Let $X_1,\cdots, X_N$ be identically independently distributed with the common continuous distribution function $F$, and let $r_1 < r_2 < \cdots < r_s$ denote the ordered ranks of the positive $X$'s among $|X_1|,\cdots, |X_N|$. Two problems are considered. The first is the location problem where the null hypothesis $H_0$ that $F$ is symmetric about zero (or more generally $F(x) \geqq 1 - F(-x)$ for all $x$) is tested against the alternative that $X$ is stochastically larger than $-X$ (i.e. $F(x) \leqq 1 - F(-x)$ for all $x$ with strict inequality for some $x$). Departure from the null hypothesis will be measured in terms of the distribution difference $D(x) = \lbrack 1 - F(-x) \rbrack - F(x)$, i.e. the difference of the distributions of $-X$ and $X$. We let $\delta(F) = \sup_x |D(x)|$ denote the Kolmogorov distance between these distributions and consider the class $\Omega(\Delta) = \{F:D(x) \geqq 0$ and $\delta (F) \geqq \Delta\}$ of one-sided alternatives with this distance at least $\Delta$. Lower bounds on the power of monotone rank tests are given for $F$ in $\Omega(\Delta)$ (Theorem 2.1), and similarly, upper bounds on the power of monotone rank tests are found for $F$ in $\bar{\Omega}(\Delta) = \{F:\delta(F) \leqq \Delta\}$ (see Theorem 2.2, Corollary 2.2 and Corollary 2.3). $\Omega(\Delta)$ and $\bar{\Omega}(\Delta)$ are of interest for paired comparison and one-sample experiments in which the location of $F$ is the main concern. Note that the above location alternatives are not necessarily shift alternatives. The second problem considered is the symmetry problem where the hypothesis that $F$ is symmetric about zero is to be tested against the alternative that it is skewed to the right. Lower (Theorem 3.1) and upper (Corollary 3.2) bounds on the power of monotone rank tests are found for $F$ in $\Omega_s(\Delta) = \{F:F \in\Omega (\Delta)$ and $F$ has median $0\}$ and $\bar{\Omega}_s(\Delta) = \{ F:\delta(F) \leqq \Delta$ and $F$ has median $0\}$, respectively. Hoeffding (1951) and Ruist (1954) considered alternative classes of the type $\Gamma(q) = \{F:F$ is symmetric and $(F(0) - \frac{1}{2}) \leqq q\}$ and found that the Sign test is minimax (maximizes the minimum power, i.e., minimizes the maximum risk = (1-power)) for $\Gamma(q)$. For the location alternative, we consider the problem of maximizing the minimum power over $\Omega(\Delta)$ and find (Theorem 4.1) that for a class of statistics of the form $\sum^S_{i=1} J_N(r_i/(N + 1))$, a solution is asymptotically given by the statistics $T(\Delta) = W + \lbrack (\frac{1}{2} \Delta (N - 1)-1)/(N + 1) \rbrack S$, where $S$ is the number of positive $X$'s and where $W = \sum^s_{i=1} r_i/(N + 1)$ is the one-sample Wilcoxon statistic. This result contrasts with the two-sample result of [6] in which the Wilcoxon statistic is asymptotically the uniformly (in $\Delta$) unique minimax solution. For the symmetry problem, a class of statistics of the form $\sum^s_{i=1} J(r_i/(N + 1))$ is considered, and it is shown (Theorem 4.2) that $V = W - \frac{1}{2} S$ asymptotically maximizes the minimum power over $\Omega_s(\Delta)$. $T(\Delta)$ and $V$ are functions of the one-sample Wilcoxon statistic $W$ and the Sign statistic $S$. Such statistics have also been considered by Ruist (1954). See also Hodges and Lehmann (1962, page 495). $V$ has been considered by Gross (1966, page 76) and is asymptotically equivalent to the statistic considered by Gupta (1967). The power bounds are tabulated exactly or estimated using Monte Carlo methods for sample sizes 10 and 20 in Section 2B. In particular, the minimum power over $\Omega(\Delta)$ of the statistics $W, T^{(1)} = T(\Delta_1)$ and $T^{(2)} = T(\Delta_2)$, where $\Delta_1 = (1.645/(3N)^{\frac{1}{2}})^{\frac{1}{2}}$ and $\Delta_2 = (2.326/(3N)^{\frac{1}{2}})^{\frac{1}{2}}$, is given or estimated using Monte Carlo power methods. These choices of $\Delta$ are discussed in Section 4, Remark 4.3. The results show that for $N = 10, T^{(1)}$ and $T^{(2)}$ do not improve on the minimum power of the Wilcoxon statistic $W$, while for $\alpha = .05$ and $N = 20$, the asymptotic results are in effect in the sense that $T^{(1)}$ and $T^{(2)}$ are improvements on $W$. The statistic $V$ is similarly compared with the statistic [Gross (1966)] $T_G = \sum^S_{i=1} r^2_i/(N + 1)^2 - (\frac{1}{3})S$ and it is found (Section 3) that in terms of minimum power over $\Omega_s(\Delta), V$ is much better than $T_G$ already for sample sizes $N = 10$ and 20. Power bounds similar to the ones obtained in this paper have been obtained by Birnbaum (1953) and Chapman (1958) for the goodness-of-fit problem, by Bell, Moser and Thompson (1966) for the two-sample problem, and by Bell and Doksum (1967) for the independence problem. In Section 5, tables of the null distributions of $T^{(1)}, V$ and $T_G$ and given for $N \leqq 10$, and the Monte Carlo powers of $T^{(1)}, T^{(2)}$ and $W$ are compared for normal, double exponential and logistic shift alternatives for $N = 10$ and 20. For the double exponential distribution, $T^{(1)}$ and $T^{(2)}$ are slightly better than $W$ when $N = 20$. For the normal distribution, $W$ is slightly better than $T^{(1)}$ and $T^{(2)}$. Thus $T^{(1)}$ and $T^{(2)}$ appear to be better than $W$ for "heavy" tail shift alternatives, while the opposite holds for "light" tail shift alternatives. Although there is essentially no difference in the power of $T^{(1)}$ and $T^{(2)}$ for the various models considered, $T^{(1)}$ is recommended because the normal approximation to the rejection limits of $(N + 1)T^{(1)} = (N + 1)W + \lbrack .487(N - 1)N^{-\frac{1}{4}}- 1\rbrack S$ are closer to the true limits.

Suppose that an experimenter is interested in determining the relationship between a response $\eta$ and several "independent" variables, $x_1, x_2,\cdots, x_K$. The $x$-variables may be controlled by the experimenter or observed without control. Suppose, further, that these $K$ "independent" variables represent all the factors that contribute to the response $\eta$, and that the exact relationship between $\eta$ and the $x$'s is \begin{equation*}\tag{1.1}\eta = \phi(x),\end{equation*} where $\mathbf{x} = (x_1, x_2,\cdots, x_K)'$. The function $\phi(\mathbf{x})$ is called the response function and, geometrically, it defines a surface in $K$-space which is called the response surface. In the real world, however, we rarely know the exact relationship, or all the variables which affect that relationship. One way of proceeding then is to graduate, or approximate to, the true relationship by a polynomial function, linear in some unknown parameters to be estimated and of some selected order in the independent variables. Under the tentative assumption of the validity of this linear model (which we can justify on the basis of a Taylor expansion of $\phi$), we can perform experiments, fit the model using regression techniques, and then apply standard statistical procedures to determine whether this model appears adequate. Since in practice we do not know all of the $K$ factors which affect the response, we usually select $k(< K)$ variables which we believe might have significant effects. This selection may be made on the basis of prior knowledge, or a preliminary experiment may be performed to screen the important variables out of a larger set of possible variables. We can write our graduating model as follows: \begin{equation*}\tag{1.2}y_u = f(x_{1u}, x_{2u},\cdots, x_{ku}\mid\beta_0, \beta_1,\cdots, \beta_p) + \varepsilon_u,\end{equation*} where $y_u$ is the observed response at the $u$th trial, $x_{iu}$ is the setting of $i$th input variable for the $u$th trial, $\beta_j$ is the $j$th parameter to be estimated, $\varepsilon_u$ is the error involved in observing $y$ on the $u$th trial, $u = 1,2,\cdots, N; i = 1,2,\cdots, k < K$; and $j = 0, 1,2,\cdots, p$. The errors $\varepsilon_u$ arise in one or more of the following ways: (i) The true response $\eta$ may be observed with error. (ii) The function $f(x_{1u}, x_{2u},\cdots, x_{ku})$ may not be the correct model. (iii) The observations on the independent variables may contain errors. Once an experimenter has chosen a polynomial model of suitable order, the problem arises as how best to choose the settings for the independent variables over which he has control. A particular selection of settings, or factor levels, at which observations are to be taken is called a design. Designs are usually selected to satisfy some desirable criteria chosen by the experimenter. In this paper we consider the problem of design selection when there are errors in the factor levels as well as in the response. This problem has received little attention; in fact it appears that only Box (1963) has tackled it. Box first gives conditions under which the variance of a linear combination of the observations, say $L = \sum \alpha_u y_u$, is estimated unbiasedly. Next, he shows that these conditions are satisfied by two-level factorial and fractional factorial designs for first order models. In Section 2, we present a criterion for selecting designs for experiments in which errors occur in the factor levels. In Section 3 we make a simplifying assumption and an approximation which leads to an approximate criterion, and this is applied to the first order model case in Section 4.

The classical Lindeberg-Feller CLT for sums of independent random variables (rv's) provides more than the convergence in distribution of the sum to a normal law. The independence of summands also guarantees the weak convergence of all finite dimensional distributions of an a.e. sample continuous stochastic process (suitably defined in terms of the partial sums) to those of a Gaussian process with independent increments, namely, the Wiener process. Moreover, the distributions of said process converge weakly to Wiener measure on $C\lbrack 0, 1\rbrack$, the latter result being known as an invariance principle, or functional CLT, an idea originating with Erdos and Kac [10] and Donsker [5], then developed by Billingsley, Prohorov, Skorohod and others. The present work contains an invariance principle for a certain class of martingales, under a martingale version of the classical Lindeberg condition. In the case of sums of independent rv's, our results reduce to the conventional invariance principle (see, for example, Parthasarathy [16]) in the setting of the classical Lindeberg-Feller CLT. Theorem 1 contains a type of martingale characteristic function convergence which is strictly analogous to the classical CLT, while Theorem 2 provides weak convergence of finite dimensional distributions to those of a Wiener process, followed by (Theorem 3) the weak convergence of corresponding induced measures on $C\lbrack 0, 1\rbrack$ to Wiener measure, thus entailing an invariance principle for martingales. Notation and results are listed in Section 2. Section 3 defines the Lindeberg condition for martingales and gives it several equivalent forms. Sections 4 and 5 contain the proofs of Theorems 1 and 2, respectively, while Theorem 3 is proved in Section 6 by use of a martingale inequality derived from an upcrossing inequality of Doob [7]. Section 7 contains brief remarks. Among the large literature on CLT's for sums of dependent rv's, mention of a martingale CLT is first made by Levy [12], [13], followed by Doob [6] page 383. Billingsley [2] and Ibragimov [11] gave a version for stationary ergodic martingales, and Csorgo [4] considered related problems. The author also knows of Dvoretsky [8]. Invariance principles for various dependent rv's were found by Billingsley [1], and in [3] for stationary ergodic martingales, the latter result being given by Rosen [17] for bounded summands. The present Theorem 3 relaxes the stationarity and ergodicity requirements of Billingsley's Theorem (23.1) of [3]. Since preparing the original version of this paper, the author's attention has been drawn to Dvoretsky [9], which announces a result strongly resembling the present Theorem 2, and to the CLT [14] and invariance principle [15] for reversed martingales, due to Lyones. The methods of Section 5 are owed to Billingsley [2] (who in turn acknowledges a "debt to Levy") and to ideas given by Dvoretsky [8]. Finally, I wish to thank Dr. C. C. Heyde, for making a vital remark, and Mr. David Scott, for helpful criticism.

Let us have a normal stationary autoregressive series $\{X_t\}^\infty_{-\infty}$ of the $n$th order with $EX_t = 0$. Denote $b$ the vector of autoregressive parameters. In this paper the Radon-Nikodym derivative $dP_b/dP$ is studied, where $P_b$ is the probability measure corresponding to the finite part (of length $N$) of the autoregressive series and $P = P_0$, i.e., $P$ corresponds to the case, when $X_t$ are independent normal random variables. The function $dP_b/dP$ may be expanded in the power series of components of vector $b$. If the norm $\|b\|$ is small, then the absolute term and the linear terms are most important. These terms are given in the paper and they are used for an approximation of the probability $P_b(A)$, where $A$ is a Borel set in the $N$-dimensional Euclidean space $R_N$. The probability that a normal stationary autoregressive series does not exceed a constant barrier is analysed as an example. A second example is devoted to the properties of the sign-test when the observations are dependent and may be described by the autoregressive model.

Limit processes for the sequences of stochastic processes defined by co-spectral and quadrature spectral distribution functions are found using the theory of weak convergence. The limit processes are shown to be Gaussian with independent increments and with covariance functions defined in terms of hypothesized spectral densities. Section 3 contains a discussion of the moments of the processes. The first and second asymptotic moments, which characterize the limit processes, are computed giving results analogous to those of Grenander and Rosenblatt [6] and Ibragimov [8] for autospectra. We also evaluate the higher asymptotic moments and put bounds on the higher moments. The latter are required in demonstrating tightness of the measures generated in $C\lbrack 0, \pi\rbrack$ by the co-spectral and quadrature spectral distribution functions. In Section 4, limit processes, under certain conditions listed in Section 2, are obtained and described in Theorem 4.5. Finally, Section 5 contains a discussion of asymptotic goodness-of-fit testing for spectral distribution functions.

Let $K$ be a compact metric space, $\Delta$ a fixed point in $K, \mathcal{O}$ a base for the topology of $K$ closed under the formation of finite unions and finite intersections, and $\mathscr{D} = \{(K - U)\cup\Delta\mid U\in\mathcal{O}\}$ (here $\Delta$ stands for $\{\Delta\}$). Let $\{ H_D(x, \cdot)\mid x\in K, D\in\mathscr{D}\}$ be a family of probability measures satisfying the obvious necessary conditions of being the hitting distributions (as suggested in the notation) of a Hunt process on $K$ with $\Delta$ as the death point and the following conditions: (a) if $x\not\in D$ there exists $D'$ such that $\sup_{\gamma\in D'-\Delta}\int H_D(y, dz)H_D'(z, D' - \Delta) < 1$; (b) if $D_n\downarrow\Delta$ and $D - \Delta$ is compact $\int H_{D_n}(x, dy)H_D(y, D - \Delta)\rightarrow 0$ uniformly on compact subsets of $K - \Delta$; (c) there is a subclass $\mathscr{D}'$ of $\mathscr{D}$ such that the sets $K - D, D\in\mathscr{D}'$, have compact closure in $K - \Delta$ and form a base for the topology of $K - \Delta$, and for $D\in\mathscr{D}'$ and real continuous $f$ on $K \int H_D(x, dy)f(y)$ is continuous on $K - D$. Then a Hunt process is constructed from the prescribed hitting distributions $H_D(x, \cdot)$. This improves an earlier result of the author in that the smoothness condition (c) is much weaker than before; in fact the smoothness condition we actually assume is somewhat weaker than (c).

We give sufficient conditions for the absolute continuity relative to Wiener measure, $P_w$, of a measure, $P_y$, induced by the sum, $y(t)$, of a Wiener process and a non-anticipating and differentiable "signal" process. When the signal process is a measurable function of $y$, we also give expressions for $dP_y/dP_w$ and $dP_w/dP_y$.

Randomization has been proposed [1] for the dilution-series problem with the purpose of producing a continuous random variable; the randomization is performed by selecting a value from the uniform distribution (0, 1]. This paper develops a structural model appropriate to a range of problems involving such randomization as part of the design. The model applies to the dilution series problem and to the bioassay problem, and produces a posterior distribution for the primary parameter and a marginal likelihood for other parameters.

In testing goodness of fit to parametric families with unknown parameters, it is often desirable to allow the cell boundaries for a chi-square statistic to be functions of the estimated parameter values. Suppose $M$ cells are used and $m$ parameters are estimated using BAN estimators based on sample. Then A. R. Roy and G. S. watson showed that in the univariate case the asymptotic null distribution of the chi-square statistic is that of $\sum^{m - m - 1}_1 Z^2_t + \sum^{m - 1}_{M - m} \lambda_t Z^2_t$, where $Z_t$ are independent standarad normal and the constants $\lambda_t$ lie between 0 and 1. They further observed that in the location-scale case the $\lambda_t$ are independent of the parameters if the cell boundaries are chosen in a natural way, and that in any case all $\lambda_t$ approach 0 as $M$ is appropraitely increased. We extend all of these results to the case of rectangular cells in any number of dimensions. Moreover, we give a method for numerical computation of the exact cdf of the asymptotic distribution and provde a short table of crticial points for testing goodness-of-fit to the univariate normal family.

Recently several authors (cf. [5], [6], [8], [9]) have established for arbitrary positive numbers $c_1,\cdots, c_k$ the inequality \begin{equation*}\tag{1}P\{|X_1| \leqq c_1,\cdots, |X_k| \leqq c_k\} \geqq \Pi^k_{i=1} P\{|X_i| \leqq c_i\}\end{equation*} valid for a random vector $X = (X_1,\cdots, X_k)$ having a multivariate normal distribution with mean values 0 and with an arbitrary covariance matrix. A question then arises whether also an analogue to (1) for multivariate Student distributions holds true, i.e. the inequality \begin{equation*}\tag{2}P\{|X_1|/S_1 \leqq c_1,\cdots, |X_k|/S_k \leqq c_k\} \geqq \Pi^k_{i=1} P\{|X_i|/S_i \leqq c_i\}\end{equation*} where $X = (X_1,\cdots, X_k)$ is as before, while $S_i = (\sum^p_{\nu=1} Z^2_{i\nu})^{\frac{1}{2}}, i = 1,\cdots, k$, where $Z_\nu = (Z_{i\nu},\cdots, Z_{k\nu}), \nu = 1,\cdots, p$, is a random sample of $p$ vectors, which are mutually independent and independent of $X$, and each of which has, in the simplest case, the same normal distribution as $X$. More generally, the $Z_\nu$'s have some normal distributions with mean values 0 and with some covariance matrices which need not coincide with that of $X$ and even need not be identical. A certain proof of (2) was presented by A. Scott [6] but we shall give here a counterexample showing that, unfortunately, this proof is incorrect. However, if the correlations between $X_i$ and $X_j$ have the form $\lambda_i\lambda_j\rho_{ij} (i,j = 1,\cdots, k; i \neq j)$ where $|\lambda_i| \leqq 1 (i = 1,\cdots, k)$ and where $\{\rho_{ij}\}$ is any fixed correlation matrix, and if the correlations between $Z_{i\nu}$ and $Z_{j\nu}$ have the form $\tau_{i\nu}\tau_{j\nu} (i,j = 1,\cdots, k; i \neq j; \nu = 1,\cdots, p)$ where $|\tau_{i\nu}| < 1(i = 1,\cdots, k; \nu = 1,\cdots, p)$, we shall prove here that the left-hand side probability in (2) is a non-decreasing function of each $|\lambda_i|$ and each $|\tau_{i\nu}|$; therefore, in this case of a special correlation structure, (2) is indeed true. The general validity of (2) still remains an open question.

The problem of obtaining sequential confidence intervals for the median of an unknown symmetric distributon based on a general class of one-sample rank-order statistics is considered. It is shown that the usual one-sample rank-order statistic possesses the martingale or sub-martingale property according as the parent distribution is symmetric about the origin or not. Certain asymptotic almost sure convergence results (with specified order of convergence) for a class of rank-order processes and the empirical distribution are derived, and these are then utilized for the study of the properties of the proposed procedures.

Raiffa and Schlaifer's theory of conjugate prior distributions is here applied to Jeffrey's theory of tests for a sharp hypothesis, for simple normal sampling, for model I analysis of variance, and for univariate and multivariate Behrens-Fisher probelms. Leonard J. Savage's Bayesianization of Jeffrey's theory is given with new generalizations. A new conjugate prior family for normal sampling which allows prior independence of unknown mena and variance is given.

Elias [9], [10] proved that group codes achieve Shannon's channel capacity for binary symmetric channels. This result was generalized by Dobrushin [7] (and independently by Drygas [8]) to discrete memoryless channels satisfying a certain symmetry condition and having a Galois field as alphabet. We prove that group codes to dnot achieve the channel capacity for general discrete memoryless channels. It therefore makes sense to introduce a group code capacity and to talk about a group coding theorem and its weak converse can be established for several reasonable channels such as the discrete memoryless channel, compound channels, and averaged channels. An example of a channel is given for which Shannon's capacity is positive and the group code capacity is zero. Using group codes, one can therefore expect high rates only for channels with a simple probabilistic structure.

Let $Q_n(n = 1,2, \cdots), Q$ be probability measures on the Borel $\sigma$-field of $R^k$. The sequence $\{Q_n\}$ converges weakly to $Q$ if for every real-valued, bounded, almost surely $(Q)$ continuous function $g$ on $R^k$ the convergence \begin{equation*}\tag{0.1} \lim_n \inf g dQ_n = \int g dQ\end{equation*} holds (cf. [9], Chapter 1). Such functions $g$ are called $Q$-continuous. If the indicator function $I_A$ of the set $A$ is $Q$-continuous, then $A$ is also called $Q$-continuous. If $\mathscr{F}$ is a class of $Q$-continuous functions $g$ over which the convergence (0.1) is uniform for every sequence $\{Q_n\}$ converging weakly to $Q$, then $\mathscr{F}$ is called a $Q$-uniformity class. A class of sets is called a $Q$-uniformity class if the indicator functions of the sets of this class form a $Q$-uniformity class of functions. A systematic study of $Q$-uniformity (in separable metric spaces) was initiated by Ranga Rao [21], who obtained a number of nice results. His studies were carried further in a very useful manner by Billingsley and Topsoe [10]. In this article the error of normal approximation $|\int g d(Q_n - \Phi)|$ is estimated for arbitrary $\Phi$-continuous $g, \Phi$ being the $k$-dimensional standard normal distribution and $Q_n$ the distribution of the appropriately normalised $n$th partial sum of a sequence of independent $k$-dimensional random vectors $\{X^{(r)}; r = 1,2,\cdots\}$. The classical central limit theorems assert weak convergence of $\{Q_n\}$ to $\Phi$ under certain moment conditions. It is shown here (Theorem 1, Section 3) that for an arbitrary real-valued, bounded, measurable $g$ on $R^k$ one has $(0.2) \quad |\int g d(Q_n - \Phi)| \leqq c(k, \delta)\omega_g(R^k) \rho^{3(1 + \delta)/(3 + \delta)}_{3 + \delta,n} n^{-\frac{1}{2}} + \int \omega_g(S(x, \varepsilon_n)) d\Phi(x)$, where $\delta$ is any positive number; $\rho_{3+\delta, n}$ is defined by (1.4), and \begin{equation*}\tag{0.3}\begin{align*} \omega_g(A) &= \sup \{|g(x) - g(y)|; x, y \epsilon A\}, S(x, \varepsilon) = \{y; |x - y| < \varepsilon\}, \\ \varepsilon_n &= c(k) \rho^{3/(3+\delta)}_{3 + \delta, n} n^{-\frac{1}{2}} \log n,\end{align*}\end{equation} $c(k), c(k, \delta)$ being positive constants depending only on their respective arguments. If $\varepsilon_n$ goes to zero as $n$ goes to infinity, then the right side of (0.2) goes to zero for every $\Phi$-continuous $g$. For the rest of this section let us assume that $\{\rho_{3+\delta,n\}}$ is bounded. By (0.2), if $\int \omega_g(S(x, \varepsilon)) d\Phi(x) = O(\varepsilon)$ as $\varepsilon$ goes to zero, then the error of approximation is $O(n^{-\frac{1}{2}} \log n)$. One may also use (0.2) to obtain uniform upper bounds for errors of approximation over arbitrary $\Phi$-uniformity classes. In particular, $(0.4) \quad \sup\{|\int g d(Q_n - \Phi)|; g \epsilon \mathscr{F}_1(\Phi; c, d, \varepsilon_0)\} = O(n^{-\frac{1}{2}}\log n)$, where $(0.5) \quad \mathscr{F}_1(\Phi; c, d, \varepsilon_0) = \{g; \omega_g(R^k) \leqq c, \int \omega_g(S(\cdot, \varepsilon)) d\Phi \leqq \varepsilon\text{for} 0 < \varepsilon \leqq \varepsilon_0\}$, $c, d, \varepsilon_0$ being arbitrary positive constants. For the largest translation-invariant subclass of this class a sharper bound $O(n^{-\frac{1}{2}})$, which is best possible, was obtained in [3], [5], (cf. [4]). A similar result (in the i.i.d. case) has been independently obtained by Von Bahr [22]. As applications one obtains precise bounds for many interesting classes of sets and functions. However, there are $\Phi$-continuous functions and $\Phi$-uniformity classes of functions for which the technique used in [5] or [22] is not effective. An example in Section 3 shows that there are Borel sets $A$ such that the upper bound for $|Q_n(A) - \Phi(A)|$ as provided by [5] (Theorem 1) is $O(1)$, while (0.2) provides the bound $O(n^{-\frac{1}{2}}\log n)$. A modification of the bound (0.2) when applied to $g = I_A$ for an arbitrary Borel set $A$ enables one to show that the Prokhorov distance between $Q_n$ and $\Phi$ is $O(n^{-\frac{1}{2}}\log n)$. It is not known whether the factor $\log n$ in the expression for $\varepsilon_n$ in (0.3) (and, hence, in (0.4) and in the estimate of Prokhorov's distance) may be dispensed with or not. However, under Cramer's condition (3.42) $\log n$ may be replaced by one. The remaining theorems are proved for the i.i.d. case, partly for the sake of simplicity and partly because of the non-availability in the existing literature of complete proofs for some of the expansions related to the characteristic function of $Q_n$ in the non-identically distributed case. Theorem 2 provides an asymptotic expansion for $\int g dQ_n$ with a remainder term which is $o(n^{-(s-2)/2})$ uniformly over all $g$ in $\mathscr{F}_1^\ast(\Phi; c, d, \varepsilon_0)$, the largest translation-invariant subclass of $\mathscr{F}_1(\Phi; c, d, \varepsilon_0)$, when $E|X^{(1)}|^s < \infty$ for some integer $s$ not smaller than three and the characteristic function of $X^{(1)}$ obeys Cramer's condition (3.42)'. Applications to the class $\mathscr{L}$ of all measurable convex sets, the class $L(c, d)$ (see (3.54)) of bounded Lipschitzian functions, etc., are immediate. Theorem 3 gives an asymptotic expansion for $\int g dQ_n$ for a very special class of functions $g$ when no restriction like (3.42)' is imposed. Theorem 4 provides some classes of functions $g$ (under varying restrictions on the distribution of $X^{(1)}$) for which the error of approximation $|\int g d(Q_n - \Phi)|$ is of the order $O(n^{-1})$. Section 1 introduces notation to be used throughout the article. Section 2 provides basic lemmas for proving the results (outlined above) of Section 3.

In this paper we will derive asymptotically optimal rank-order tests for independence against suitable classes of nonparametric alternatives and give asymptotic relative efficiencies (ARE's) of such tests under general contiguous alternatives of positive quadrant dependence (cf. Lehmann [14]). From Lehmann [14] one can also see that such alternatives in some respects are more general than the alternatives considered in Bhuchongkul [1], Konijn [12], Hajek and Sidak [8] page 221), and others. For the problem of symmetry and for the two-sample problem we can get completely analogous results with similar proofs. Details are omitted. The paper is based on the theory of contiguity that was introduced by LeCam [13] and Hajek [6]. The results of this paper complement results obtained by Hodges and Lehmann [9], [10], Chernoff and Savage [3], Hajek [6], van Eeden [4], Bhuchongkul [1], Gokhale [5], and others.

Let $X$ be a random variable with distribution function $F(x\mid\theta)$ over a sample space $\mathscr{H}$ for each $\theta$ in $\Theta$, the parameter space. Consider the decision problem with loss function $L(\theta, a)$ for each $a$ in $\mathscr{A}$, the action space. For a decision rule $\delta(x)$ mapping $\mathscr{H}$ into $\mathscr{A}$ the risk function is $R(\theta, \delta) = \int L(\theta, \delta(x)) dF(x \mid \theta)$. If $\tau(\theta)$ is a distribution over $\Theta$, the expected risk of using rule $\delta(x)$ is then $r(\tau, \delta) = \int_\Theta R(\theta, \delta)d\tau(\theta)$. In many problems a priori information will be incomplete. Suppose that our prior information consists of a class $\Upsilon$ of distributions over $\Theta$. One method of utilizing such partial prior information to obtain a decision rule is given by Blum and Rosenblatt [1]. DEFINITION 1.1. The rule $\delta_0(x)$ is a $\Upsilon$-minimax decision rule if $\sup_{\tau \epsilon \Upsilon} r(\tau, \delta_0) = \inf_\delta\sup_{\tau \in\Upsilon} r(\tau, \delta)$. The use of partial prior information by Menges [5] and Hodges and Lehmann [2] may also be considered as satisfying the $\Upsilon$-minimax criterion for suitable choices of $\Upsilon$. In this paper the $\Upsilon$-minimax principle is applied to the problem of selecting treatment populations which have larger translation parameters than that of a control population. Let $S_0, S_1, \cdots, S_k$ be $k + 1$ independent random variables with respective probability density functions $f_0(s - \theta_0), f_1(s - \theta_1), \cdots, f_k(s - \theta_k)$. The random variables $S_0, S_1,\cdots, S_k$ may represent sufficient statistics from the control and $k$ treatment populations, respectively. We assume that each $f_i(s), i = 0, 1, \cdots, k$, is a Polya frequency function of order two $(PF_2)$, that is, if $x_1 < x_2$ and $y_1 < y_2$ then \begin{equation*}\begin{vmatrix}f_i(x_1 - y_1) & f_i(x_1 - y_2) \\ f_i(x_2 - y_1) & f_i(x_2 - y_2)\end{vmatrix} \geqq 0.\end{equation*} Hence $f_i(s - \theta_i)$ has a monotone likelihood ratio in its translation parameter. In Section 2, necessary notation, the loss function, and the incomplete prior information are introduced. In Section 3, a $\Upsilon$-minimax decision rule is found, for the case in which the control population parameter $\theta_0$ is known, by finding a rule which is Bayes with respect to a least favorable prior in the class of prior distributions $\Upsilon$. The case in which the control population parameter is unknown is treated in Section 4. Rules are derived which are $\Upsilon$-minimax among procedures for which the decision to select or reject the $i$th population depends only on $S_i$ and $S_0$. When specialized to normal populations with common known variance $\sigma^2$, a $\Upsilon$-minimax rule selects (rejects) the $i$th population as $\bar{X}_i - \bar{X}_0 \geqq (<)$ a constant, where the constant depends on $\Delta$ (a known constant used to define "positive" and "negative" populations), $\sigma^2$, the sample sizes, the ratio of the losses for the two kinds of incorrect decisions, and the ratio of the prior probabilities of negative and positive populations. (An analogous result is found in Section 3 for the known control case.) Section 5 gives comparisons of a $\Upsilon$-minimax rule with a Bayes competitor based on independent normal priors for the case of normal populations with common known variance. Some comments on a theorem by Y. L. Tong [7] are given in Section 6. Whenever a new criterion is being considered by the statistical community, it is important to see if the criterion is fruitful in a variety of situations. In the treatment versus control problems discussed here, the $\Upsilon$-minimax criterion leads to simple explicit rules which compare favorably with Bayes rules that require stronger assumptions on the prior distribution. We hope that this work will encourage others to obtain and apply $\Upsilon$-minimax rules in different settings.

The asymptotic distribution of the log-likelihood ratio is shown to provide a method of determining approximate confidence bounds for the reliability function of any coherent system when each component has an exponential life with unknown failure rate and component performance data are provided in the form: number of failures (minimum of one) and total operating time. Thus the method applies under all general types of censoring. This extends the results of the authors, Ann. Math. Statist. (1968), on confidence limits for coherent structures with binomial data on the component's reliability. Methods similar to those previously utilized are combined with some special properties of the exponential distribution to obtain the results.

Consider the classical homogeneity of variances model. That is, suppose we have a one way layout, with independent random samples of equal sizes in each column. We assume the samples are from normal populations with unknown means and unknown variances. We wish to test the hypothesis that all the variances are equal. R. V. Laue (1965) defined a two parameter family of statistics to test the homogeneity hypothesis. The family is defined as a function of the ratio of mean value functions of the sample variances. Included in the family are many of the well-known tests for homogeneity. In this paper we investigate which of the tests is unbiased. Although we are unable to resolve the question for every test in the family, we can demonstrate the unbiased character for several subfamilies, which include some of the better known tests.

In this note we discuss a characteristic property of the exponential distribution which is closely related to the "lack-of-memory" characterization.

For stationary point processes in $R^n$ a (proper) distribution of group size is shown to exist and a simple proof of the result: intensity = parameter $\times$ mean group size is given. A generalization of this result to higher order moments is proved by similar methods.

Let $X_0, X_1, X_2, \cdots$ be a sequence of independent random variables with common distribution function $F(x)$. Let $\nu$ be a discount factor $(0 < \nu < 1)$. Then we define \begin{equation*}\tag{1.1} X_\nu = \sum^\infty_{k=0} \nu^k X_k,\end{equation*} which may be interpreted as the present value of a sum of certain periodic and identically distributed payments $X_k$. We assume that the first three moments of $X_k$ are finite: \begin{align*}\tag{1.2}\mu &= \int^{+\infty}_{-\infty} x dF(x) < \infty,\quad \sigma^2 = \int^{+\infty}_{-\infty} (x - \mu)^2 dF(x) < \infty, \\ \rho &= \int^{+\infty}_{-\infty}|x - \mu|^3 dF(x) < \infty.\\ \end{align*} It will be shown that the normalized random variable \begin{equation*}\tag{1.3} Z_\nu = \frac{(1 - \nu)^{\frac{1}{2}}}{\sigma} \big(X_\nu - \frac{\mu}{1 - \nu}\big)\end{equation*} is asymptotically normal for $\nu \rightarrow 1$. The analogue of the Berry-Esseen theorem (see [1], [2], [3], [4]) will be established for the difference $F_\nu(x) - \mathcal{N}_\nu(x), F_\nu(x)$ being the distribution of $Z_\nu$, whereas $\mathcal{N}_\nu(x)$ is the normal distribution with zero mean and variance $(1 + \nu)^{-1}$: \begin{equation*}\tag{1.4} \mathcal{N}_\nu(x) = \big(\frac{1 + \nu}{2\pi} \big)^{\frac{1}{2}}\int^x_{-\infty} \exp \big{-\frac{1 + \nu}{2}t^2\big)dt.\end{equation*} The proof uses those Fourier techniques which are masterfully presented in Feller's book [3] for the proof of the "ordinary" Berry-Esseen theorem. In the last section the corresponding estiamte is established for the compound Poisson process. Other aspects of the Discounted Central Limit Theorem are treated in a paper by Whitt [5].

Let $X_1, X_2, \cdots$ be independent random variables defined on a probability space $(\Omega, F, P)$ and with mean 0 and variance1. Let $F_0 = \{\Omega, \phi\}$, and for $n \geqq 1$ let $S_n = X_1 + X_2 + \cdots + X_n$ and $F_n = \sigma(X_1, X_2,\cdots, X_n)$. Denote by $T$ the class of all stopping rules with respect to $\{F_n\}$, i.e., the class of all $t: \Omega \rightarrow \{1, 2,\cdots, \infty\}$ such that $\{t = k\}\in F_k$ for $k = 1, 2,\cdots$, and let $T_n = \{t \in T: t\geqq n \operatorname{a.s.}\}$. If $t \in T$ we adopt the convention of the author of [3] that $|S_t|/t = \lim \sup_n |S_n|/n$ if $t = \infty$. We remind the reader that for this sequence of random variables $X_1, X_2,\cdots, \lim \sup_n |S_n|/n = 0 \operatorname{a.s}.$ Since $E(\sup_n|S_n|/n) < \infty$ (Lemma 9 [1]), it follows at once from the results of [3] that LEMMA 1. If $f_n = \operatorname{ess} \sup_{t\in T_n} E(|S_t\mid/t| F_n)$, then $f_n = \max (|S_n\mid/n, E(f_{n+1} |F_n))$ a.s. and $\lim \sup_nf_n = \lim \sup_n|S_n|/n = 0$ a.s. For each $c > 0$ define $t(c) =$ first $n \geqq 1$ for which $|S_n| > cn^{\frac{1}{2}}, = \infty$ if no such $n$ exists. In this note we prove THEOREM 1. If for each $c > 0, P(t(c) < \infty) = 1$, then $P(|S_n| = nf_n \operatorname{i.o.}) = 1$. Although this theorem will not be a surprise to readers of the recent literature on optimal stopping problems related to $S_n/n$, this proof may be of some interest.

The Strong Law of Large Numbers, valid for independent, identically distributed (i.i.d.) random variables $\{X_n, n \geqq 1\}$ with finite first moment, may be regarded as merely one of a host of summability methods applicable to the divergent sequence $\{X_n\}$. Here, a subclass of regular (Toeplitz) summability methods will be considered and concern will focus on the almost certain (a.c.) convergence to zero of the transformed sequence \begin{equation*}\tag{1}T_n = A_n^{-1} \sum^n_{j=1} a_jX_j\end{equation*} when centered where \begin{equation*}\tag{i}a_n \geqq 0,\quad A_n = \sum^n_{j=1} a_j \rightarrow \infty,\end{equation*} thereby ensuring regularity. If $T_n - C_n \rightarrow_{\operatorname{a.c.}} 0$ for some choice of centering constants $C_n$, the i.i.d. random variables $\{X_n\}$ will be called $a_n$-summable with probability one or simply $a_n$-summable. The Strong Law is the special case $(a_n \equiv 1)$ of Cesaro-one summability with $C_n \equiv EX$. Of course, if $X^\ast_n = X_n - X'_n, n \geqq 1$ are the symmetrized $X_n$ (i.e., $\{X'_n\}$ is i.i.d., independent of $\{X_n\}$ with the same distribution), then $a_n$-summability of $\{X_n\}$ implies $a_n$-summability of $\{X^\ast_n\}$ with vanishing centering constants, i.e. \begin{equation*}\tag{2}T^\ast_n = A_n^{-1} \sum^n_{j=1} a_jX^\ast_j \rightarrow_{\operatorname{a.c.}} 0.\end{equation*} It will be shown, on the one hand, that no such choice of $\{a_n\}$ and $\{C_n\}$ will render i.i.d. $\{X_n\}$ with the St. Petersburg (mass $2^{-n}$ at the point $2^n, n \geqq 1$) or Cauchy distribution $a_n$-summable. On the other hand, necessary and sufficient conditions for certain types of $a_n$-summability more refined than (implied by) Cesaro-one will be proffered. The prototype of these appears in Corollary 1 and Corollary 2.

In a paper entitled "Ergodic theorems for infinite probabilistic tables," Paz studies the ergodic properties of non-homogeneous Markov chains with countably infinite state space. This note is to draw attention to the fact that Paz's theory is easily extended to arbitrary state spaces.