In this paper, we relate properties of a distribution function $F$ (or its density $f$) to properties of the corresponding hazard rate $q$ defined for $F(x) < 1$ by $q(x) = f(x)/\lbrack 1 - F(x)\rbrack$. It is shown, e.g., that the class of distributions for which $q$ is increasing is closed under convolution, and the class of distributions for which $q$ is decreasing is closed under convex combinations. Using the fact that $q$ is increasing if and only if $1 - F$ is a Polya frequency function of order two, inequalities for the moments of $F$ are obtained, and some consequences of monotone $q$ for renewal processes are given. Finally, the finiteness of moments and moment generating function is related to limiting properties of $q$.

Many problems in the theory of decision processes give rise to an unending sequence of cycles whose lengths are given by a sequence of independent and identically distributed positive random variables $X_1, X_2, \cdots$, constituting a renewal process. Typically $X_i$ will represent the number of items passing an inspection point in a production process or the length of time elapsing in some continuous time process until a decision point is reached. In many such problems another sequence of random variables, say $Y_1, Y_2, \cdots$, arises where $Y_i$ is the profit or loss associated with the $i$th cycle. Examples of problems having this structure may be found, for instance, in [1], [3], [6], [9] and [10]. In most cases $X_i$ and $Y_i$ will not be independent for the same index $i$ and in this note we will permit $Y_i$ to depend on $X_{i-q}, \cdots, X_{i-1}, X_i$ for any fixed finite $q$. In all such problems the appropriate index of merit for the decision procedure under consideration is the "average" profit (or loss) per unit time (or per item) for a large number of cycles. The notion of "average" profit rate can be mathematically defined in four distinct and apparently equally plausible ways, and it is the purpose of this note to show that the various definitions are not necessarily equivalent and to determine the conditions under which they are equivalent. First, the profit rate up to time $t$ may be defined in terms of $N(t)$, the number of cycles completed by time $t$, as $(1/t) \sum^{N(t)}_{i=1} Y_i$. Then the average profit rate may be defined as either the limit of the expected value of the profit rate, or as the almost sure limit of the profit rate, as $t \rightarrow \infty$. The use of the expected value definition is consistent with the principles of utility theory and the notions of risk that underlie the formulation of problems in decision theory. The almost sure limit definition has, of course, considerable intuitive appeal. It is shown in this note that, as an immediate consequence of well-known results from renewal theory, both of the above definitions lead to the average profit rate $\eta/\mu$ where $EX = \mu$ and $EY = \eta$, assuming that $\eta$ is finite. Alternately, the profit rate may be defined for $n$ cycles as $\sum^n_{i=1} Y_i/ \sum^n_{i=1} X_i$ and again the average profit rate may be defined as either the limit of the expected values, or the almost sure limit of this ratio, as $n \rightarrow \infty$. By the strong law of large numbers the almost sure limit is always $\eta/\mu$ but in Section 2 an example is given for which the limit of the expected values is not $\eta/\mu$. In Section 3 necessary and sufficient conditions are obtained for the limit of the expected values of the ratios to be $\eta/\mu$ if $\eta$ is finite.

In the present paper we consider the one dimensional random walk of a particle restricted by a partially reflecting barrier. The reflecting barrier is described by a coefficient of reflection $r$. The probability of finding a particle at a lattice point $m$ after $N$ steps is calculated and expressed in terms of hypergeometric functions of the $_2F_1$-type. Other theorems are deduced concerning the one dimensional random walk. For instance the number of paths leading from one lattice point to another lattice point in $N$ steps and showing a given number of reflections at the barrier is calculated.

Let $F$ be a (nonempty) set of distribution functions (d.f.'s) of random variables (r.v.'s) with zero means and positive, finite variances. Let $\mathfrak{F}(F)$ be the set of all sequences of independent r.v.'s (independent within each sequence) whose d.f.'s belong to $F$ but are not necessarily the same from term to term of the sequence. No assumptions are made on the interrelations between the joint probability spaces of the r.v.'s of different sequences of $\mathfrak{F}(F)$. Accordingly, dependence or independence between r.v.'s of different sequences needs not be specified. A generic member of $\mathfrak{F}(F)$ will be denoted by $\epsilon = \{\epsilon_k; k = 1, 2, \cdots\}$, or, when we discuss sequences of members of $\mathfrak{F}(F)$, by $\epsilon(n) = \{\epsilon_{nk}; k = 1, 2, \cdots\}, n = 1, 2, \cdots$. In the following, $\mathfrak{F}(F)$ plays the role of a parameter space, the parameter points being $\epsilon, \epsilon(1), \epsilon(2), \cdots$. Since only the d.f.'s of a sequence of (independent) r.v.'s are relevant to the central limit theorem, instead of $\mathfrak{F}(F)$, the set of sequences of d.f.'s corresponding to the elements of $\mathfrak{F}(F)$ may also be regarded as the parameter space. It clearly is a map of $\mathfrak{F}(F)$. The inverse mapping subdivides $\mathfrak{F}(F)$ into certain equivalence classes. From elementary set theory it follows that $\mathfrak{F}(F)$ as well as its map has the cardinality of the continuum if $F$ contains more than one element. Further, let $\{a_{nk}; n = 1, 2, \cdots; k = 1, 2, \cdots\}$ be a double sequence of real constants, and let $\{k_n\}$ be a sequence of positive integers such that $a_{nk_n} \neq 0$ and $a_{nk} = 0$ for $k > k_n, n = 1, 2, \cdots$. We denote the variance of $\sum^{k_n}_{k=1} a_{nk}\epsilon_{nk}$ by \begin{equation*}\tag{1}B_n^2 = \sum^{k_n}_{k=1} a^2_{nk}\sigma^2_{nk}, \quad\sigma^2_{nk} = \operatorname{var} \epsilon_{nk},\end{equation*} and put \begin{equation*}\tag{2}\zeta_n = B_n^{-1} \sum^{k_n}_{k=1} a_{nk}\epsilon_{nk},\quad n = 1, 2, \cdots,\end{equation*} where $B_n = + (B_n^2)^{\frac{1}{2}}$. If all sequences $\epsilon(n)$ are identical we denote them by $\epsilon$ and write instead of (2) $\zeta_n(\epsilon) = B_n^{-1}(\epsilon) \sum^{k_n}_{k=1} a_{nk}\epsilon_k, \quad n = 1, 2, \cdots,$ which shows the dependence on the parameter $\epsilon$ more clearly; $B_n^2(\epsilon)$ is defined analogously to (1). This note deals with necessary and sufficient conditions on the set $F$ and on the double sequence $\{a_{nk}\}$ in order that the d.f.'s of the $\zeta_n$ tend to the standard normal d.f. (denoted by $\Phi(x)$) and that the $B^{-1}_na_{nk}\epsilon_{nk}$ are infinitesimal for every sequence of sequences $\epsilon(1), \epsilon(2), \cdots \varepsilon \mathfrak{F}(F)$ (Theorem 1). For the case of identical sequences $\epsilon(n) (= \epsilon)$, which is of particular interest in applications, a normal convergence theorem (Theorem 3) holding uniformly for $\epsilon$ on $\mathfrak{F}(F)$ is obtained under the same necessary and sufficient conditions as in Theorem 1. Theorem 3 yields, e.g., conditions for the asymptotic normality of the least squares estimators of the parameters in a linear regression with independent and not necessarily identically distributed errors whose d.f.'s are unknown but belong to a certain class $F$ (Eicker (1963)). From the necessity of the conditions it follows that these conditions are the best possible ones under the limited information about the error terms provided by the model assumptions. Frequently, when limit theorems for families of sequences of random variables are met in statistics and probability theory, the emphasis is on the uniformity of the convergence of the sequences with respect to the family parameter which assumes values in a given a priori set. (For an example, compare Parzen (1954), p. 38. That paper also cites some of the earlier publications on the subject.) The present note emphasizes not primarily the uniformity of the convergence, but the necessity of the conditions (including one on the parameter space $\mathfrak{F}(F))$ for the convergence on the parameter space. Accordingly, the set $F$ cannot be prescribed arbitrarily. The uniformity of the convergence on $\mathfrak{F}(F)$ is shown without difficulty to be implied by the ordinary convergence on $\mathfrak{F}(F)$. Another difference from earlier work may be seen in the particular structure of $\mathfrak{F}(F)$, which is comparable to an infinite-dimensional vector space. Although many of the earlier convergence theorems are (or can be) formulated for abstract parameter spaces, in the applications such as estimation theory these spaces are usually specialized to intervals on the line or in a finite-dimensional vector space.

This paper deals with linear regressions \begin{equation*}\tag{1.1}y_k = x_{k1}\beta_1 + \cdots + x_{kq}\beta_q + \epsilon_k, \quad k = 1, 2, \cdots\end{equation*} with given constants $x_{km}$ and with error random variables $\epsilon_k$ that are (a) uncorrelated or (b) independent. Let $E\epsilon_k = 0, 0 < E\epsilon^2_k < \infty$ for all $k$. The individual error distribution functions (d.f.'s) are not assumed to be known, nor need they be identical for all $k$. They are assumed, however, to be elements of a certain set $F$ of d.f.'s. Consider the family of regressions associated with the family of all the error sequences possible under these restrictions. Then conditions on the set $F$ and on the $x_{km}$ are obtained such that the least squares estimators (LSE) of the parameters $\beta_1, \cdots, \beta_q$ are consistent in Case (a) (Theorem 1) or asymptotically normal in Case (b) (Theorem 2) for every regression of the respective families. The motivation for these theorems lies in the fact that under the given assumptions statements based only on the available knowledge must always concern the regression family as a whole. It will be noticed moreover that the conditions of the theorems do not require any knowledge about the particular error sequence occurring in (1.1). Most of the conditions are necessary as well as sufficient, with the consequence that they cannot be improved upon under the limited information assumed to be available about the model. Since the conditions are very mild, the results apply to a large number of actual estimation problems. We denote by $\mathfrak{F}(F)$ the set of all sequences $\{\epsilon_k\}$ that occur in the regressions of a family as characterized above. Thus, $\mathfrak{F}(F)$ comprises all sequences of uncorrelated (Case (a)) or independent (Case (b)) random variables whose d.f.'s belong to $F$ but are not necessarily the same from term to term of the sequence. For each $G \varepsilon F$ the relations $\int x dG = 0$ and $0 < \int x^2 dG < \infty$ hold. In this paper, $\mathfrak{F}(F)$ may be looked upon as a parameter space. A parameter point then is a sequence of $\mathfrak{F}(F)$. Correspondingly, we say that a statement holds on $\mathfrak{F}(F)$ (briefly on $F$) if it holds for all $\{\epsilon_k\} \varepsilon \mathfrak{F}(F)$. The statements of Theorems 1 and 2 are of this kind. The proof of Theorem 1, as well as the proof of the sufficiency in Theorem 2, is elementary and straight forward. Theorem 2 is a special case of a central limit theorem (holding uniformly on $\mathfrak{F}(F)$) for families of random sequences [3]. Some similarity between the roles of the parameter spaces $\mathfrak{F}(F)$ in our theorems and of the parameter spaces that occur, e.g., in the Gauss-Markov and related theorems may be seen in the fact that these theorems remain true only as long as the conclusions in the theorems hold for every parameter point in the respective spaces. As is well known, the statements in the Gauss-Markov and related theorems hold for every parameter vector $\beta_1, \cdots, \beta_q$ in a $q$-dimensional vector space (see e.g. Scheffe 1959, p. 13, 14). A result in the theory of linear regressions that bears some resemblance with the theorems of this paper has been obtained by Grenander and Rosenblatt (1957, p. 244). Let the error sequence $\{\epsilon_k\}$ in (1.1) be a weakly stationary random sequence with piecewise continuous spectral density, and let the regression vectors admit a joint spectral representation. Under these assumptions Grenander and Rosenblatt give necessary and sufficient conditions for the regression spectrum and for the family of admissible spectral densities in order that the LSE are asymptotically efficient for every density of the family. In Sections 3 and 6 we discuss some examples relevant to Theorems 1 and 2.

Let $\{X_i\}, i = 1, 2, \cdots$ be a sequence of nonnegative, independent, identically distributed random variables, and let $F_n(x) = P\{X_1 + \cdots + X_n < x\}$. The purpose of this note is to investigate $\lim_{n\rightarrow\infty} \lbrack F_n(a_1)/F_n(a_2)\rbrack,$ where $0 < a_1 < a_2 < \infty$. In order to do so, estimates are required of the extreme lower tail of the distribution $F_n(x)$, for large $n$. There are a number of known results on the probability in the tail of a convolution (see [1], [2], [3], [4]), but none are appropriate for the present problem. Let $X$ denote a typical $X_i$. In the case when $P\{0 < X < x_0\} = 0$ for some $x_0 > 0$, the required limit is trivial to evaluate. This result is stated in Theorem 1. In the contrary case it is shown in Theorem 3 that a sufficient condition for $\lbrack F_n(a_1)/F_n(a_2)\rbrack \rightarrow 0$ is that there exist real $\gamma \geqq 0$ and $k > 0$ such that \begin{equation*}\tag{1}0 < \lim_{x\rightarrow0+}x^{-\gamma} P\{0 < X < x\} = k < \infty.\end{equation*} This result is achieved by means of an estimate of $F_n(x)$ which is given in Theorem 2. The Condition (1) is satisfied by a wide class of random variables including those with density functions $f(x)$ such that for some $\alpha \geqq 0, 0 < \lim_{x\rightarrow 0+} x^{-\alpha}f(x) = k < \infty.$ It is easy to verify that most "textbook" densities satisfy this condition, as do all densities with a positive right-continuous $k$th derivative $(0 \leqq k < \infty)$ at zero. Although (1) is not necessary for $\lbrack F_n(a_1)/F_n(a_2)\rbrack \rightarrow 0$ (a class of examples will be given later to illustrate this point), one would expect that some regularity condition near the origin would be required. In attempting to weaken the sufficient Condition (1), one might conjecture that conditions such as the existence of a bounded density $f(x)$, such that $0 < f(x)$ on an open interval $(0, \epsilon), 0 < \epsilon$, is sufficient for $\lbrack F_n(a_1)/F_n(a_2)\rbrack \rightarrow 0$. This conjecture is at the present neither proved, nor disproved by a counterexample.

Estimators of the form $\hat f_n(x) = (1/n) \sum^n_{i=1} \delta_n(x - x_i)$ of a probability density $f(x)$ are considered, where $x_1 \cdots x_n$ is a sample of $n$ observations from $f(x)$. In Part I, the properties of such estimators are discussed on the basis of their mean integrated square errors $E\lbrack\int(f_n(x) - f(x))^2dx\rbrack$ (M.I.S.E.). The corresponding development for discrete distributions is sketched and examples are given in both continuous and discrete cases. In Part II the properties of the estimator $\hat f_n(x)$ will be discussed with reference to various pointwise consistency criteria. Many of the definitions and results in both Parts I and II are analogous to those of Parzen [1] for the spectral density. Part II will appear elsewhere.

Although a great deal has been written concerning the theory of tests, decisions and inference for statistical problems invariant under the action of some group, (see for example [4]-[7], [9], [12]-[14], [16]), no great amount of literature exists concerning the problem of discerning whether or not a given problem is actually invariant under some group. In fact the literature seems to consist of one abstract [8] and one paper [15]. In this paper necessary and sufficient conditions are developed that a statistical problem must satisfy in order that it be invariant, in a precise sense to be defined later, under a fairly general class of transformation groups, Lie transformation groups. It must be added, however, that the sufficient conditions are to some extent tautological. In addition two methods of actually constructing the group, if it can be shown to exist, are given, and the main theorem is illustrated by a variety of examples. One of the examples yields the interesting result that the fiducial distribution of the correlation coefficient derived from a sample from a bivariate normal distribution by R. A. Fisher is not a Bayes' distribution for any prior distribution.

The present paper is concerned with those pairwise comparison experiments for which ranking in descending order of the "row sums" is the appropriate ranking procedure. More precisely, assume that $n$ items are compared pairwise, omitting no pairs, and that the result of the comparison between item $i$ and $j$ is expressed as a real number $x_{ij}$ satisfying \begin{equation*}\tag{1}x_{ij} + x_{ji} = 0\quad\text{for all} i, j.\end{equation*} $x_{ij}$ might be some measured difference between $i$ and $j$, or it might take the values $1, 0, -1$ according as item $i$ is judged to be superior, equal or inferior to item $j$, or it might be a statistic summarizing the results of several comparisons between the two items, etc. It will be shown that ranking in descending order of the scores \begin{equation*}\tag{2}s_i = \sum^n_{j=1} x_{ij}\end{equation*} (row sum procedure) uniformly minimizes the risk among all permutation invariant procedures, and for all "reasonable" loss functions, provided the $x_{ij} (i < j)$ are independent random variables distributed according to an exponential distribution of the type \begin{equation*}\tag{3}P(x_{ij} \leqq t) = c(\vartheta_i - \vartheta_j) \int^t_{-\infty} e^{(\vartheta_i-\vartheta_j)\tau} \mu(d\tau),\end{equation*} where $\mu$ is a symmetric probability measure on the real line. Then $s = (s_1, \cdots, s_n)$ is a sufficient statistic for the joint distribution of the $x_{ij}$. Under suitable regularity conditions, the converse also holds: if the distributions of the $x_{ij}$ are not of this form, then $s$ is not sufficient, and the row sum procedure is not optimal. Thus, the model just described seems to constitute the natural domain of the row sum ranking procedure. Fortunately, this domain contains the important case where the $x_{ij} (i < j)$ are independent normal variables with mean $\vartheta_i - \vartheta_j$ and equal variance $\sigma^2$. Another particular case--the case of tournaments without draws--where the $x_{ij}$ can take only two values, has been treated in the joint paper [1]. Incidentally, the present paper grew out of an attempt to cover the case of tournaments with draws (see example (iv) in Section 4 below). Although one cannot expect strict optimality properties, one may still ask whether the row sum procedure has "nice" (e.g., minimax) properties even outside the above model. A modest result in this direction is the following. Take a particular joint distribution of the $x_{ij}$ belonging to the above model, and consider the class of all joint distributions of the $x_{ij}$ that lead (up to permutations) to the same joint distribution of the $s_i$ as this particular one. Then this latter will be least favorable for the ranking problem, and the row sum procedure will have minimax properties in each such class (modified minimax principle of Wesler [2]). For instance, if the $x_{ij} (i < j)$ are independent normal variables with mean $\vartheta_{ij}$ and equal variance, then the joint distribution of the $s_i$ depends only on the $\vartheta_i = (1/n) \sum^n_{j=1} \vartheta_{ij}$, and the linear model $\vartheta'_{ij} = \vartheta_i - \vartheta_j$ is least favorable, if the $\vartheta_i$ are kept fixed (up to permutations). There is an immediate generalization of the results to the case of comparison in $k$-tuples instead of in pairs, as indicated in the last section of this paper.

It is shown that, under certain hypotheses, the dual of a symmetric partially balanced incomplete block design with $m$ associate classes is also a PBIB design with $m$ associate classes and all parameters the same as before. In the case $m = 1$, and in the case that the design is group divisible, these hypotheses coincide with assumptions previously known to be sufficient to ensure duality. To show that some hypotheses are needed for duality, an example is given of a group divisible design whose dual is not a group divisible design.

A sequential procedure is given for deciding to which of $k$ non-overlapping intervals the unknown mean $\theta$ belongs which satisfies the requirement that the probability of making an incorrect decision is less than some preassigned value $\alpha$. The sequential procedure is worked out explicitly for the following two cases: (1) when $\theta$ is the mean of a normal distribution with a known variance, and (2) when $\theta$ is the mean of a normal distribution with an unknown variance. A brief discussion is also given of a related but apparently new problem, to find a sequential procedure which will simultaneously select one of the $k$ intervals and also yield a confidence interval for $\theta$ of a specified width.

For the problem of inference about a real parameter $\mu$ on the basis of $n$ independent observations $x_1, \cdots, x_n$ (or x) each distributed as $N(\mu, \sigma^2)$ with $\sigma^2$ "unknown", it is commonly asserted, for example in [2] p. 465, that the Bayesian method is close to other forms of inference (significance tests, confidence and fiducial intervals) since it too may be based on $s_{n-1}(t)$, the probability density function (pdf) of Student's $t$ with $n - 1$ degrees of freedom. The Bayesian role of $s_{n-1}(t)$ is that of the posterior pdf of $t = \lbrack n(n - 1)/S\rbrack^{\frac{1}{2}} (\bar x - \mu)$, where $\bar x = n^{-1} \sum x_i$ and $S = \sum (x_i - \bar x)^2$ are the sufficient statistics for $\mu$ and $\sigma^2$. It results from formal use in Bayes's Theorem of the improper prior pdf for $\mu$ and $\sigma^2$ described by "independence of $\mu$ and $\log \sigma$ and their uniform distributions on $R^1$". More convincing support for $s_{n-1}(t)$ as a posterior pdf could be obtained by detailed examination of the product space of proper (integrable) prior pdfs and $(\bar x, S)$ and the determination of the essential features of the region where replacement of the posterior pdf of $\mu$ by that derived from $s_{n-1}(t)$ does not seriously affect inference about $\mu$. In this note, attention will be confined to prior pdfs in the following class. Let $\omega$ denote the Fisher information $\sigma^{-2}$ and let $I\{ \}$ denote the 0-1 indicator function of a set. Consider prior pdfs for $\mu$ and $\omega$ drawn from the sequence \begin{equation*}\tag{1.1}p_\alpha(\mu, \omega) \propto \omega^{-1}I\{\mu, \omega\mid\mu_{1\alpha} < \mu < \mu_{2\alpha}, \omega_{1\alpha} < \omega < \omega_{2\alpha}\} \alpha = 1, 2, \cdots.\end{equation*} For each member of this sequence, $\mu$ and $\omega$ are independent while $\mu$ and $\log \omega$ (or $\log \sigma$) have rectangular distributions (from which it is clear that the choice of (1.1) is motivated by the improper prior pdf for $\mu$ and $\sigma^2$ above). The posterior pdf of $\mu$ obtained by combining $p_\alpha(\mu, \omega)$ with the likelihood function $p(\mathbf{x}\mid\mu, \omega) \propto \omega^{\frac{1}{2}n} \exp \lbrack-\frac{1}{2}n\omega(\bar x - \mu)^2 - \frac{1}{2}\omega S\rbrack$ is proportional to $\int^{\omega_{2\alpha}}_{\omega_{1\alpha}} \omega^{\frac{1}{2}n-1} \exp \lbrack-\frac{1}{2}n\omega(\bar x - \mu)^2 - \frac{1}{2}\omega S\rbrack d\omega\cdot I\{\mu\mid\mu_{1\alpha} < \mu < \mu_{2\alpha}\}$ giving, with the change of variable $u = \omega\lbrack 1 + t^2/(n - 1)\rbrack S$ \begin{equation*}\tag{1.2}p_\alpha(t \mid \mathbf{x}) \propto s_{n-1}(t) \int^{\lbrack 1+t^2/(n-1)\rbrack S\omega_{2\alpha}}_{\lbrack 1+t^2/(n-1)\rbrack S\omega_{1\alpha}} u^{\frac{1}{2}(n-2)}e^{-\frac{1}{2}u}du\end{equation*} $I\{t\mid\lbrack n(n - 1)/S\rbrack^{\frac{1}{2}}(\bar x - \mu_{2\alpha}) < t < \lbrack n(n - 1)/S\rbrack^{\frac{1}{2}}(\bar x - \mu_{1\alpha})\}.$ To obtain $s_{n-1}(t)$, Jeffreys (p. 68 of [1]) uses a convergence argument which, in our specialisation, would involve letting \begin{equation*}\tag{1.3}\mu_{1\alpha} \rightarrow - \infty, \mu_{2\alpha} \rightarrow \infty, \omega_{1\alpha} \rightarrow 0, \omega_{2\alpha} \rightarrow \infty \text{as} \alpha \rightarrow \infty\end{equation*} as necessary and sufficient conditions for \begin{equation*}\tag{1.4}\lim p_\alpha(t\mid\mathbf{x}) \equiv s_{n-1}(t)\end{equation*} for all values of $\mathbf{x}$. In (1.4), $\mathbf{x}$ is kept fixed. However, in changing $\alpha$, we are changing the prior distribution used, so that keeping $\mathbf{x}$ fixed has no obvious relevance. To emphasize that a different $\mathbf{x}$ would normally be associated with a different prior pdf, we will, except in the proofs of Section 2, write $\mathbf{x}_\alpha, \bar x_\alpha, S_\alpha, t_\alpha$ for the $\mathbf{x}, \bar x, S, t$ associated with $p_\alpha(\mu, \omega)$. A radically different justification of $s_{n-1}(t)$ is provided as follows. Let us suppose that the person who is to make the inference about $\mu$ has the prior pdf $p_s(\mu, \omega)$ for $s$ some positive integer, that is, a pdf that happens to be a member of the sequence (1.1). Examination of (1.2) shows that he can take $s_{n-1}(t_s)$ as a good approximation to his posterior pdf provided \begin{equation*}\tag{1.5}S_s\omega_{1s} \ll 1, S_s\omega_{2s} \gg 1, S^{-\frac{1}{2}}_s(\mu_{2s} - \bar x_s) \gg 1, S^{-\frac{1}{2}}_s(\bar x_s - \mu_{1s}) \gg 1.\end{equation*} Now a person holding the prior pdf $p_s(\mu, \omega)$ would expect to obtain $\mathbf{x}_s$'s according to the marginal pdf $p_s(\mathbf{x}_s) = \int \int p(\mathbf{x}_s\mid\mu, \omega)p_s(\mu, \omega) d\mu d\omega$. The probability of (1.5) under $p_s(\mathbf{x}_s)$ is therefore the person's prior probability of being able to use $s_{n-1}(t_s)$ as a basis for inference about $\mu$. In the light of this, if, for the sequence (1.1), we were to have $\operatorname{plim} S_\alpha\omega_{1\alpha} = 0, \quad \operatorname{plim} S_\alpha\omega_{2\alpha} = \infty,$ \begin{equation*}\tag{1.6} \operatorname{plim} S^{-\frac{1}{2}}_\alpha(\bar x_\alpha - \mu_{1\alpha}) = \infty, \operatorname{plim} S^{-\frac{1}{2}}_\alpha(\bar{x}_\alpha - \mu_{1\alpha}) = \infty \end{equation*} with the $\operatorname{plim}$ evaluated with respect to the sequence of marginal distributions $p_\alpha(\mathbf{x}_\alpha)$, we would, by proceeding down the sequence, be able to invest $s_{n-1}(t)$ with an asymptotic justification. (By $\operatorname{plim} z = \infty$, we mean that $\lim \operatorname{Prob} (z < K) = 0 \text{for all} K.)$ In Lemma 1 be $\mathrm{(a)} \rho_{2\alpha}/\rho_{1\alpha}\rightarrow \infty$ \begin{equation*}\tag{1.7} \mathrm{(b)} \rho_{2\alpha} \rightarrow \infty \end{equation*} $\mathrm{(c)} \lim \inf \lbrack\log \rho_{1\alpha}/\log \rho_{2\alpha}\rbrack \geqq 0 \quad \text{as} \quad \alpha \rightarrow \infty. $ Lemma 2 then shows that (1.6) is equivalent to \begin{equation*}\tag{1.8}\operatorname{plim} p_\alpha(t\mid\mathbf{x}_\alpha) \equiv s_{n-1}(t)\end{equation*} where the $\operatorname{plim}$ is again evaluated with respect to the sequence $p_\alpha(\mathbf{x}_\alpha), \alpha \rightarrow \infty$. Hence (1.7) is necessary and sufficient for (1.8) which, since it allows direct comparison with the Jeffreys approach in (1.3) and (1.4), we state as the principal theorem. The interpretation of the conditions (1.3) is superficially straightforward; it is that the prior pdfs for $\mu$ and $\omega$ should (separately) approach conditions representing "complete ignorance". (1.7) is apparently more complex. In the requirement $\rho_{2\alpha}/\rho_{1\alpha} \rightarrow \infty$, it agrees with (1.3); its principal divergence from (1.3) lies in the existence of the joint conditions, (b) and (c), on the developments of the prior pdfs of $\mu$ and $\omega. \rho_{1\alpha}$ and $\rho_{2\alpha}$ may be regarded as measures of the information about $\mu$ in the least and most informative conditional distribution $p(\mathbf{x}\mid\mu, \omega)$ allowed by $p_\alpha(\mu, \omega)$, relative to the prior information about $\mu$ measured by the quantity $(\mu_{2\alpha} - \mu_{1\alpha})^{-1}$. (1.7) (c) requires that, although there is no necessity for $\rho_{1\alpha}$ to approach zero at all, if it does so, it should not do so too rapidly that is, loosely speaking the least informative conditional distribution should not be too uninformative. For the case $\mu_{1\alpha} = -\alpha, \mu_{2\alpha} = \alpha, \omega_{1\alpha} = \alpha^\lambda, \omega_{2\alpha} = \alpha$, (1.3) requires $-\infty < \lambda < 0$, while (1.7) requires $-2 \leqq \lambda < 1$. The case $\mu_{1\alpha} = -1, \mu_{2\alpha} = 1, \omega_{1\alpha} = 1, \omega_{2\alpha} = \alpha$ satisfies (1.7) but not (1.3). The comparison of (1.3) and (1.7) is assisted by noting that $t_\alpha$ is invariant with respect to the simultaneous transformations of $x$ and $\mu, x \rightarrow a_\alpha x + b_\alpha, \mu \rightarrow a_\alpha\mu + b_\alpha$. We would therefore expect that any reasonable condition on the sequence (1.1) for the asymptotic relevance of $s_{n-1}(t)$ would be unaffected by these transformations, when coupled with $\omega \rightarrow a^{-2}_\alpha\omega$. (1.7) agrees with such expectation while (1.3) does not.

This paper obtains and explores a family of confidence procedures for the mean of a normal distribution which are, in a certain sense, more efficient than the usual procedure. Let $X$ (possibly a sample mean) have a normal density $\varphi(x; \theta, \sigma^2)$ with unknown mean $\theta$, and known variance $\sigma^2$. Let $R(X)$ be a confidence region for $\theta$ at level $1 - \alpha$. Let $m(R)$ be the length of $R$ if $R$ is an interval; more generally for any region $R$ let \begin{equation*}\tag{1}m(R) = \int_R d\theta,\end{equation*} which we will also call the length of $R$. Then, following [3], we have \begin{align*}E_{\theta'} \{m(R(X))\} &= \int\int_{\theta\varepsilon R(x)} d\theta\varphi(x; \theta', \sigma^2) dx = \int P_\theta' \{\theta \varepsilon R(X)\} d\theta \\ \tag{2} \\ &= \int_{\theta\neq\theta'} P_\theta' \{\theta \varepsilon R(X)\} d\theta.\end{align*} Thus the expected length of the confidence region $R(X)$ may also be interpreted as the integral over all false values $\theta$ of the probability of covering $\theta$, where the expected length and the probability are both computed under the true value $\theta'$. Whether we are interested in length or in the probability of covering false values, we would like to make (2) small. For a particular $\theta'$, we can minimize (2) as follows. Let $A(\theta)$ be the acceptance region of the family of tests corresponding to $R(X)$, that is \begin{equation*}\tag{3}X \varepsilon A(\theta)\quad\text{if and only if} \theta \varepsilon R(X).\end{equation*} Substituting (3) in (2) gives \begin{equation*}\tag{4}E_{\theta'}\{m(R(X))\} = \int P_{\theta'}\{X \varepsilon A(\theta)\} d\theta = \int_{\theta\neq\theta'} P_{\theta'} \{X \varepsilon A(\theta)\} d\theta.\end{equation*} For $\theta \neq \theta', 1 - P_{\theta'}\{X \varepsilon A(\theta)\}$ is the power of the test of the null hypothesis value $\theta$ against the alternative $\theta'$. Thus we see that the expected length of the confidence region is minimized, when $\theta'$ is the true value, by choosing the test of each null hypothesis value $\theta$ which is most powerful against the alternative $\theta'$. This gives the confidence interval \begin{equation*}\tag{5}\min \{\theta', X - \xi_\alpha\sigma\} \leqq \theta \leqq \max \{\theta', X + \xi_\alpha\sigma\},\end{equation*} where $\xi_\alpha$ is the upper $\alpha$-point (not $\frac{1}{2} \alpha$-point) of the standard normal distribution. Table 1 shows what happens if we guess $\theta = \theta'$ and use the foregoing confidence procedure: our expected length will be considerably less than that of the usual procedure if we guess correctly, but greater if we are wrong by much over $2\sigma$ (for $\alpha = .05$). Since we do not know the true value of $\theta$, we may prefer to minimize not the expected length under a particular $\theta'$, but a weighted average of this. Consider then, for some $W \geqq 0$, the weighted average \begin{equation*}\tag{6} \int W(\theta') E_{\theta'} \{m(R(X))\} d\theta' = \int\int P_{\theta'} \{X \varepsilon A(\theta)\}W(\theta') d\theta' d\theta,\end{equation*} where the equality follows from (4). If $W$ is interpreted as the prior density of $\theta$, then (6) is the prior (marginal) expected length of the confidence region $R(X)$, but this interpretation need not be made. The procedure minimizing (6) also corresponds to a most powerful test of each null hypothesis value $\theta$ against a certain alternative distribution, as we shall see explicitly in Section 2. We are concerned with two kinds of question: (A) If we use the minimizing confidence procedure for some $W$, what is the effect of $W$ on the expected length and on the efficiency to be gained by giving up the usual procedure? In particular, how diffuse does $W$ have to get before the gain over the usual procedure is small in terms of the weighted average of the expected length? (B) Are the minimizing confidence intervals for $W$ more like posterior probability intervals obtained from the prior $W$ than are the usual intervals? Does the use of $W$ in selecting a confidence procedure largely eliminate the difference between confidence intervals and posterior probability intervals? Specifically, we will introduce a normal weight function $W(\theta') = \varphi(\theta'; \theta_0, \omega^2)$ and obtain the minimizing confidence procedure $R_\omega(X)$, which reduces to (5) with $\theta' = \theta_0$ when $\omega = 0$ and to the usual procedure when $\omega = \infty$ and is given by Figure 1 or Table 5 when $\omega = \sigma$ and $\alpha = .05$. Then to answer (A), Table 2 gives the expected length of the minimizing procedure for $\alpha = .05$ and various values of $\omega$. Notice that, as $\omega$ increases, so does the value of $\theta - \theta_0$ at which the minimizing procedure has the same expected length as the usual procedure: for $\omega = 4\sigma, \theta - \theta_0$ can be about $1.5\omega = 6\sigma$ before the usual procedure is better; for $\omega = 2\sigma$, about $1.6\omega = 3.2\sigma$; for $\omega = 0$, about $2\sigma$ (from Table 1). Table 2 also gives the weighted average (6) of the expected length of the minimizing procedure. Thus for $\omega = 4\sigma$, the minimizing confidence procedure has weighted average expected length 1.5{\tt\%} less than the usual procedure, and in this sense saves 3{\tt\%} of the observations, so that the usual procedure is 97{\tt\%} efficient. For $\omega \leqq 2\sigma$, however, the usual procedure wastes more than 10{\tt\%} of the observations, in the same sense. To answer (B), Table 3 gives, for $\alpha = .05$ and $\omega = \sigma$, the posterior probability of the usual interval and the minimizing confidence interval when the prior density is normal with variance $\omega^2 = \sigma^2$. The $X$-scale reflects that the fact that, with this prior, the prior (marginal) variance of $X$ is $2\sigma^2$. Even for a priori3 probable $X$'s, the confidence level .95 is a poor approximation to the posterior probability of either interval, though much poorer for the usual interval. Thus, by using a prior distribution, one can obtain a confidence procedure with prior expected length substantially less than that of the usual procedure (11{\tt\%} less in this instance, corresponding to a 23{\tt\%} smaller sample size), but considerable discrepancy remains between confidence and posterior probability. In Section 2, we obtain the minimizing procedure and the formulas used in calculating the tables. Section 3 concerns conditioning on the event that the confidence region covers the true value of $\theta$. Section 4 consists of remarks, which are largely independent of Sections 2 and 3. I am grateful to the referee for a number of suggestions and comments.

A serious objection to many of the classical statistical methods based on linear models or normality assumptions is their vulnerability to gross errors. For certain testing problems this difficulty is successfully overcome by rank tests such as the two Wilcoxon tests or the Kruskal-Wallis H-test. Their power is more robust against gross errors than that of the $t$- and $F$-tests, and their efficiency loss is quite small even in the rare case in which the suspicion of the possibility of gross errors is unfounded. For the corresponding problems of point estimation a beginning has been made to attack the difficulty by modifying the classical estimates either through removal or Winsorization of outlying observations; see for example Tukey (1960) and Anscombe (1960). It is the purpose of the present paper to explore a different approach to these problems of point estimation. In Sections 2-5 point estimates of location or shift parameter are defined in terms of rank test statistics such as the Wilcoxon or normal scores statistic, which are successful in providing robust power for the corresponding testing problems. In Sections 6 and 7, certain regularity and invariance properties of these estimates are proved. The distributions of the estimates are shown in Section 8 to be symmetric with respect to the parameter being estimated--and hence in particular to be unbiased--if the underlying distribution of the observations on which the estimate is based is symmetric. Without this assumption, the estimates are shown in Section 9 to be either exactly or approximately median unbiased for small samples and in Section 10 to be approximately normally distributed about the true parameter value for large samples. The variance of this asymptotic distribution depends of course on the underlying distribution of the observations, so that the estimates are not "distribution-free." In Section 9 there is also established a close relationship between the estimates and the corresponding upper and lower confidence bound for the parameter at confidence level $\frac{1}{2}$, with which the estimate coincide in many cases. Finally, in Section 11, it is proved that the asymptotic relative efficiency of the estimates to the classical linear estimates is the same as the Pitman efficiency of the rank tests on which they are based to the corresponding $t$-tests.

Most of the nonparametric tests available in the literature use the assumption that the distributions concerned are absolutely continuous. Under this assumption the power efficiencies of these tests for parametric families of distribution are relatively well known. A problem arises, however, when we assume that the distributions are discrete. One would then need to solve the problem of ties, which occur with positive probabilities. One way of getting round this difficulty would be to randomize the order of the tied observations and employ randomized tests. Putter (1955) has presented some interesting results comparing the relative efficiencies of these randomized tests with suitably modified non-randomized tests, when the distributions are discrete. One particular aspect of the nonparametric test, however, has to the best of the author's knowledge, not been discussed so far. This concerns the efficiency of non-parametric tests when the distributions are discrete relative to the most powerful tests available for such distributions when the latter belong to parametric families. One has the impression that the relative efficiencies for such situations are indirectly related to the degree of "discreteness" of the distributions. In other words, the fewer the number of points of probability concentration, the higher is the efficiency. To determine how far this conjecture agrees with the facts, we have discussed the special case of the two-sample Wilcoxon or Mann-Whitney test (Mann and Whitney (1947)). The power efficiency of this test has been worked out for the class of discrete distributions which are of the exponential type, and finally three examples have been discussed.

Small sample power and efficiency are computed for the one sample Wilcoxon and normal scores tests for normal shift alternatives. A recursive scheme is given which reduces the problem of power computation permitting investigations up to sample size $N = 10$. Local efficiencies for the two nonparametric tests are computed for small samples using the values of the normal scores statistic. In addition, efficiencies for large shifts are obtained by comparing the exponential rate of convergence to zero of the type two error.

The main purpose of the paper is to obtain the lower bound on the number of integrals to be evaluated in order to know the first, second and mixed (linear) moments of the normal order statistics ().S.) in a sample of size $N$ assuming that these moments are available for sample sizes less than $N$. Towards this, the recurrence relationships, identities, etc. among the moments of the normal order statistics, which have appeared in the literature have been collected with appropriate references. Also, these formulae are listed and stated in the most general form wherever possible. Simple and alternate proofs of some of these formulae are given. These results are also supplemented with new formulae or relationships. It is shown that it is sufficient to evaluate at most one single integral and ($N$-2)/2$ double integrals when $N$ is even and one single integral and ($N$-3)/2$ double integrals when $N$ is odd, in order to know the first, second and mixed (linear) moments of normal O.S. However, for these moments of O.S. in samples drawn from an arbitrary population symmetric about zero, one has to evaluate one more double integral in addition to the number of integrals required in the case of normal O.S. Also, a possible scheme of computing these moments which will be useful especially for small sample sizes, is presented in Section 5. The lower moments of quasi-ranges in samples drawn from an arbitrary population symmetric about zero are expressed in terms of the moments of the0 corresponding O.S. Simple recurrence formulae among the expected values of quasi-ranges in samples drawn from an arbitrary continuous population are obtained. A modest list of references is provided at the end which is by no means exhaustive.

In case the underlying distribution of a sample is normal, a substantial literature has been devoted to the distribution of quantities such as $(X_{(i)} - u)/v$ and $(X_{(i)} - u)/w$, where $X_{(i)}$ denotes the $i$th ordered observation, $u$ and $v$ are location and scale statistics of the sample, or one is a location or scale parameter and $w$ is an independent scale statistic. The case $i = 1$ or $n$ has been frequently studied in view of the great importance of extreme values in physical phenomena and also with a view to testing outlying observations or the normality of the distribution. Bibliographical references will be found in Savage [10] and, as far as the general problem of testing outliers is concerned, in Ferguson [4]; references to recent literature include Dixon [1], [2], Grubbs [5], Pillai and Tienzo [9]. Thompson [12] has studied the distribution of $(X_i, - \bar{X})/s$ where $X_i$ is one observation picked at random among the sample, and this statistic has been used in the study of outliers; Laurent has generalized Thompson's distribution to the case of a subsample picked at random among a sample [7], then to the multivariate case and the general linear hypothesis [8]. Thompson's distribution is not only the marginal distribution of $(X_i - \bar{X}/s$ but its conditional distribution, given the sufficient statistic $(\bar{X}, s)$, hence it provides the distribution of $X_i$ given $\bar{X}, s$, and, using the Rao-Blackwell-Lehmann-Scheffe theorem, gives a way of obtaining a minimum variance unbiased estimate of any estimable function of the parameters of a normal distribution for which an unbiased estimate depending on one observation is available, a fact that has been exploited in sampling inspection by variable. The present paper presents an analogue to Thompson's distribution in case the underlying distribution of a sample is exponential (the exponential model is nowadays widely used in Failure and Queuing Theories). Such a distribution makes it possible to obtain minimum variance unbiased estimates of functions of the parameters of the exponential distribution. Here an estimate is provided for the survival function $P(X > x) = S(x)$ and its powers. As an application of these results the probability distribution of the "reduced" $i$th ordered observation in a sample and that of the reduced range are derived. For possible applications to testing outliers or exponentially the reader is invited to refer to the bibliography.

Expected values and standard deviations of order statistics from a logistic distribution are given for sample sizes $n = 1,2, \cdots, 10$ and $n = 15, 20, 50$, and 100, and summarized in graphs to facilitate interpolation to other sample sizes. For $n \leqq 10$ the results are compared with asymptotic approximations. Related distribution theory is discussed briefly, and related investigations and applications are cited.

The line-graph, $L(G)$, of any given simple graph, $G$, is that graph whose points can be put in a one-to-one correspondence with the two edges of $G$ in such a way that two points in $L(G)$ are adjacent if, and only if, their corresponding edges in $G$ are adjacent. The complete $m$ by $n$ bigraph, $B_{mn}$, contains $mn$ edges which join each point in one set of $m$ points to each point in a second set, disjoint from the first, containing $n$ points. As general references on terminology see, e.g., Harary [4] and Ore [8]. If we suppose that $m \geqq n \geqq 1$ it can be seen that $L(B_{mn})$ has the following three properties: (1) The graph has $mn$ points each of which is adjacent to $m + n - 2$ other points. (2) $n \binom{m}{2}$ of the pairs of adjacent joints are mutually adjacent to $m - 2$ other points and the remaining $m \binom{n}{2}$ pairs of adjacent points are mutually adjacent to $n - 2$ other points. (03) Any two distinct nonadjacent points are mutually adjacent to two points. The object of this note is to show that if any graph satisfies these three conditions then it is isomorphic to $L(B_{mn})$ except possibly when $(m, n) = (4, 4), (4, 3)$ or $(5, 4)$. This will generalize a result of Shrikhande [10] (see also Mesner [7]) who, using different terminology, has already shown this for the case that $m - n$. The corresponding problem for the line-graph of the ordinary complete graph of $n$ points has been treated by Connor [3], Shrikhande [9], Hoffman [5], and Chang [1], [2].