This paper reconsiders the usual sequential decision problem of choosing between two simple hypotheses $H_0$ and $H_1$, in terms of iirv when there is a time delay, assumed to have a known exponential distribution, in obtaining observations. A basic assumption underlying much of the current analysis, is that results of taking observations are considered as immediately available. In other words, it is assumed that there is no time delay between the decision to take an observation and obtaining the result of the observation. This, of course, can be a tremendous limitation to the applicability of the theory. In actuality, such time lags can be substantial and taking an observation often involves experimentation. One important example, where this time delay often considerably inhibits the use of sequential analysis is medical experimentation. Here, a long time may elapse between the application and the result of a treatment. The theory of sequential analysis is considered, explicitly taking account of time lags. At any point in time the decision maker must decide whether to stop and take action now or to continue and schedule more experiments. If he continues he must also decide how many more experiments to schedule. The problem basically then is to find an optimal procedure incorporating a stopping, scheduling and terminal action rule. There is an interesting interplay among these three; and optimal stopping rules, currently used for some problems, may not be optimal when scheduling factors are considered. The usual losses related to decision errors are specified and linear cost assumptions, with regard to amount of experimentation and time until a final decision, are made. Time until a terminal decision is an important variable. If it is decided to continue observation then scheduling many experiments will result in a small expected waiting time until the next result. However, this advantage must be balanced with the cost of scheduling these experiments. Finally, all the previous must be weighed with the loss of taking immediate action with only the currently available information. Bayes procedures are derived. The information state, at any time, because of the exponential assumption, will be described by $(n, \pi)$ where $\pi$ is the current posterior probability of $H_0$ and $n$ the numbers of results outstanding from tests already scheduled. As indicated in Section 2, when a known exponential time delay distribution is assumed, possible decision changes should be made only when test results are obtained. In Section 3, the usually used Sequential Probability Ratio Test (SPRT) stopping rule is studied. Here there are two values $0 < \pi_1 \leqq \pi_2 < 1$ and SPRT specifies that $(n, \pi)$ is a continuation state as long as $\pi_1 < \pi < \pi_2$. It is shown that there is a bounded function $z(\pi)$ such that the optimal scheduling quantity, for a continuation state $(n, \pi)$, is $y(n, \pi) = \max \lbrack 0, z(\pi) - n\rbrack$. That is, if $n > z(\pi)$ we schedule no experiments. On the other hand, if $n \leqq z(\pi)$ then $z(\pi) -n$ more experiments are scheduled. The functional equation approach of Dynamic Programming is used and provides a computational method for approximating $z(\pi)$. The general case, where the problem is to find an optimal stopping and scheduling rule, is studied in Section 4. Various results about the optimal stopping region, in the $(n, \pi)$ plane, are derived and it is shown that the optimal procedure stops with probability one. The optimal stopping region is a kind of generalized SPRT described by functions $0 < \pi_1(n) \leqq \pi_2(n) < 1$ such that $(n, \pi)$ is a continuation state as long as $\pi_1(n) < \pi < \pi_2(n)$. Also, it is shown that there exist two bounded functions $z_1(\pi) \leqq z_2(\pi) \leqq \mathbf{M} < \infty$ such that if $(n, \pi)$ is a continuation point the optimal scheduling quantity is $y(n, \pi) = z_1(\pi) - n$ if $n \leqq z_1(\pi)$, and $y(n, \pi) = 0$ if $n \geqq z_2(\pi)$. When a continuous approximation to the problem is used, allowing $n$ to take on a continuous range of values, a stronger result is proven. Here, the two functions $z_1(\pi)$ and $z_2(\pi)$ may be taken as equal. The results for optimal scheduling rules have similarities to some problems studied in Inventory theory, see [5].

Consider a sample $(x_1, x_2, \cdots, x_N)$ from a population with a distribution function $F_\theta(x), (\theta \epsilon \mathbf{\Omega})$ for which a complete sufficient statistic, $s(x)$, exists. Then any parametric function $g(\theta)$ possesses a unique minimum variance unbiased estimator U.M.V.U.E., which may be obtained by the Rao-Blackwell theorem provided an unbiased estimator of $g(\theta)$ with finite variance for each $\theta \epsilon \mathbf{\Omega}$ is available. In this paper we will consider the Koopman-Darmois class of exponential densities and develop a method for obtaining the U.M.V.U.E., $t_g$, of $g(\theta)$ without explicit knowledge of any unbiased estimator of $g(\theta)$. The U.M.V.U.E. $t_g$ is given as the limit in the mean (l.i.m.) of a series and a convergent series is also given for the variance. For any arbitrary but fixed $\theta_0 \epsilon \mathbf{\Omega}$, it can be verified that the complete sufficient statistic $s(x)$ has moments of all orders and that these moments determine its distribution function. Hence the set of polynomials in $s(x)$ is dense in the Hilbert space, $V$ (with the usual inner product), of Borel measurable functions of $s(x)$. Since $t_g$ is an element of $V$, we may obtain a generalized Fourier series for it by constructing a complete orthonormal set $\{\varphi_n\}$ for $V$. Such a set $\{\varphi_n\}$ may be obtained from the density function and its derivatives with respect to $\theta$. For a subclass of the exponential family, Seth [18] has obtained $\{\varphi_n\}$ in a form which is convenient for our purposes. We will study this case in Section 3 and use Seth's results to give an explicit construction of $t_g$. Criteria for the pointwise convergence of the series will also be given. In Section 4 examples illustrating the use of the method are given and some related results are discussed. The general theory for the representation of minimum variance unbiased estimates, both local and uniform, has been developed in depth, for example in [5], [18], [19], [16], [3], and [4]. The present remarks, though founded in the general theory (in particular [3] and [4]), are tailored specifically to the exponential family.

Assume $E(X_i) \equiv 0$. For $\nu \geqq 2$, bounds on the $\nu$th moment of $\max_{1 \leqq k \leqq n}|\sum^{a + k}_{a + 1} X_i|$ are deduced from assumed bounds on the $\nu$th moment of $|\sum^{a + n}_{a + 1} X_i|$. The inequality due to Rademacher-Mensov for $\nu = 2$ and orthogonal $X_i$'s is generalized to $\nu \geqq 2$ and other types of dependent $\operatorname{rv's}.$ In the case $\nu > 2$, a second result is obtained which is considerably stronger than the first for asymptotic applications.

Let $X(t), 0 \leqq t \leqq 1$, be separable measurable Gaussian process with mean 0, stationary increments, and $\sigma^2(t) = E(X(t) - X(0))^2$. If $\sigma^2(t) \sim C|t|^\alpha, t \rightarrow 0$, for some $\alpha, 0 < \alpha < 2$, then the Hausdorff dimension of $\{s: X(t) = X(s)\}$ is equal to $1 - (\alpha/2)$ for almost all $t$, almost surely. Under further variations and refinements of this condition there is a jointly continuous local time for almost every sample function. This extends the author's previous results for stationary Gaussian processes and for continuity in the space variable alone. The result on joint continuity of the local time is used to prove that the sample function has an "approximate derivative" of infinite magnitude at each point (and so is nowhere differentiable); and that the set of values in the range of at most countable multiplicity is nowhere dense in the range.

In the regression model $Y = \beta_1x_1 + \beta_2x_2 + Z$ a class of asymptotically distribution free (ADF) tests for testing $H_0:\beta_1 = 0$ when $\beta_2$ is unknown is given. It turns out that if one uses Wilcoxon type tests then there is no reasonable distribution of $Z$ under which the test would be ADF unless $x_1$ and $x_2$ are orthogonal to each other when centered. On the other hand if one is sampling from double exponential, then the class of tests is reasonably large. The tests of the Freund-Ausari type, Mood-type, among others, are in the class. Section 1 consists of introduction, notation and assumptions. In Section 2, we prove a uniform continuity Theorem 2.2 for rank statistics. Theorem 2.1 and Lemma 2.3 are proved before Theorem 2.2. These latter two results are based on the work done in [4]. Theorem 2.4 gives the desired result. Finally generalization to the situation where one has multiple linear regression model and has more than one parameter under $H_0$ with more than one unknown is mentioned.

It is known that any functional of Brownian motion with finite second moment can be expressed as the sum of a constant and an Ito stochastic integral. It is also known that homogeneous additive functionals of Brownian motion with finite expectations have a similar representation. This paper extends these results in several ways. It is shown that any finite functional of Brownian motion can be represented as a stochastic integral. This representation is not unique, but if the functional has a finite expectation it does have a unique representation as a constant plus a stochastic integral in which the process of indefinite integrals is a martingale. A corollary of this result is that any martingale (on a closed interval) that is measurable with respect to the increasing family of $\sigma$-fields generated by a Brownian motion is equal to a constant plus an indefinite stochastic integral. Sufficiently well-behaved Frechet-differentiable functionals have an explicit representation as a stochastic integral in which the integrand has the form of conditional expectations of the differential.

Let $Z_i = (X_i, Y_i), i = 1,2,\cdots, m$ be a random sample from a bivariate population with continuous $\operatorname{cdf} F(x, y)$ and let $Z_i = (X_i, Y_i), i = m + 1, \cdots, m + n$ be an independent sample from the population with continuous $\operatorname{cdf} G(x, y)$. The present paper is concerned with the development of a nonparametric procedure for testing the null hypothesis $H:F = G$ against a class of one-sided alternatives which are, generally, those $\operatorname{cdf's}$ which are shifted from $F$ towards large values of $x$ and $y$. That is, large values of $x$ and $y$ are more likely under $G$ than $F$. A formal definition of the alternatives, tied to a concept of two-dimensional ordering, is introduced in Section 2. A simple example is given by the one-sided translation alternatives \begin{equation*}\tag{1.1}K_0 = \{(F, G): G(x, y) = F(x - \theta_1, y - \theta_2), \mathbf{\theta} \geqq \mathbf{0} \text{and} \mathbf{\theta} \neq \mathbf{0}\}\end{equation*} where inequality between vectors means coordinate-wise inequality. Another interesting class of directed shifts, which has not been previously studied, is composed of the bivariate Lehmann-type alternatives \begin{equation*}\tag{1.2}K_1 = \{(F, G):G(x, y) = F^\theta(x, y) \theta > 1\}.\end{equation*} All $\operatorname{cdf's}$ are assumed to be continuous unless otherwise stated. Among other areas, problems involving ordered alternatives arise in reliability studies of two-component systems when the components are interdependent and prior information on the mechanism involved ensures that the average lifetimes of the components in one system are no smaller than those of the other. Another example would be the comparison of a new with a standard method for treating a disease when two responses are measured. The usefulness of incorporating prior information in the form of an ordered parameter space was first recognized by Bartholomew [2], [3], [4] in the one-way analysis of variance model. He derived the likelihood ratio test for the hypothesis that the means of several normal distributions are equal against the alternative that they are ordered. Maximum likelihood estimation of ordered parameters in other distributions has been studied by Brunk [5] and Van Eeden [17]. Chacko [6] and Shorack [16] proposed rank analogs of Bartholomew's test and showed that the asymptotic Pitman efficiency equals that of the Wilcoxon test relative to the $t$ test. Employing some results from nonlinear programming theory, Kudo [9] and Nuesch [12] derived the likelihood ratio procedure for testing $\mathbf{\mu} = \mathbf{0}$ against $\mathbf{\mu} \geqq \mathbf{0}$ on the basis of a single sample from a multivariate normal $\mathcal{N}(\mathbf{\mu}, \mathbf{\Sigma})$ with $\mathbf{\Sigma}$ known. The case of unknown $\mathbf{\Sigma}$ has recently been investigated by Perlman [13]. Other tests when $\not\sum$ is known, namely most stringent somewhere most powerful tests, have been studied by Schaafsma and Smid [15] and these have been compared with most stringent tests by Schaafsma [14]. No distribution-free or even asymptotically distribution-free test is yet available in the literature for detecting ordered shifts in bivariate and multivariate distributions. As groundwork for the bivariate two-sample problem, we propose and study a nonparametric test based on the concept of two-dimensional layer ranks which were introduced by Barndorff-Nielsen and Sobel [1]. Let $N = m + n$ and define \begin{align*}L(i, j) &= 1 \quad\text{if}\mathbf{Z}_i \geqq \mathbf{Z}_j; \\ &= 0\quad \text{otherwise}.\end{align*}\begin{equation*}\tag{1.3}\begin{align*}L_i &= \sum^N_{j = 1} L(i, j),\quad L_j^\ast = \sum^N_{i = 1} L(i, j);\quad 1 \leqq i \leqq N, 1 \leqq j \leqq N, \\ \mathbf{L} &= (L_1, \cdots, L_N),\quad \mathbf{L}^\ast = (L_1^\ast, \cdots, L_N^\ast).\end{align*}\end{equation*} Then $L_i(L_i^\ast)$ is called the 3rd (1st) quadrant layer rank of $\mathbf{Z}_i$ in the combined sample $\{\mathbf{Z}_1, \cdots, \mathbf{Z}_N\}$. Geometrically, $L_i$ and $L_i^\ast$ are the number of points $\mathbf{Z}_j$ in the closed 3rd and 1st quadrant, respectively, with reference to Cartesian coordinates having origin $\mathbf{Z}_i$. For intuitive motivation of our test statistic, consider plotting the first sample as dots and the second as crosses in the same diagram [Fig. 1]. Under $H$, the $\mathbf{Z}_i$ are independent identically distributed and the dots and crosses are expected to be well mixed. Under an ordered alternative, $G$ has more mass in the upper right-hand corner than $F$ and the second sample layer ranks $L_{m + 1}, \cdots, L_N$ are expected to be larger than $L_1, \cdots, L_m$ on the average. High values of the statistic $\lbrack m\sum^N_{i = m + 1} L_i - n \sum^m_{i = 1} L_i\rbrack$ should then lead to rejection of $H$ in favor of an ordered shift. Similarly, a small value of $\lbrack m \sum^N_{i = m + 1} L_i^\ast - n \sum^m_{i = 1} L_i^\ast\rbrack$ indicates a one-sided shift. Incidentally, for the univariate two sample problem the diagram would be one of dots and crosses on a line and the layer ranks would reduce to ordinary ranks making each of the two statistics equivalent to the one-sided Wilcoxon statistic. Although $\mathbf{L}$ and $\mathbf{L}^\ast$ are invariant under a natural group of transformations which leave the problem invariant, it is difficult to characterize a maximal invariant under this group in a manageable form. A permutation test based upon the 3rd quadrant layer ranks $\mathbf{L}$ is proposed in Section 2 and is shown to be unbiased. A large sample unconditional test is proposed in Section 3 as an approximation to the permutation test and it is shown to be consistent. Section 4 contains results on the asymptotic distribution under local ordered shift alternatives and the Pitman efficacy of the test. Modifications of the test for some variants of the basic problem and a parametric test under normal theory are discussed in Section 5.

The purpose of this paper is to try to show that certain moment conditions are essential for the admissibility of various "good" translation invariant statistical procedures. In 1951 Blackwell [1] first gave an example in which he proved that a best invariant estimate may be inadmissible. Since then many papers dealing with the admissibility of the best invariant procedures have been published. The problem of admissibility in the case of point estimation of a location parameter was treated by Blyth [2], Blackwell [1], Stein [9], [10], [11], Fox and Rubin [6], Farrell [4], Brown [3]. The problem of admissibility of certain confidence intervals was treated by Joshi [7]. The problem of admissibility in the case of a best invariant test involving a location parameter was treated by Lehmann and Stein [8]. In each paper cited above, admissibility requires the existence of one more moment than what is needed for finite risk. The first two examples of this paper indicate that, without this extra moment, inadmissibility may result. In Section 3, we show, by example, a unique best translation invariant estimate may be inadmissible if a certain moment condition fails to be satisfied. In Section 4 we prove a theorem which gives a set of sufficient conditions for the admissibility of certain translation invariant confidence interval procedures and we also give an example which shows that a certain translation invariant confidence interval procedure may be inadmissible if a certain moment condition fails to hold. In Section 5 we show by example a best translation invariant test may be inadmissible if the test is non-unique.

We are motivated by Stein's proof (Stein (1946), Wald (1947), pages 157-158) of the termination of a sequential probability ratio test in the case of independent and identically distributed random variables. Extending his ideas to take certain "dependencies" into account we examine the rank-order sequential probability ratio test based on a Lehmann alternative studied in a paper with the above title by I. R. Savage and the author (1966) (referred to as SS I in the rest of this paper). We prove that this test terminates with probability one and that the stopping time has a finite moment generating function under a very mild condition on the bivariate random variables which resembles the Stein-condition, namely that a certain random variable $V(X_1, Y_1)$, defined in (32), is not identically equal to 0. Finally the asymptotic normality of the logarithm of the likelihood ratio of the rank order is established using the well-known Chernoff-Savage Theorem.

In a recent paper R. M. Dudley [2] obtained very interesting results on the speed of mean Glivenko-Cantelli convergence where the underlying random variables are supposed to be independent identically distributed (i.i.d.) taking their values in a separable metric space. In the present paper we shall show that his method of proof also applies for the case of Markov processes with stationary transition probabilities fulfilling Doeblin's condition $(D_0)$. The main additional tool in the treatment of this more general case is the determination of an appropriate upper estimate for the variance of the empirical $p$-measure for the transitions performed in a given set (Lemma 3.2 and Corollary 3.3 (i)) using a mixing property (Lemma 3.1). As indicated by Dudley, the results obtained in this way are applicable to problems in testing statistical hypotheses. An application concerning the speed of convergence of asymptotically normal estimates for Markov processes will be given in a separate paper [3].

Let $x_1, x_2, \cdots, x_m$ be a random sample from a $p$-dimensional random variable $X = (X_1, X_2, \cdots, X_p)$ with probability distribution $P$. It is assumed that $P$ is absolutely continuous with respect to Lebesgue measure, and that the corresponding probability density function is denoted by $f$. If $z = (z_1, z_2, \cdots, z_p)$ is a point at which $f$ is both continuous and positive, an estimator for $f(z)$ based on statistically equivalent blocks is suggested and its consistency is shown. This estimator grew out of work on the nonparametric discrimination problem. Fix and Hodges [2] showed how density estimation could be used in this problem and demonstrated a consistent estimator at points such as $z$. Loftsgaarden and Quesenberry [4] proposed another estimator which is consistent at points such as $z$; their estimator was based on statistically equivalent blocks. Although this estimator is easier to use in practice than that suggested by Fix and Hodges, it does require separate calculations if the sample is to be used to estimate the density at two or more points, and gives complex regions on which the estimate is constant if it is desired to estimate $f$ on some subset of the entire space. The estimator suggested in this paper is consistent at all points at which the two above estimators are consistent and allows the investigator to estimate the density at every point of $p$-dimensional Euclidean space from one construction, as well as providing rectangular regions on which the estimate is constant.

Let $X_1, X_2, \cdots$ be independent identically distributed random variables having a common distribution function $F$ and let $f_n(x) = (na_n)^{-1} \sum^n_{i = 1} k((x - X_i)/a_n)$ where $\{a_n\}$ is a sequence of positive numbers converging to zero and $k$ is a probability density function. If $\sum^\infty_{n = 1} \exp(- cna_n^2)$ is finite for all positive $c$ and if $k$ satisfies: (i) $k$ is continuous and of bounded variation on $(-\infty, \infty)$. (ii) $uk(u) \rightarrow 0$ as $u \rightarrow + \infty$ or $-\infty$. (iii) There exists a $\delta$ in (0, 1) such that $u(V^{-u^\delta}_{-\infty} (k) + V^\infty_{u^\delta} (k)) \rightarrow 0$ as $u \rightarrow \infty$. (iv) $\int|u| dk(u)$, the integral of $|u|$ with respect to the signed measure determined by $k$, is finite. Then the author [2] has established the following: THEOREM. A necessary and sufficient condition for $\lim_{n\rightarrow\infty} \sup_x|f_n(x) - g(x)| = 0$ with probability one for a function $g$ is that $g$ be the uniformly continuous derivative of $F$. The purpose of this note is to show that this theorem remains true if conditions (i)-(iv) on $k$ are replaced by the condition that $k$ is of bounded variation on $(-\infty, \infty)$.

For statistical problems based on groups, it is shown that there is convergence in probability to an invariant posterior distribution if and only if the posterior corresponds to right-invariant Haar prior and the group can support an asymptotically right-invariant sequence of proper prior distributions.

The distributions considered in this paper have the probability density functions ($\operatorname{pdf's})$ \begin{equation*}\tag{1.1}\frac{(1 - x^2)^{\frac{1}{2}(n - 1)}(1 + r^2 - 2rx)^{-\frac{1}{2}n}}{B\lbrack\frac{1}{2}(n + 1), \frac{1}{2}\rbrack}, \quad -1 \leqq x \leqq 1,\end{equation*} and \begin{equation*}\tag{1.2}\frac{(n + 1)(1 - x^2)^{\frac{1}{2}n - 1}(1 + r^2 - 2rx)^{\frac{1}{2}(1 - n)}(1 - x)}{(n - nr + 1 + r)B(\frac{1}{2}n, \frac{1}{2})}, \quad -1 \leqq x \leqq 1,\end{equation*} and are respectively the Madow-Leipnik approximation distribution for the serial correlation coefficient, circularly defined with known non-null mean, and Daniels' (1956) modified approximation distribution for the coefficient, circularly defined with fitted mean. (Note correction of misprints in (1.2).) General expressions for the uncorrected moments (u.m's) of (1.1) were given by Kendall (1957) and by White (1957); these may also be obtained from Leipnik's (1958) Neumann-type series for the characteristic function (ch.f.). They have the form of polynomials in $r$, with coefficients involving the Hermite polynomial coefficients. The first four central moments (c.m's) of (1.1) have been derived from the u.m's by Jenkins (1956), Kendall (1957) and White (1957). General expressions for the u.m's and c.m's of (1.2) do not seem to appear in the literature. In this paper we consider firstly the u.m's of (1.1) about $x = 1$; these are shown to be proportional to Gaussian hypergeometric functions, and to lead to representations of the c.m's and of the ch.f. by hypergeometric functions in two variables. The u.m's of (1.2) about $x = 1$ are closely related to those of (1.1), yielding corresponding formulas for its c.m's and ch.f.