Sections 1-5 are concerned with finding lower bounds for the expected sample sizes of sequential multihypothesis tests in the presence of a constraining error matrix. We consider $K$ simple hypotheses corresponding to $K$ density functions $f_i, i = 1, \cdots, K,$ and fix all of the entries of the $K \times K$ error matrix $A = (\alpha_{ij})$, where $\alpha_{ij} = P$ [accepting $f_j \mid f_i$ true]. Lower bounds are found for $E(N \mid f)$, first, when $f$ is one of the $K$ densities, and then, for a $K + 1$st density $f_0$. In Section 6, lower bounds are found when the error constraints arising from the error matrix are relaxed and/or modified. Section 7 finds lower bounds for average sample size when the test is not constrained by an error matrix but rather by a lower bound for the probability of a "correct decision" as a function of the true state of nature. The reader will find that many of the results of this paper extend immediately to a decision theory context with a finite number (not necessarily $K$) of actions or terminal decisions, with $\alpha_{ij}$ denoting the probability of the $j$th action given density $f_i$.

This paper examines a sequential testing procedure for choosing one of three simple hypotheses concerning the unknown mean $\mu$ of the normal distribution when the variance is known. The test is conducted by plotting $S_n$, the sum of the observations, versus $n$, the current sample size, until the point $(n, S_n)$ is contained within one of three triangular regions. When this occurs, sampling is terminated and the region containing $(n, S_n)$ determines which state of nature is accepted. Although we shall formally view the problem as one with only three states of nature $(\mu = \mu_1, \mu_2$ or $\mu_3)$, we shall proceed with the usual understanding that the performance of the test procedure should be evaluated for a wider class of states $(- \infty < \mu < \infty)$. The test is approximated by a corresponding exact test for the Wiener process. Formulas are derived which approximate the operating characteristics (OC) and the average sample size (ASN) for all values of $\mu$. The ASN function is compared with theoretical lower bounds. The testing procedure is compared with a modification of a three hypothesis testing procedure proposed by Sobel and Wald [5].

Let $X(t) \sim N(\mu t, \sigma^2t)$ be a Wiener process for $t \geqq 0$. Suppose $\sigma^2$ is known and that $H_1, \cdots, H_K$ are $K$ hypotheses concerning the unknown drift parameter $\mu$. A general sequential testing procedure presents itself in the following form: Continue to observe $X(t)$ until, for the first time $s$, the point $(s, X(s)) \varepsilon B$, a subset of the right half plane. $B$ will be called the boundary. Let $B$ be partitioned into $K$ disjoints subsets $B_i (i = 1, \cdots, K)$ called boundary sets. Then, if $(s, X(s)) \varepsilon B_i$, one accepts $H_i$. In this paper, we will assume that each boundary set $B_i$ is composed of a finite number of straight lines or line segments called boundary lines. Any test using such a procedure (for some $K$) will be called a boundary test. We shall derive recursive methods for computing the exact OC functions, the average sample time (AST), and, in fact, all the moments of the sampling time $\tau$. Even though the study of tests for Wiener processes seems to be of intrinsic interest, their study usually has been motivated by other considerations. T. W. Anderson [1] has developed an approximate test for the unknown mean of the normal distribution by relating his testing procedure with a corresponding boundary test. In a similar manner, one may derive Wald's sequential probability ratio test, the Neyman-Pearson fixed sample size test, and multistage tests, for the mean of the normal distribution. The restriction to a finite number of (straight) boundary lines may seem unpleasant but, of course, curvilinear boundaries can be approximated with polygonal lines. One consequence of the restriction is that there will exist at most a finite number of times at which an end-point of a boundary line, an intersection of two boundary lines, or a vertical boundary line occurs. These times, coupled with times $t = 0$ and $t = \infty$, will be called critical times. The recursive methods developed below will allow us to reduce the computations of the OC functions and the moments of $\tau$ to the computation of certain "fundamental probabilities" involving events which occur between critical times. We shall denote the critical times as $0 = t_0 < t_1 < \cdots < t_m < t_{m+1} = \infty.$ The points $(s, x)$, with $s \geqq 0$, which are not in the boundary $B$ will be denoted by $B_0$. A point in $B_0$ which can be reached by a continuous sample path before time $\tau$ will be called an accessible point. Otherwise, the point is an inaccessible point. An important subclass of the class of boundary tests is the class of boundary tests which are closed for all values of $\mu$. This class has three characterizations (See [3].): \begin{align*}\tag{i}P_\mu\lbrack\tau < \infty\rbrack = 1 \text{for all} \mu. \\ \tag{ii}E_\mu(\tau^r) < \infty \text{for all} \mu, \text{for} r = 1, 2, \cdots.\end{align*} (iii) Every accessible point of the form $(t_m, x)$ is contained between two parallel, non-vertical boundary lines extending to infinity. Sections two and three will present the recursive procedures for computing the OC functions and the moments of the stopping time $\tau$, respectively. Section four discusses the computation of the "fundamental probabilities." The reader will quickly perceive that the recursive methods presented below need not be limited to straight line boundaries. The limitation arises from the rather restricted class of computable fundamental probabilities (at the present time).

Lindley [6] studies the topic in our title. By using Fisher's conditional-Poisson approach to the multinomial and the logarithmic transformation of gamma variables to normality, he showed that linear contrasts in the logarithms of the cell probabilities $\theta_i$ are asymptotically jointly normal and suggested that the approximation can be improved by applying a "correction" to the sample. By studying the asymptotic series for the joint distribution in Section 2 an improved correction procedure is found below. A more detailed expansion is given in Section 3 for the distribution of a single contrast in the $\log \theta_i$. In many problems a linear function of the $\theta_i$ is of interest. The exact distribution is obtained and is of a form familiar in the theory of serial correlation coefficients. A beta approximation is given. For three cells, a numerical example is given to show the merit of this approximation. A genetic linkage example is considered which requires the joint distribution of two linear functions of the $\theta_i$. The exact joint distribution is found but is too involved for practical use. A normal approximation leads to Lindley's results [7].

A special group structure common to many statistical problems is presented. In the context of equivariant estimation, this leads to a characterization of equivariant estimators (Section 3). Under further conditions, measurable equivariant estimators are characterized (Section 4). Some examples are presented in Section 5 and Section 6 contains a brief discussion of related work in which the special group structure is useful.

Let $Y_1, \cdots, Y_n$ be an ordered sample from a density with mode $\theta$. We propose to estimate $\theta$ by suitable points in the interval formed by the first and the last of those $s$ consecutive $Y_i$'s which are closest together. Choices of $s$ which yield consistency of these estimates, the speed of convergence and asymptotic distributions are discussed in this paper.

The familiar application of the normal linear model involves a response variable that is assumed normally distributed with constant variance and with location linear in a vector parameter. In other applications a response variable may occur in a form that suppresses an underlying normal linear structure. Sometimes in these applications the context may suggest a logarithmic or square root transformation which reveals the normal linear form. Box and Cox (1964) consider a parametric class of transformations on the response variable and derive a method for estimating the class parameter based on Bayesian and likelihood techniques. In this paper a more comprehensive statistical model is proposed; it is a revision of the structural model (Fraser 1966). It gives stronger inference statements in the context for the linear model. It can handle normal or nonnormal error. And in the context for the transformed linear model it gives directly a method for estimating the class parameter. This estimation avoids an approximation in the Bayesian prior and it includes additional sensitivity to the data.

It is customary to consider that two random variables $S$ and $T$ are associated if $\operatorname{Cov}\lbrack S, T\rbrack = EST - ES\cdot ET$ is nonnegative. If $\operatorname{Cov}\lbrack f(S), g(T)\rbrack \geqq 0$ for all pairs of nondecreasing functions $f, g$, then $S$ and $T$ may be considered more strongly associated. Finally, if $\operatorname{Cov}\lbrack f(S, T), g(S, T)\rbrack \geqq 0$ for all pairs of functions $f, g$ which are nondecreasing in each argument, then $S$ and $T$ may be considered still more strongly associated. The strongest of these three criteria has a natural multivariate generalization which serves as a useful definition of association: DEFINITION 1.1. We say random variables $T_1,\cdots, T_n$ are associated if \begin{equation*}\tag{1.1}\operatorname{Cov}\lbrack f(\mathbf{T}), g(\mathbf{T})\rbrack \geqq 0\end{equation*} for all nondecreasing functions $f$ and $g$ for which $Ef(\mathbf{T}), Eg(\mathbf{T}), Ef(\mathbf{T})g(\mathbf{T})$ exist. (Throughout, we use $\mathbf{T}$ for $(T_1,\cdots, T_n)$; also, without further explicit mention we consider only test functions $f, g$ for which $\operatorname{Cov}\lbrack f(\mathbf{T}), g(\mathbf{T})\rbrack$ exists.) In Section 2 we develop the fundamental properties of association: Association of random variables is preserved under (a) taking subsets, (b) forming unions of independent sets, (c) forming sets of nondecreasing functions, (d) taking limits in distribution. In Section 3 we develop some simpler criteria for association. We show that to establish association it suffices to take in (1.1) nondecreasing test functions $f$ and $g$ which are either (a) binary or (b) bounded and continuous. In Section 4 we develop the special properties of association that hold in the case of binary random variables, i.e., random variables that take only the values 0 or 1. These properties turn out to be quite useful in applications. We also discuss association in the bivariate case. We relate our concept of association in this case to several discussed by Lehmann (1966). Finally, in Section 5 applications in probability and statistics are presented yielding results by Robbins (1954), Marshall-Olkin (1966), and Kimball (1951). An application in reliability which motivated our original interest in association will be presented in a forthcoming paper.

Let $X_{ij} (j = 1, 2,\cdots, n_i; i = 1, 2,\cdots, k)$ be $k$ independent samples of sizes $n_1, n_2,\cdots, n_k$ respectively from $k$ populations with continuous cumulative distribution functions $F_1, F_2,\cdots, F_k$ respectively. We assume that the $F_i$'s belong to a family $\mathscr{F}$ of distribution functions indexed by a parameter $\theta$. Let all the $N = \sum^k_{i=1} n_i$ observations be put together and ordered to form a single sequence and suppose that only the first $r$ ordered observations are available. That is, let us have a combined (right) censored sample of total size $r$. Such a censored sample occurs naturally in many physical situations as for example, in problems of life testing where we are interested in comparing the mean life of several physical systems, or in clinical trials or bio-assay problems where we want to compare the efficacy of several drugs but we cannot afford to wait indefinitely to get information on all the sampling units put on test. For details see Basu [2]. For facility of discussion, we shall use the terminology of life testing. Any test based on the first $r$ ordered observations (out of a combined sample of size $N$) will be termed an $r$ out of $N$ test. In this paper we propose two $k$-sample $r$ out of $N$ rank tests which generalize the rank tests proposed by Kruskal [7], Jonckhere [6] and Terpstra [10], [11]. In the first part of the paper we propose the statistic $B_r^{(N)}$ (large values being critical) to test the null hypothesis \begin{equation*}\tag{1.1}H_0:F_1(x) = F_2(x) = \cdots = F_k(x)\end{equation*} (or equivalently, $H_0:\theta_1 = \theta_2 = \cdots = \theta_k = 0$ say, under location alternatives) against the alternative hypothesis \begin{equation*}\tag{1.2}H_1:F_i(x) = F(x, \theta_i) \quad (i = 1, 2, \cdots, k)\end{equation*} In Section 2 we define the statistic $B_r^{(N)}$ and show its relationship with other statistics. The mean and variance of $B_r^{(N)}$ under the null hypothesis is derived in Section 3. In Section 4 we find the asymptotic distribution of $B_r^{(N)}$ both under the null and the non-null case. The computation of $B_r^{(N)}$ has been illustrated by an example in Section 5. In the second part of the paper we consider the statistic $V(N, r)$ (to be defined later) for testing the hypothesis (1.1) against the ordered alternative \begin{equation*}\tag{1.3}H_2:F_1(x) < F_2(x) < \cdots < F_k(x)\end{equation*} for all $x$, where the $F_i(x)$ are labeled in such a way that (1.3) is the ordered alternative being considered. The second formulation is useful in life testing where the $k$ cdf's might be associated with $k$ different processes and the experimenter wishes to test whether the $k$ processes give rise to units with the same life-time distributions against the alternative that the processes can be ordered in a particular manner, in the sense that the $k$ life-time distributions can be ordered uniformly with respect to $x$, (that is, according to their reliabilities). In Section 6 we define the statistic $V(N, r)$ and in Section 7 we have derived the mean and variance of $V(N, r)$ under $H_0$. Section 8 is devoted to investigate the extreme values of $V(N, r)$ and the asymptotic normality of $V(N, r)$ is proved in Section 9. Both the statistics $B_r^{(N)}$ and $V)N, r)$ may be considered as $k$-sample extensions of the $V_r$ statistic proposed by Sobel [9]. The definitions of $B_r^{(N)}$ and $V(N, r)$ are slightly different from what is given earlier in Basu [2].

Tables are given for the asymptotic distribution function of the extremal quotient, which is valid for a wide class of initial distributions. Selected values of the shape parameter $\lambda$ are considered. This parameter depends upon the initial distribution, and the size of the sample from which the quotient is drawn. This distribution function is not too different from that of a logarithmic normal distribution. The large sample approach of the distribution function to the logistic one is very slow. A Monte Carlo study shows an unexpectedly good fit to the theory.

The present paper gives sufficient conditions on a linear process $\{X_t\}$ and its spectral density $f(\cdot)$ for the following limit relation to hold: \begin{equation*}\tag{0.1}\parallel W \parallel^{-1}_2(N/2m_N \log m_N)^{\frac{1}{2}} \max_{-\pi\leqq\lambda\leqq\pi} \lbrack |f_N (\lambda) - f(\lambda)| / f(\lambda)\rbrack \rightarrow 1\end{equation*} in probability as $N \rightarrow \infty$ where $f_N(\cdot)$ is the usual windowed sample spectral density, $m_NW(m_N\cdot)$ is the (varying) window, and $m_N \uparrow \infty$ as $N \rightarrow \infty$ at a suitable rate. Under the same conditions it is shown that \begin{align}\tag{0.2}P(a_N^{-1}\lbrack N^{\frac{1}{2}}m_N^{-\frac{1}{2}} \parallel W\parallel^{-1}_2 \max_{|i|\leqq m_N} \lbrack |f_N (\lambda^\ast_{N,i}) - f(\lambda^\ast_{N,i})|/f(\lambda^\ast_{N,i})\rbrack \\ - b_N\rbrack \leqq x) \rightarrow \exp (-\exp(-x))\end{align} as $N \rightarrow \infty \text{for} - \infty < x < \infty$ where $\lambda^\ast_{N,i}, a_N$, and $b_N$ are defined by (2.1) and (2.2). Observe that the difference between the maximum deviation and the deviation at a single $\lambda$ point [5] manifests itself in the factor $(\log m_N)^{-\frac{1}{2}}$. Thus in practice a confidence band for all $\lambda$ is $O((\log m_N)^{\frac{1}{2}})$ times that for a finite set.

Let $X_1, X_2, \cdots, X_n, \cdots$ be a sequence of independent, identically distributed random variables with common distribution function $F$. Let $Z_n = \max \{X_1, X_2, \cdots X_n\}$. Conditions for the stability and relative stability of such sequences with the various modes of convergence have been given by Geffroy [3], and Barndorff-Nielsen [1]. The principal result of this paper is Theorem 2.1, which is an analogue for maxima of the law of the iterated logarithm for sums (Loeve [6] pages 260-1). In Section 3, it is indicated that the theorem is satisfied by a wide class of distributions, and specific forms are given for the normal and exponential distributions.

It has been pointed out in the literature that the maximum likelihood (ML) estimator may be misleading in the presence of prior information. Many of these examples assume extreme sizes: one or infinity. In the present paper an example is considered where sample size may be any odd positive integer. This example is an amplification of the one given by Lehmann (1949) where he considers estimation of the probability of "success" $p$, based on a single observation of a Bernoulli random variable $X$. He states that with the prior information $\frac{1}{3} \leqq p \leqq \frac{2}{3}$ the ML estimator has uniformly larger expected squared error than any estimator $\delta(X)$ which is symmetric about $\frac{1}{2}$ and is such that $\frac{1}{3} \leqq \delta(0) \leqq \frac{1}{2} \leqq \delta(1) \leqq \frac{2}{3}$. In particular, the ML estimator is uniformly worse than the trivial estimator $\delta(X) \equiv \frac{1}{2}$. A natural question arises: does the same phenomenon occur for larger samples? In the following it has been shown that with $(2n + 1)$ observations if $p$ is known to be in a small interval around $\frac{1}{2}$ then the trivial estimator is uniformly better than the ML estimator [now] based on $(2n + 1)$ observations. The interval having this property shrinks as $n$ becomes large. The proof is based on a monotone convergence of certain binomial probabilities which itself may be of some interest.

The object of the present investigation is to generalize the results of Greenberg (1966) to a wider class of robust estimators which includes her estimator as a special case. As in Greenberg (1966), we consider an incomplete block design $D$ consisting of $J$ blocks of (constant) size $b$ to which $c(> b)$ treatments are applied, there being $n_j$ replications of the $j$th block, for $j = 1, \cdots, J$. Let $n = \sum^J_{j=1} n_j$ and let $S_j$ consist of the numbers of the $b$ treatments applied in the $j$th block, for $j = 1, \cdots , J$. The observable random variables are then \begin{align}\tag{1.1}X_{ij\alpha} = v + \xi_i + \mu_j + \beta_{j\alpha} + U_{ij\alpha}, \\ \alpha = 1, \cdots, n_j; \quad i \varepsilon S_j; \quad j = 1, \cdots, J,\end{align} where $\xi_i$ is the $i$th treatment-effect, $\mu_j$ the $j$th replication effect, $\beta_{j\alpha}$ the effect of the $\alpha$th block in the $j$th replication set and $U_{ij\alpha}$'s are independent and identically distributed residual error components with a common distribution $F(u)$. We may set (without any loss of generality) that \begin{equation*}\tag{1.2}\sum^c_{i=1} \xi_i = 0, \quad \sum^J_{j=1} \mu_j = 0; \quad \sum^{n_j}_{\alpha=1} \beta_{j\alpha} = 0 \quad\text{for all} j = 1, \cdots, J.\end{equation*} Our intention is to provide some robust estimators of contrasts among $\xi_i$'s and to study their various properties.