sequential interval estimation for the mean can accomplish in the case of a normal distribution. However, as we hope to show in this paper, by suitably redefining the problem, we can find sequential procedures either for obtaining a confidence interval for the difference in the means of two normal populations or for obtaining simultaneous confidence intervals for the means of $k$ normal populations, which promise to be useful in some applications. The basic idea in the reformulation of the problem is that the requirement to be put on the width of the confidence interval should depend on the location of the confidence limits, either with respect to some standard value or with respect to the confidence limits for the means of other populations. When this reformulation is appropriate to the problem at hand, Tables I and II indicate that a substantial saving is possible with the sequential procedure developed here at the risk of a relatively small (about 15 or 20%) increase in the average sample size if the least favorable parameter configuration should occur. In the next section we will find a sequence of random intervals $(J_n)$ such that $P\lbrack m \varepsilon J_n$ for all $n, n_0 \leqq n \leqq T\rbrack \geqq 1 - \alpha$. When $T = \infty$, sequences of this type, which might be called "confidence sequences" seem to have been first introduced into statistics by Wald in Chapter 10 of [12]. Such sequences were used by the present writer [7], [8], [9] as an important tool in finding sequential solutions to problems involving the selection of one of a finite number of possible decisions. Recently Darling and Robbins [2], [3], [4], [5] Robbins and Siegmund [10] have introduced and studied the properties of some new types of confidence sequences. Their work will be commented on briefly at the end of the paper.

Let $X, X_1, X_2, \cdots$ be independent random variables with $E(X_n) = 0 (n \geqq 1)$, and put $S_n = X_1 + \cdots + X_n (n \geqq 1)$. Marcinkiewicz and Zygmund [5] and Wiener [8] have shown that if the $X$'s have a common distribution, then \begin{equation*}\tag{1}E\{\sup_n|S_n/n|\} < \infty\end{equation*} provided that \begin{equation*}\tag{2}E\{|X|U (|X|)\} < \infty,\end{equation*} where we have put $U(x) = \max (1, \log x) (U_2(x) = U(U(x))$, etc.). Burkholder [2] has extended this result by showing that (1), (2), and \begin{equation*}\tag{3}E\{\sup_n|X_n/n|\} < \infty,\end{equation*} are equivalent. More recently, motivated by certain optimal stopping problems Teicher [7] and Bickel [1] under various assumptions on the distributions of $X_1, X_2, \cdots$ have shown that \begin{equation*}\tag{4}E\{\sup_n c_n|S_n|^\alpha\} < \infty\end{equation*} for certain sequences $(c_n)$ and positive constants $\alpha$. The interesting special case \begin{equation*}\tag{5}c_n = (nU_2(n))^{-\alpha/2}\end{equation*} is not covered by the results of these authors. This note gives a method which seems suitable for proving statements like (4) in a variety of cases. The method involves modifications of standard techniques used in the study of the law of the iterated logarithm. In particular, for each $\alpha = 1, 2, \cdots$ we are able to establish necessary and sufficient conditions for (4) when the $X$'s are identically distributed and the sequence $(c_n)$ satisfies (5). In Section 2 we state and prove one such theorem. Section 3 is devoted to explaining in somewhat more detail the scope of our results and their relation to the previously mentioned literature.

A phase free estimate of the coherence of a bivariate Gaussian process is presented. The technique is based on the usual independent, complex normal approximation to the distribution of the finite Fourier transform of a multivariate stationary time series, and the complex Wishart approximation to the distribution of spectrum estimates. If the spectral densities and coherence can be assumed to be constant over a wider frequency band than the phase can be assumed to be constant, the concept of inner and outer spectral windows would seem appropriate. Maximum likelihood estimates of the coherence are obtained using phase free marginal distributions at the inner window level. The results of simulations are presented showing the likelihood for various inner windows.

The results described here have their roots in two areas, for in a certain sense we combine on the one hand the work of Girshick, Mosteller and Savage [5] and Wolfowitz [11] and [12] on sequential estimation of the binomial parameter, and on the other the result of Hoeffding [7] concerning the consistency of $U$-statistics. The link between the two is the Blackwell [2] procedure for obtaining another (better) estimator from a given one by taking expectations conditional on a sufficient statistic. The main result is that if from a given estimator $T$ of $\theta = ET$ we construct new estimators by the Blackwell procedure corresponding to a sequence of stopping-rules $N_i$, then this sequence of estimators is consistent provided $N_i$ tends to infinity in probability; in fact it has also to be assumed that the $N_i$ have a certain structural property.

Suppose that a fixed total number $N$ of observations are to be made in a population $Pi$ which is composed of two strata $\Pi_1$ and $\Pi_2$. For $i = 1, 2$, it is assumed that each observation in the stratum $\Pi_i$ has a normal distribution with unknown mean $\theta_i$ and specified precision $r_i (r_i > 0)$. It should be kept in mind that the precision of any normal distribution is the reciprocal of the variance. Let $p$ denote the unknown proportion of the total population $\Pi$ which is included in the stratum $\Pi_1$. Then the mean $\bar\theta$ of the population $\Pi$ is given by the equation $\bar{\theta} = p\theta_1 + q\theta_2$, where $p + q = 1$. In this paper, the problem of estimating the value of $\bar{\theta}$ will be studied. It will be assumed that the loss which results from any estimate $\delta$ is the squared error $(\delta - \bar{\theta})^2$. It is well known that for any prior distribution of $\theta_1, \theta_2$, and $p$, the Bayes estimate of $\bar{\theta}$, after all of the observations have been taken, will be the mean of the posterior distribution of $\bar{\theta}$. Furthermore, the expected loss from this estimate will be the variance of the posterior distribution of $\bar{\theta}$. Therefore, we must find a sampling procedure for which the expected value with respect to the prior distribution of this posterior variance will be minimized. Throughout this paper, it will be assumed that the joint prior distribution of $\theta_1, \theta_2$, and $p$ is as follows: $\theta_1, \theta_2$, and $p$ are independent; the distribution of $p$ is a beta distribution with parameters $\alpha$ and $\beta(\alpha > 0, \beta > 0)$; and for $i = 1, 2$, the distribution of $\theta_i$ is a normal distribution with mean $\mu_i$ and precision $h_i.$ This joint distribution has the following fundamental property: After any number of observations have been taken from $\Pi, \Pi_1,$ or $\Pi_2$, the posterior joint distribution of $\theta_1, \theta_1$, and $p$ will again be of the same form and, in particular, $\theta_1, \theta_1$, and $p$ will again be independent under their posterior distribution. We assume that sampling will be carried out in two stages. At the first stage, a random sample of size $m (0 \leqq m \leqq N)$ will be taken from the whole population $\Pi$. At the second stage, the remaining observations $N - m$ are to be allocated between the two strata $\Pi_1$ and $\Pi_2$. Hence, at the second stage, $n_i$ observations are taken from the stratum $\Pi_i$, where $n_1 + n_2 = N - m$. The problem is to find an optimal choice of the design constants $m, n_1,$ and $n_2$. Note that the value of $m$ must be chosen in advance of any sampling, whereas the constants $n_1$ and $n_2$ need not be chosen until the values of the first $m$ observations obtained from the whole population have been studied. In this paper, we shall develop effective approximations to the optimal sampling procedure for situations in which the total number $N$ of available observations is large and, therefore, the optimal number $m$ of observations which should be obtained at the first stage will also be large. The techniques which will be presented can be extended for studying populations which are composed of $k$ strata $(k \geqq 2)$, in each of which the observations have a normal distribution. However, although the theory can be extended without difficulty, the actual computations become somewhat more complex, and we shall not consider these extensions. The optimal allocation of observations from the Bayesian point of view has also been studied by Ericson [2], [3] and Draper and Guttman [1]. Ericson [2] studied a related optimal one-stage stratified sampling scheme in which the proportion in each stratum is known. Draper and Guttman considered the optimal allocation at the second stage of a two-stage process, extending [2]. Ericson [3] investigated an optimal two-stage design different from ours in a nonresponse context. Here we shall study the basic problem of finding the optimal choice of $m$ at the first stage.

It is shown that maximum likelihood estimators of parameters of a linear model with autoregressive disturbances have an asymptotic multivariate normal distribution with mean vector equal to the true parameter values. Inspection of the variance matrix shows that the estimators are asymptotically efficient and that the estimates of coefficients of the independent variables have the same variance matrix as the best unbiased estimates for a modified model in which the autocorrelation parameter is known. It is conjectured that the asymptotic distribution of the estimates of coefficients of independent variables may be a useful approximation for moderate sized samples. Alternative approximations for the estimates of the autoregression coefficient and the variance are suggested for further study.*

Certain tests of constant failure rate versus failure rate increasing on the average are unbiased when complete samples of observations are available, as pointed out by Bickel and Doksum in the Annals of Mathematical Statistics (1968). In the present note, unbiasedness is proved when incomplete samples of failure data are available. A similar result is obtained for monotone tests of constant versus increasing failure rate. Finally, a table of percentiles is given to facilitate application of the total time on test statistic for testing constant failure rate versus failure rate increasing on the average.

The purpose of the present investigation is to develop a class of rank order tests for the equality of treatment-effects in the presence of a set of concomitant variates for the analysis of covariance (ANOCA) model relating to completely randomized layouts. The proposed procedures are shown to be conditionally distribution-free and to have some desirable large sample properties.

This paper is concerned with single-sample multiple-decision procedures based on the ranks of the observations for selecting from $c$ continuous populations (a) the "best $t$" populations without regard to order, (b) the "best $t$" populations with regard to order, and (c) a subset which contains all populations "as good or better than a standard one." The "bestness" of a population is characterised by its location parameter; the best population being the one having the largest location parameter; the second best being the one having the second largest location parameter, etc. Large-sample methods are provided for computing the sample sizes necessary to guarantee a preassigned probability of correct grouping (or ranking) under specified conditions on location parameters. It is shown that the asymptotic efficiency of these procedures relative to the normal theory procedures (see, for example, Bechhofer [1] and Gupta and Sobel [3]) is the same as that of the associated tests in one-way analysis of variance model I problem. If the ratio of the sample sizes is equal to this efficiency, the two procedures being compared are shown to have the same asymptotic performance characteristic. Finally, in the case of problem (c) two alternative rank-score procedures are proposed which are asymptotically equi-efficient.

Let $x_1, x_2, \cdots$ be independent random variables with mean 0 and variance 1. Let $s_n = x_1 + \cdots + x_n, n \geqq 1$, and for each $c > 0$ define \begin{align*}\tau_1 &= \tau_1(c) = \text{first} n \geqq 1 \text{for which} s_n > cn^{\frac{1}{2}} \\ &= \infty \text{if} s_n \leqq cn^{\frac{1}{2}} \text{for all} n, \\ \end{align*} and \begin{align*}\tau_2 = \tau_2(c) &= \text{first} n \geqq 1 \text{for which} |s_n| > cn^{\frac{1}{2}} \\ &= \infty \text{if} |s_n| \leqq cn^{\frac{1}{2}} \text{for all} n. \\ \end{align*} The stopping times $\tau_1$ and $\tau_2$ have received considerable attention recently (see, for example, [1], [2], [3], [6], and [8]), and naturally there has arisen the question as to whether $P\{\tau_k < \infty\} = 1, k = 1, 2$. One contribution of this note is (1) THEOREM. If $s_n/n^{\frac{1}{2}}$ does not tend in probability to 0, then for each $c > 0, P\{\tau_2 < \infty\} = 1$. We show by examples that (1) is no longer true with $\tau_2$ replaced by $\tau_1$. The final section contains two remarks bearing on the converse to (1).

Nonparametric analysis of variance tests for factorial experiments, available in the literature, relate mostly to main effects in two-way layout linear models without interactions. Although the approach of Lehmann (1964) (see also Puri and Sen (1967)) can be adapted to construct tests for interactions, such tests are rather tedious to apply and are only asymptotically distribution-free. To obviate these drawbacks, the theory of permutationally distribution-free rank order tests for main effects, developed independently by the two authors, Sen (1968), Mehra (1968) (see also Mehra and Sarangi (1967)), is further extended in this paper to provide suitable tests for interactions in factorial experiments. The asymptotic power-efficiencies of the proposed tests are also studied. [The results of this paper were obtained independently by the two authors using, respectively, extensions of the Hajek (1962) and the Chernoff-Savage (1958) theorems on the asymptotic normality of rank order statistics.]

Joshi (1965), (1966) studied in great detail admissible estimation, in relation to survey-sampling. He (1966) also established a property more demanding than admissibility namely uniform admissibility (previously called global admissibility by Godambe (1966)) for the conventional sample mean while estimating the population total. In this paper we establish uniform admissibility of a class of Bayes estimators. Using the notation similar to that of Godambe and Joshi (1965) we denote the population units by integers $1, 2, \cdots, N$. Any subset $s$ of the integers $1,\cdots, N$ is called a sample. If $S$ denotes the set of all possible samples, $(s\varepsilon S)$, any real function $p$ on $S$ such that $\sum_s p(s) = 1$ and $1 \geqq p(s) \geqq 0$, for all $s \varepsilon S$ is called a sampling design. Next we denote by $x_i$ the real value associated with the unit $i (i = 1, \cdots, N)$ of the population. $\mathbf{x} = (x_1, \cdots, x_i, \cdots, x_N)$ is a vector in the $N$-dimensional Euclidean space $R_N$. Any real function $e(\mathbf{x}, s)$ on the product space $R_N \mathbf{\times} S$, such that $e$ depends on $\mathbf{x}$ only through those $x_i$ for which $i \varepsilon s$, is called an estimator. Since in this paper we would be concerned with estimation of the population total $T(\mathbf{x}) = \sum^N_1 x_i$, the terms such as estimator, admissibility, uniform admissibility etc. used subsequently are to be understood in relation to estimation of $T$. Now to distinguish `admissibility' from `uniform admissibility' we introduce the following four definitions. DEFINITION 1.1. For a given sampling design $p$, an estimator $e'$ is said to be superior to the estimator $e$ if for all $\mathbf{x} \varepsilon R_N$, $\sum_s p(s)\lbrack e'(s, \mathbf{x}) - T(\mathbf{x})\rbrack^2 \leqq \sum_s p(s)\lbrack e(s, \mathbf{x}) - T(\mathbf{x})\rbrack^2$ strict inequality being true for at least one $\mathbf{x}$. DEFINITION 1.2. For a given sampling design $p$, an estimator $e$ is said to be admissible if no estimator $e'$ is superior (Definition 1.1) to $e$. DEFINITION 1.3. A pair $(e', p')$ of an estimator $e'$ and a sampling design $p'$ is said to be uniformly superior to another pair $(e, p)$ if for all $\mathbf{x} \varepsilon R_N$, $\sum_s p'(s)\lbrack e'(s, \mathbf{x}) - T(\mathbf{x})\rbrack^2 \leqq \sum_s p(s)\lbrack e(s, \mathbf{x}) - T(\mathbf{x})\rbrack^2$ strict inequality holding for at least one $\mathbf{x}$. DEFINITION 1.4. With respect to a class $C$ of sampling designs, a pair $(e, p)$ of an estimator $e$ and a sampling design $p$ is said to be uniformly admissible if no other pair $(e', p')$ such that $p' \varepsilon C$, is uniformly superior to $(e, p)$, (Definition 1.3). For the discussion of the practical significance of the notion of uniform admissibility, especially if in Definition 1.4, the class $C = C_n$, where \begin{equation*}\tag{1.1}C_n = \{p:\sum_s p(s) \cdot n(s) = \operatorname{const.} = n\}\end{equation*} $n(s)$ being the number of units $i$ such that $i\varepsilon s$, we refer to Joshi ((1966), Section 7). Obviously $C_n$ above is the class of all sampling designs having a fixed `average sample size.' The main result of this paper is the following THEOREM 1.1 With respect to the class $C_n$ in (1.1) of sampling designs the pair $(e^\ast, p^\ast)$, where $e^\ast$ is the estimator given by, \begin{equation*}\tag{1.2}e^\ast(s, \mathbf{x}) = \sum_{i\varepsilon s} x_i + \sum_{i\not\varepsilon s} \lambda_i,\end{equation*} $\lambda_1, \cdots, \lambda_i, \cdots, \lambda_N$ being any arbitrarily fixed numbers and $p^\ast$ is any sampling design belonging to the class $C_n$ in (1.1), is uniformly admissible. (Definition 1.4).

Bounds on the distribution of the maximum of the absolute values of the "partial sums" constructed from a multi-dimensional array of independent random variables are derived and used to establish the weak convergence of certain random functions defined in terms of such "partial sums".

Let $\xi(t), t \geqq 0, \xi(0) = 0$ be a separable stochastic process with stationary independent increments whose sample functions are continuous on the right. Write $\bar{\xi}(t) = \sup_{0\leqq s\leqq t} \xi(s)$ and $T_t = \min \lbrack u:0 \leqq u \leqq t; \xi(u) = \bar{\xi}(t)\rbrack.$ The object of this paper is to establish the following theorem: THEOREM. The limiting distribution \begin{equation*}\tag{1}\lim_{t\rightarrow\infty} \operatorname{Pr}(t^{-1} T_t < x) = F(x)\end{equation*} exists if and only if \begin{equation*}\tag{2}\lim_{t\rightarrow\infty} t^{-1} \int^t_0 \operatorname{Pr}(\xi(u) > 0) du = \alpha,\quad 0 \leqq \alpha \leqq 1,\end{equation*} and then $F(x)$ is related to $\alpha$ by \begin{equation*}\begin{align*}\tag{3}F(x) &= F_\alpha(x) = (\sin \pi\alpha) \pi^{-1} \int^x_0 \nu^{-(1-\alpha)} (1 - \nu)^{-\alpha} dv, 0 < \alpha < 1, 0 \leqq x \leqq 1; \\ F_0(x) &= 0 \text{if} x < 0, 1 \text{if} x \geqq 0; \\ F_1(x) &= 0 \text{if} x < 1, 1 \text{if} x \geqq 1\end{align*} .\end{equation*} This theorem is the exact counterpart to a theorem of Spitzer ([4], Theorem 7.1) for sums of independent and identically distributed random variables. It contains as a special case the well-known arc-sine limit theorem for the Brownian motion process. An earlier version of the theorem was obtained by Heyde [3] under the additional condition $\int^1_0 t^{-1} \operatorname{Pr}(\xi(t) > 0) dt < \infty$, the violation of which leads to $\operatorname{Pr}(\bar{\xi}(t) = 0) = 0, t > 0$. It should be remarked that $T_t$ has the same distribution as $N_t = \mu\{u:0 \leqq u \leqq t; \xi(u) > 0\}, \mu$ denoting ordinary Lebesgue measure. This follows from the well-known corresponding result of Sparre-Andersen for sums of independent and identically distributed random variables by a straightforward limiting argument.

A general method for determining stopping rules to obtain a fixed-width confidence interval of prescribed coverage probability for an unknown parameter of a distribution is obtained. Asymptotic theory in the sense of Chow and Robbins [4] is discussed. The sequential procedure obtained is asymptotically consistent and efficient in the sense of Chow and Robbins [4].