The main purpose of the paper is to prove a combinatorial extension of a recent theorem of Baxter [6] concerning partial sums of random variables. The method of proof is closely related to that used by Sparre Andersen [1] in his basic 1949 paper which initiated the combinatorial approach to problems in fluctuation theory. The method does not seem to have been used since. The central idea behind Sparre Andersen's method, as it is used below, is conceptually very simple. It consists of verifying the validity of two operations on the finite sequences of real numbers under consideration. The first operation is referred to as "shrinking", and the second as "counting". In order to prove an invariant combinatorial result for such finite sequences of numbers, one shows first that if the result is true for a given sequence, it remains true as one decreases (or shrinks) the smallest number in the sequence (the shrinking Lemma 2.1) and, second, that if one inserts a sufficiently small number into a sequence for which the theorem holds, then the result also holds for the new sequence (the counting Lemma 2.2). Section 2 contains the fundamental combinatorial theorem concerning the joint behavior of the number of partial sums greater than zero and the number of them less than the last partial sum. Section 3 presents a probabilistic framework for the results of Section 2, as well as some further results.

In general, the class of mixtures of the family of normal distributions or of Gamma (Type III) distributions or binomial distributions is not identifiable (see [3], [4] or Section 2 below for the meaning of this statement). In [4] it was shown that the class of all mixtures of a one-parameter additively-closed family of distributions is identifiable. Here, attention will be confined to finite mixtures and a theorem will be proved yielding the identifiability of all finite mixtures of Gamma (or of normal) distributions. Thus, estimation of the mixing distribution on the basis of observations from the mixture is feasible in these cases. Some separate results on identifiability of finite mixtures of binomial distributions also appear.

The distribution of the linear discriminant function $W$, Anderson's classification statistic (1951), is investigated by several authors: Bowker (1960), Bowker and Sitgreaves (1961), Sitgreaves (1952, 1961), etc. Since the exact distribution is too complicated to be used numerically, as indicated by Sitgreaves (1961), we present here an asymptotic expansion of the distribution with respect to three numbers $N_1, N_2$ and $n$ representing degrees of freedom. This is a generalization of the result of Bowker and Sitgreaves who deal with a special case where $N_1 = N_2 = N$ and $n = 2N - 2$.

Investigations have been made in the past by several people on the possibility of extending the content of the classical central limit Theorem when the basic random variables are no longer independent. Several interesting extensions have been made so far. Hoeffding and Robbins (1948) have established asymptotic normality for the distribution of mean of a sequence of $m$-dependent random variables, where $m$ is a finite positive constant. The result has been proved by Diananda (1953) under more general conditions and has been extended to cover situations where the random variables $\{X_t\} (t = 1, 2, 3, \cdots)$ are of the type $X_t - E(X_t) = \sum^\infty_{j = 0} g_j Y_{t-j}$ where $\{Y_t\} (t = 0, \pm 1, \cdots)$ is an $m$-dependent stationary process and $E(Y_t) = 0,\quad\sum^\infty_{j = 0} g_j < \infty$. Walker (1954) has established asymptotic normality for the distributions of serial correlations based on $X_t$ of the above form. However, so far no attempt has been made to investigate whether the type of asymptotic expansions as discussed by Cramer (1937), Berry (1941) and Hsu (1945) for the distributions of means of independent random variables could also be extended to apply to situations where the random variables are not independent. Chanda (1962) has made a start in this direction, but the results are understandably incomplete. An attempt has been made in this paper to investigate this problem more systematically. The conclusion is that an extension is possible under conditions precisely similar to those under which Cramer, Berry and Hsu proved their results.

In comparison with the vast number of distribution functions available for describing non-normal populations (see Haight [11]) a very few sampling distributions of the mean, when sampling from these, are known in exact form. The more important results have been derived by Baker [1], [2], Baten [3], [4], Bose [6], Church [7], Hall [12], Irwin [13], [14], Rao [16] and Shrivastava [18]. Except in the case of the well-known result which follows when sampling from a Pearson's Type III population, many of these are of exceptional form and hardly any of them have been tabulated. Furthermore, since most of these results are only of practical use in the case of very small sample sizes, the need for approximations to the sampling distribution of the mean have been felt. In the case of the normal approximation, which has been widely used in many cases in virtue of the Central Limit Theorem, Berry [5] has shown that it is not sufficient for moderate sample sizes. For the latter case, Esseen [9] has obtained approximations in terms of the normal distribution and its derivatives. These results have been extended by Gnedenko and Kolmogorov [10]. Other forms of approximations have been obtained by Daniels [8] and Welker [21]. Another approach to find approximations to fill the gap between the exact sampling distribution and its ultimate normal approximation is presented in this paper. A method developed by Steyn [19] in deriving a differential equation of the moment generating function of the sample mean and variance respectively for samples from a normal population, is used in deriving approximations to sampling distributions of the mean for samples from a number of Pearson's Type populations. Only first order approximations are considered and it will be shown that in the case of sampling from certain skew populations, the results lead to the Pearson's Type III-distribution before approaching normality, whereas, in the symmetrical cases the sampling distribution approaches normality directly.

The problem of estimating desired characteristics of a response $X$, where $X = h(y)$ and $y$ is a vector random variable with statistically independent components from known distributions, may be handled by standard Monte Carlo techniques. We are interested in the generalization where several distributions are of interest, one at a time, for each component of $y$ and the desired characteristics of $X$ must be estimated for each combination of component distributions. A relatively small number of observations, compared to the total number required if each combination were posed as a separate Monte Carlo problem, may be used instead by sampling from a fictitious distribution, calculating an estimate by appropriately weighting the observations, and then reusing the sample. These techniques are standard; the contribution here is to find the fictitious distribution which is best for the characteristics desired, the distributions of interest, and allied considerations. Various concepts of the meaning for "best" are examined in the paper. Finally, a quantitative evaluation is made under restricted conditions.

Suppose a single contrast $y = \sum c_j u_j$, where $\sum c_j = 0$, is to be tested as a basis for detecting differences among unknown parameters $\mu_j$, where $y_j = \mu_j + \epsilon_j$, and the $\epsilon_j$ are independent and normally distributed with mean zero and variance $\sigma^2$. Write $\mu_j = \alpha + \beta x_j$. Then the problem is to detect $\beta \neq 0$. If $\sum x_j = 0$, and $\sum x^2_j = 1$, the noncentrality of $y$, referred to its standard deviation, is $(\beta/\sigma)$ times the formal correlation coefficient $r$ between the $c_j$ and the $x_j$. If the $x_j$ are known, the $c_j$ can be chosen to make the correlation unity. If the $x_j$ are wholly unknown, no single contrast can guarantee power in detecting $\beta \neq 0$. Intermediate situations, where we know something but not everything about the $x_j$, occur frequently. If our knowledge can be placed in the form of linear inequalities restricting the $\mu_j$ (equivalently the $x_j$) the problem of choosing a contrast $\{c_j\}$ which will give relatively good power against the unknown (latent) configuration $\{x_j\}$ is a relatively manageable one. The problem is to obtain a large value of $r^2$ between $\{c_j\}$ which is at our choice, and $\{x_j\}$, which is only partially known. A conservative approach is to try to select the $\{c_j\}$ so that the minimum value of $r^2$ compatible with the restrictions on $\{x_j\}$ is maximized, or nearly so. The maximization of minimum $r^2$ when response patterns are constrained by linear homogeneous inequalities leads to the mathematical problem of finding the geometric direction whose maximum angle with a given set of directions is least. The solution to this problem is characterized and proven unique (Sections 8, 17-20). No useful algorithm which is absolutely certain to reach the solution in a few steps appears to exist. However, procedures are discussed (Sections 10 and 11) which reach a solution relatively rapidly in the instances we have considered. The procedures are illustrated on selected examples (Sections 15-16). The general theory is applied (Sections 13-14) to the latent configuration defined by $x_1 \leqq x_2 \leqq x_3 \leqq \cdots \leqq x_n$, which we call simple rank order. A formula is found for the maximum contrast which maximizes minimum $r^2$, and its coefficients are given for $n \leqq 20$. The "linear-2-4" contrast, constructed from the usual linear contrast by quadrupling $c_1$ and $c_n$, and doubling $c_2$ and $c_{n-1}$, is a reasonable approximation to the maximum contrast for small or medium $n$, and its minimum $r^2$ remains above 90{\tt\%} of the maximum possible for $n \leqq 50$ (Table 2). Knowing only simple rank order for the $\mu_j$, good practice seems to indicate the use of "maximum" or "linear-2-4" contrasts in careful work. If more information or insight about the $x_j$ is available, some other contrast may be preferable.

The empirical Bayes approach is described in Section 2. In Section 3 "optimal" empirical Bayes rules are given for the problem of testing a simple hypothesis against a simple alternative. In Section 4 a limit theorem is proved which is used in Section 5 to obtain "optimal" empirical Bayes rules for testing one and two-sided hypotheses about the parameters in the Poisson, geometric, negative binomial and binomial distributions. The same methods are used in Section 6 to obtain "optimal" empirical Bayes rules for testing hypotheses about parameters in continuous distributions of the exponential family. Examples of areas of applications are given in Section 7, and the last Section discusses uses of the above methods in the compound decision problem.

Doob (1949) obtained a very general result on the consistency of Bayes' estimates. Loosely, if any consistent estimates are available, then the Bayes' estimates are consistent for almost all values of the parameter under the prior measure. If the parameter is thought of as being selected by nature through a random mechanism whose probability law is known, Doob's result is completely satisfactory. On the other hand, in some circumstances it is necessary to identify the exceptional null set. For example, if the parameter is thought of as fixed but unknown, and the prior measure is chosen as a convenient way to calculate estimates, it is important to know for which null set the method fails. In particular, it is desirable to choose the prior so that the null set is in fact empty. The problem is very delicate; considerable work [8], [9], [12] has been done on it recently, in quite general contexts and under severe regularity assumptions. It might therefore be of interest to discuss the simplest possible case, that of independent, identically distributed, discrete observations, in some detail. This will be done in Sections 3 and 4 when the observations take a finite set of possible values. Under this assumption, Section 3 shows that the posterior probability converges to point mass at the true parameter value among almost all sample sequences (for short, the posterior is consistent; see Definition 1) exactly for parameter values in the topological carrier of the prior. In Section 4, the asymptotic normality of the posterior is shown to follow from a local smoothness assumption about the prior. In both sections, results are obtained for priors which admit the possibility of an infinite number of states. The results of these sections are not entirely new; see pp. 333 ff. of [7], pp. 224 ff. of [10], [11]. They have not appeared in the literature, to the best of our knowledge, in a form as precise as Theorems 1, 3, 4. Theorem 2 is essentially the relevant special case of Theorem 7.4 of Schwartz (1961). In Sections 5 and 6, the case of a countable number of possible values is treated. We believe the results to be new. Here the general problem appears, because priors which assign positive mass near the true parameter value may lead to ridiculous estimates. The results of Section 3 (let alone 4) are false. In fact, Theorem 5 of Section 5 gives the following construction. Suppose that under the true parameter value the observations take an infinite number of values with positive probability. Then given any spurious (sub-)stochastic probability distribution, it is possible to find a prior assigning positive mass to any neighborhood of the true parameter value, but leading to a posterior probability which converges for almost all sample sequences to point mass at the spurious distribution. Indeed, there is a prior assigning positive mass to every open set of parameters, for which the posterior is consistent only at a set of parameters of the first category. To some extent, this happens because at any stage information about a finite number of stages only is available, but on the basis of this evidence, conclusions must be drawn about all states. If the prior measure has a serious prejudice about the shape of the tails, disaster ensues. In Section 6, it is shown that a simple condition on the prior measure (which serves to limit this prejudice) ensures the consistency of the posterior. Prior probabilities leading to posterior distributions consistent at all and asymptotically normal at essentially all (see Remark 3, Section 3) parameter values are constructed. Section 5 is independent of Sections 3 and 4; Section 6 is not. Section 6 overlaps to some extent with unpublished work of Kiefer and Wolfowitz; it has been extended in certain directions by Fabius (1963). The results of this paper were announced in [5]; some related work for continuous state space is described in [3]. It is a pleasure to thank two very helpful referees: whatever expository merit Section 5 has is due to them and to L. J. Savage.

Suppose a $p \times p$ dispersion matrix $\mathbf{T}$ is considered to have the Wishart distribution $W (\Sigma, n)$, c. f. Anderson (1958) p. 158, where $\Sigma$ is an arbitrary full rank covariance matrix and $n \geqq p$. Suppose $\mathbf{T}$ is observable but $\Sigma$ is unknown, and suppose a posterior distribution is to be assigned to $\Sigma$ given $\mathbf{T}$, where the term posterior is meant in a wide sense to allow the use of a Bayesian or fiducial or any other form of reasoning in arriving at the posterior distribution. A lemma is proved in Section 3 giving a property of all such posterior distributions which possess a natural linear invariance property. The paradoxical nature of this property is discussed in Section 4.

The basic problem is to decide none of $k$ experimental categories is better than the control or decide a certain category is better. For this problem three sequential procedures are examined with specification of how procedures are carried out in practice. With a definite loss function and a cost $c > 0$ per observation the three sequential procedures and fixed sample size procedures are compared in a certain asymptotic sense as $c \rightarrow 0$. In particular, one of the procedures is shown to be optimal in this asymptotic sense. By appealing to asymptotic results a discussion of the relative merits of the three sequential procedures as considered in practice is given.

As an alternative to the standard analysis of variance, procedures are developed for linear models with several observations per cell which are asymptotically distribution-free and whose asymptotic efficiency relative to the standard procedures is the same as that of the Wilcoxon test relative to Student's $t$-test. Specific procedures discussed are (i) tests of linear hypotheses, (ii) confidence intervals for any contrast, (iii) simultaneous intervals for all contrasts.

In this paper we prove the asymptotic normality of two statistics which have been proposed to test the hypothesis that two samples come from the same parent population. One statistic is the number of runs of $X$'s and $Y$'s in the combined sample of $X$'s and $Y$'s; the other is the sum of squares of "$S_i$'s" where $S_i$ is the number of $X$'s falling between the $i$th and $(i - 1)$st largest $Y$'s. Both statistics have been studied previously, both lead to consistent tests, and both were known to be asymptotically normal under the null distribution. Here we prove limiting normality under a fairly wide class of alternatives. By means of limiting power against a sequence of alternatives which approach the null hypothesis, we compare these tests with one another and with the Smirnov test based on the sample c.d.f.'s. Against a rather large class of alternatives, the Smirnov test is seen to be considerably more powerful. The method of proving limiting normality used here is based on studying conditional moments and can be used to prove limiting normality of "combinatorial" statistics other than the ones studied herein.

In the first nontrivial case, dimension $p = 2$ and sample size $N = 3$, it is proved that Hotelling's $T^2$ test of level $\alpha$ maximizes, among all level $\alpha$ tests, the minimum power on each of the usual contours where the $T^2$ test has constant power. A corollary is that the $T^2$ test is most stringent of level $\alpha$ in this case.

Existence and uniqueness questions concerning SPRT's for testing one distribution against another in the case of independent and identically distributed random variables are considered under general conditions. Tests that need not take an observation $(n \geqq 0)$, as well as tests that are required to take at least one observation $(n \geqq 1)$ are being considered. SPRT's are allowed to have arbitrary stopping rule at the stopping bounds. The error point $\alpha$ of a test is the vector of error probabilities, and $A(u, v)$ is the set of all error points of SPRT's with stopping bounds $u, v(u \leqq v)$. For given $u < v$ there are four nonrandomized SPRT's (three if $u = v$), where $u$ may be a stopping point or a continuation point, and similarly $v$. It is shown in Section 4, Theorem 6, that the error points of these four tests are the extreme points of the convex set $A(u, v)$. A special mixture of these four tests is denoted $R(s, t)$, its error point $\alpha(s, t)$, where $s = (u, \lambda)$ and $t = (v, \mu)$ are the randomized stopping bounds, and $s, t$ may be considered points in an ordered topological space $Z$. Let $D = \{\alpha: \alpha_i \leqq \alpha_i(s, s)$ for some $s \varepsilon Z$ and $i = 1, 2\}$. Using the characterization of SPRT's as Bayes tests, it is shown in Theorem 2 that there is a SPRT with given error point $\alpha^\ast$ if and only if $\alpha^\ast_1 + \alpha^\ast_2 \leqq 1$ in the case $n \geqq 0$, and $\alpha^\ast \varepsilon D$ in the case $n \geqq 1$. In Theorem 1 a somewhat stronger result is claimed, namely that the SPRT in Theorem 2 can be taken to be a test of the form $R(s, t)$. The main tool in dealing with the uniqueness question is the monotonicity Theorem 3, stating that if of a test $R(s, t) s$ is decreased or $t$ increased, and the new $s, t$ have positions $u, v$, then the change $\Delta\alpha$ in the error point satisfies $u\Delta\alpha_1 + \Delta\alpha_2 \leqq 0$ and $\Delta\alpha_1 + \Delta\alpha_2/v \leqq 0$, with at least one strict inequality unless the new test is equivalent to the old one. Theorem 4 says that a test of the form $R(s, t)$ with given error point is unique up to an equivalence. On the other hand, a SPRT is in general not unique. Theorem 5 claims only uniqueness of the stopping bounds $u, v$, up to an equivalence, in the case of a SPRT with given error point.

Second-order stationary random sequences, especially if Gaussian, may be of statistical interest because their covariances may, under certain conditions, be consistently estimated with finite realizations of the sequences. It is shown that there is a class of non-stationary random sequences, namely sequences of orthogonal random variables with zero means and variances $f_t$ which form a uniformly almost periodic sequence, which are of statistical interest, at least in the Gaussian case, in the following sense. $f_t$ admits a generalized Fourier series expansion, and the Fourier coefficients $\gamma_s$ of this expansion can be consistently estimated with finite realizations of the sequences. In certain situations, the nonconstant variance sequence $f_t$ may be directly estimated. The sequences of interest may be converted, via Fourier transformations, into second-order stationary random functions, and the Fourier coefficients $\gamma_s$, of the expansion of $f_t$, are shown to form a sequence of stationary covariances. A multiplicative representation is given for the nonstationary sequences considered.

While the analysis of variance test statistic $F$ in a balanced incomplete block design without randomization is a constant under the Fisher model, i.e., a linear model without technical errors, and it has no distribution at all, under the Neyman model, i.e., a model with technical errors, its null-distribution is a non-central $F$-distribution whose non-centrality parameter being a quadratic form of unit-errors. This is carried out in Section 1. The mean value and the variance of $\theta$ with respect to the permutation distribution due to the randomization are calculated in Section 2, and in Section 3, the null-distribution of the $F$-statistic after the randomization is shown to be approximated by the familiar central $F$-distribution under the Neyman model assuming no interaction between treatments and experimental units, if the following two conditions are satisfied: (i) the variances of unit-errors within blocks are sufficiently uniform from block to block, and (ii) the number of blocks is sufficiently large. Since the unit-errors are not directly observable, how one can group the experimental units into blocks in such a way as the above Condition (i) would be satisfied is another problem, which is left open in this paper. R. A. Fisher [2] initiated the use of the so-called "randomization procedure" in order to control the unit-errors in block designs. Mathematical treatments of the Fisher randomization in randomized block and the Latin-square designs were made by B. L. Welch [12], E. J. G. Pitman [11] and M. B. Wilk [14]. Underlying models in those works may be called the "Fisher models", i.e., containing no technical errors. J. Neyman et al. [7] and M. B. Wilk [13] pointed out that there are instances in which a model with technical errors is more adequate by the very nature of the problem under consideration, and the present author calls this sort of models the "Neyman models" for convenience. M. D. McCarthy [6] investigated the null-distribution of the analysis of variance test statistic in a randomized block design under the Neyman model, and he came out with rather pessimistic results. J. Ogawa [10] treated the same problem, and his result turned out to be supporting the usual approximation by the familiar central $F$-distribution. The purpose of this article is the treatment of the same null-distribution problem for a randomized balanced incomplete block design under the Neyman model. Since a randomized block design is a limiting case of a randomized BIBD, this article should be regarded as a generalization of the earlier work.

The present paper envisages a generalized situation of the balanced incomplete block design in the sense of allowing for the sampling of sources of experimental material, of blocks within sources, of experimental units within blocks, and of treatments under consideration. A model for an arbitrary observation of a generalized balanced incomplete block design is derived explicitly from the physical way in which the experiment is performed, i.e., from the way in which the sampling and randomization procedures are carried out. The correlational structure of the observations is therefore implicit in the model. The model initially uses no assumptions of additivity of treatments with experimental material. It is shown that expected values of squares of partial observational means, as well as the expected values of products of individual observations, admit simple and easily specifiable expressions in terms of quantities called cap sigmas and denoted by $\Sigma$'s. The expected values of mean squares in the analysis of variance table are then derived. Consequences of the presence of various types of nonadditivity on the usual test of no treatment effects are discussed for fixed, mixed and random situations. For example, when the blocks actually used in the experiment form a random sample from an infinite population of blocks then the presence of interactions of blocks with treatments produces no bias in the comparison of the error and adjusted treatment mean squares. The correlational structure of the observations under the simplifying additivity assumption is examined for the standard balanced incomplete block design. It is shown that the usual forms of estimators of treatment comparisons are appropriate and that the $\Sigma$'s play the roles which the block and plot variances have in the corresponding assumed infinite model. In the presence of nonadditivities of treatments with the experimental material the usual forms of the linear estimators are no longer best.

The distribution of the non-homogeneous quadratic form $Q = \sum^n_1 a_i(x_i - b_i)^2$, where the $x_i$ are independent standardized normal variables and the $a_i$ and $b_i$ are real constants with $a_i > 0$, has recently [1] been obtained as an infinite linear combination in scaled central and noncentral $\chi^2$ distribution functions in the form \begin{equation*}\tag{1.1}P\lbrack Q \leqq t\rbrack = \sum^\infty_0 c_j(p) F_{n+2j}(t/p) = \sum^\infty_0 d_j(p) G_{n+2j;k}(t/p).\end{equation*} Here $p$ is an arbitrary positive constant, $F_{n + 2j}(\cdot)$ is the distribution function of $\chi^2$ with $n + 2j$ degrees of freedom and $G_{n + 2j;k}(\cdot)$ is the distribution function of $\chi^2$ with $n + 2j$ degrees of freedom and non-centrality parameter $\kappa = (\sum^n_1 b^2_i)^{\frac{1}{2}}.$ The main purpose of the present paper is to rederive the first of the two expansions in (1.1), for the special case $p \leqq \min_ia_i$ when the expansion is a proper mixture representation, by a simple conditional probability argument which may be of some general interest. At the same time the $c_j(p)$ will be expressed in simpler and more appealing form than in [1]. In essence, the distribution of $Q$ (including that of $\sum^n_1 a_ix^2_i$) is found to be almost a direct consequence of the distribution of the special non-homogeneous form $\sum^n_1 (x_i - b_i)^2$, that is, of non-central $\chi^2$ with $n$ degrees of freedom and non-centrality parameter $(\sum^n_1 b^2_i)^{\frac{1}{2}}.$ Specifically, the distribution of $Q$ can be expressed as a weighted non-central chi-square in the sense that the non-centrality parameter is not fixed but is rather a random variable with a given distribution depending on the $a_i$ and $b_i$.

Wald's Fundamental Identity in sequential analysis [15] has been widely used for various applications apart from sequential sampling. Bartlett [1] used it for the insurance risk problem and also for the random walk and gambler's ruin problem. It is the purpose of this paper to show how the Fundamental Identity can be used to derive certain results in Dam Theory. The paper is purely of an expository nature and considers only simple examples; more difficult problems will be dealt with in a future paper. We consider here a continuous time dam model due to Gani and Prabhu ([5], [6], [7]), based on Moran's ([10], [11]) discrete time model. Briefly it is as follows: (a) The dam has finite capacity $K$. (b) Let $X(T)$ represent the input (including the amount overflowed, if any) during a time interval of length $T$; we assume that $X(T)$ is an additive process with stationary increments. It is known that for such processes the m.g.f. is given by $E\{e^{\theta X(T)}\} = e^{-T\xi(\theta)}$ where $\xi(\theta)$ is a function of a specified type (Levy [9]). (c) The release is continuous and occurs at a unit rate except when the dam is empty. It then follows that the net input (including the amount overflowed, if any) in the dam during the time interval $(0, T)$ is $Y(T) = X(T) - T$ whose m.g.f. is $M_T(\theta) = e^{-\theta T-T\xi(\theta)}$. For $T = 1$ this gives $M_1(\theta) = e^{-\theta-\xi(\theta)}$. Let us denote the dam content at time $t$ by $Z(t)$, with the initial content $Z(0) = u (0 < u < K)$; we have, then $Z(t) = u + X(t) - t$. It follows from the above assumptions that the stochastic process $Z(t)$ is a temporally homogeneous Markov process. Following Moran's first paper, attention was concentrated mainly on investigating the stationary properties of the system. Later, Kendall [8], Gani [4], Prabhu [13] and Weesakul [16] considered the problem of emptiness of the dam; it is with this second problem that the present paper is concerned. Considering the dam process described above as a random walk with barriers at $Z = 0$ and $Z = K$, the process starting at $Z(0) = u$, we obtain the probability that the dam becomes empty (i.e. the process terminates at 0) before it overflows (i.e. the process terminates at $K$) and the probability of the reverse situation; further, we derive the probability distribution of the time at which the dam becomes empty. We do this by making use of the following extension of Wald's Fundamental Identity to continuous time parameter by Dvoretzky, Kiefer and Wolfowitz [3]. If $\{Y(t); T \geqq 0\}, Y(0) = 0,$ is a process with stationary independent increments and if $M_1(\theta) = E\lbrack e^{\theta Y(1)\rbrack}$ exists for all $\theta$, then \begin{equation*}\tag{1.1}E\lbrack e^{\theta Y)t + 1)} M_1(\theta)^{-t}\rbrack = 1,\end{equation*} where $t$ is such that $Y(t) \geqq b$ or $Y(t) \leqq a$, while $a < Y(\tau) < b \text{and all} \tau < t$. (Here $a$ and $b$ are constant, $a < 0, b > 0$.) We also require a lemma due to Bartlett [1]. Lemma. Let $Y$ be a random variable such that: (a) $E(Y)$ exists and is not equal to zero; (b) $M(\theta) = E\lbrack^{Y\theta}\rbrack$ exists for $\theta$ in a finite interval (c) $M(\theta) \rightarrow \infty$ as $\theta \rightarrow \pm \infty$. Then there exists one and only one real $\theta_0 \neq 0$ such that $M(\theta_0) = 1$. Since the Identity as stated above is applicable only when the process starts from zero, we put $b = K - u$, and $a = -u$. Applying the above lemma to the input process of the dam model, we find that there exists a nonzero real solution $\theta_0$ of the equation \begin{equation*}\tag{1.2}\theta + \xi(\theta) = 0;\end{equation*} we then have, putting $\theta = \theta_0$ in (1.1) \begin{equation*}\tag{1.3}E\lbrack^{\theta_0Y(t + 1)}\rbrack = 1.\end{equation*} If the boundaries $a$ and $b$ are exactly reached at stage "$t$" we put $Y(t) = a$ with probability $P_a$ and $Y(t) = b$ with probability $P_b = 1 - P_a$; then (1.3) gives $P_ae^{a\theta_0} + (1 - P_a)e^{b\theta_0} = 1,$ so that \begin{equation*}\tag{1.4}P_a = (1 - e^{b\theta_0})/(e^{a\theta_0} - e^{b\theta_0}).\end{equation*} Using (1.3) the characteristic function (c.f.) of the time "$t$" at which the dam becomes empty for the first time before touching the barrier $K$ can also be determined. It is known that the eqauation $\theta + \xi(\theta) = i\phi$ has two roots $\theta_1(\phi)$ and $\theta_2(\phi)$ such that $\theta_1(\phi) \rightarrow 0$ and $\theta_2(\phi) \rightarrow \theta_0$ as $\phi \rightarrow 0$. Hence, in the case where the boundaries "$a$" and "$b$" are exactly reached, we obtain from (1.1) \begin{align*} P_ae^{a\theta_1(\phi)} C_a(\phi) &+ P_be^{b\theta_1(\phi)} C_b(\phi) = 1,\\ \tag{1.5} \\ P_ae^{a\theta_2(\phi)} C_a(\phi) &+ P_be^{b\theta_2(\phi)} C_b(\phi) = 1,\end{align*} where $C_a(\phi)$ is c.f. of $t$ conditional on $Y(t) = a$, and $C_b(\phi)$ is defined similarly, so that $E\lbrack^{i\phi t}\rbrack = P_aC_a(\phi) + P_bC_b(\phi)$. Equations (1.4) and (1.5) give $P_aC_a(\phi)$, the c.f. of "$t$" the time at which the dam becomes empty for the first time without overflowing, (cf. Bartlett [1] pp. 18-19). Before we use the Fundamental Identity in any problem let us consider the advantages of this method. The main advantage is its simplicity and easy applicability. Moreover, by a slight modification we can derive the more important probability distribution of the times of emptiness, no matter how often the dam has overflowed in the meantime. In random walk terminology this means that we have an absorbing barrier at 0 and a reflecting barrier at $K$. Another advantage of this method is that it can be used even when the inputs are not independent, as assumed in the above model, but Markovian. In this case the extension of Wald's Fundamental Identity to Markov Processes (Bellman [2], Tweedle[14], Phatarfod [11]) has to be applied; this problem will be considered in a future paper.

Heretofore, the ordinary Wilcoxon statistic for the two-sample problem [9], [5] has been used only to test the null hypothesis that the two parent populations are identical. This paper presents a technique for utilizing the Wilcoxon statistic to test a broader type of null hypothesis, like that encountered in the Behrens-Fisher problem: we show that the usual Wilcoxon test, with $(m + n + 1)/12mn$ replaced by $1/\lbrack 4 \min(m, n)\rbrack$, may be used to test the null hypothesis of the equality of the medians of two symmetrical (continuous) distributions which are of the same form but which have different (unknown) scale parameters; more generally, the test still works for testing the equality of the medians of any two symmetrical distributions.