In a variety of standard multivariate normal testing problems, it is shown that certain procedures, often fully invariant, similar, and/or likelihood ratio, are admissible Bayes procedures. The problems include the multivariate general linear hypothesis (where some of the procedures considered were previously shown to be admissible by other methods), the testing of independence of sets of variates (where the likelihood ratio test is shown, for the first time, to be admissible), tests about only some components of the means, classification procedures (for any number of populations), Behrens-Fisher problem, tests about values of or proportionality or equality of covariance matrices, etc. A general technique is developed for obtaining certain Bayes procedures for such problems from the corresponding Bayes procedures relative to a priori distributions of a certain type for problems where nuisance parameter means have been deleted.

Let $X_1, X_2, \cdots, X_n$ be a sequence of independent random variables (r.v.'s) with zero mean and finite standard deviation $\sigma_i, 1 \leqq i \leqq n$. According to the central limit theorem, the normed sum $Y_n = (1/s_n) \sum^n_{i=1} X_i,$ where $s_n = \sum^n_{i=1} \sigma^2_i$, is under certain additional conditions approximatively normally distributed. We will here examine the convergence of the moments and the absolute moments of $Y_n$ towards the corresponding moments of the normal distribution. The results in this general case are stated in Theorem 3 and Theorem 4, but, in order to avoid repetition and unnecessary complication, explicit proofs will only be given in the case of equally distributed random variables. (Theorem 1 and Theorem 2).

During the past 15 years various approaches have been proposed to deal with the lack of robustness of the sample mean as an estimate of the population mean when the distribution sampled is contaminated by gross errors, i.e., has heavier tails than the normal distribution. First, Tukey and the Statistical Research Group at Princeton in [9] suggested and investigated the properties of "trimmed" and "Winsorized" means. More recently, Hodges and Lehmann [6], proposed estimates related to the well-known robust Wilcoxon and normal scores tests, among others. Finally Huber in [7] considered essentially the class of maximum likelihood estimates and found those members of this class which minimize the maximum variance over various classes of contaminated distributions. For a review of work in these directions in testing as well as estimation the reader is referred to Elashoff [3]. In Theorems 3.1 and 3.2 we state the main results of the asymptotic theory of the Winsorized and trimmed means and outline the proof. An alternative method of trimming and Winsorizing (not equivalent to that of Tukey) which encompasses the efficient estimates proposed by Huber and which generalizes to higher dimensions is discussed in Section 2. The fourth section (Theorem 4.1) gives the minimum efficiency with respect to the families of all symmetric and symmetric unimodal distributions, of the Winsorized and trimmed means with respect to the mean. The lower bounds found for the trimmed means (for small trimming proportions) in the unimodal case compare well with that found by Hodges and Lehmann in [5] for the median of averages of pairs, the Hodges-Lehmann estimate. However, the Winsorized mean (for unimodal distributions) has minimum efficiency $\frac{1}{3}$ with respect to the mean whatever be the trimming proportion used. For all distributions, the minimum efficiency is 0. Also in the fourth section (Theorem 4.2) we compare the trimmed mean to the $H-L$ estimate and find that while the latter can be infinitely more efficient than the former, the $H-L$ estimate, for small trimming proportions, $\alpha = .05$, is at least 90 per cent (approximately) as efficient. This would suggest that unless the computations involved are prohibitive, the $H-L$ estimate is to be preferred in any situation where the degree of contamination and type of distribution is not known with great precision. The same remarks apply to the Winsorized mean with only somewhat less force since the lower bounds involved are .74 for all symmetric distributions and .79 for symmetric unimodal distributions. Finally we compare the principal estimate proposed by Huber in [7] (Proposal 2) to the mean and the Hodges-Lehmann estimate, both for all symmetric densities and for the symmetric unimodal family. Results similar to those already mentioned in connection with the trimmed mean are obtained in Theorems 5.1 and 5.2.

Sufficient conditions for a Bayes sequential procedure to be truncated contained in Wald [12] and in Blackwell and Girshick [3] have been used by Sobel [11] and Mikhalevich [7] to show that under certain conditions a Bayes sequential test is truncated when the cost of observation does not depend on the unknown parameter. When this latter condition is not satisfied as illustrated by some sequential testing problems considered by Kiefer and Weiss [5] and Weiss [14], and by Anscombe [2] (in connection with ethical cost of medical trials) the sufficient condition referred to (see Corollary 3.1) is not always applicable. When it is applicable it generally provides an upper bound on the maximum sample size that a Bayes sequential procedure can take. For the purpose of computation of the Bayes procedure by backward induction it is desirable to know the exact stage of truncation as close as possible. Section 3 contains two theorems (Theorem 3.2 and 3.3) which provide better sufficient conditions for a Bayes sequential procedure to be truncated and simultaneously provide better bounds on the exact stage of truncation. Sufficient conditions are also provided for the upper bounds involved to be exact. Section 4 contains some results with the help of which the theorems of Section 3 may be applied to some special sequential testing problems in Section 5. Finally Section 6 contains proofs of the results of Sections 3 and 4.

This paper deals with the sequential compound decision problem, when the component problem is an estimation problem. The aim is to find sequential compound rules which have Property (A) or (B), as described in Section 2. That is, one seeks rules the compound risk or loss of which do not exceed the value of the Bayes envelope functional at the "empirical distribution" of the parameters, by more than $\epsilon$, for $n$ sufficiently large. In Section 4 some general results for the estimation problem are obtained, which are applied in Sections 5 and 6 to several discrete and continuous distributions to obtain rules with Property (A) or (B). In Section 3 some results are obtained which hold for the sequential compound decision problem with any general component problem. These are applied for the estimation problem in the later sections.

The three familiar definitions of statistical information, Fisher (1925), Shannon (1948), and Kullback (1959), are closely tied to asymptotic properties, hypothesis testing, and a general principle that information should be additive. A definition of information is proposed here in the framework of an important kind of statistical model. It has an interpretation for small samples, has several optimum properties, and in most cases is not additive with independent observations. The Fisher, Shannon and Kullback informations are aspects of this information. A variety of definitions of information may be found in the statistical literature. Perhaps the oldest and most familiar is that due to Fisher (1925). For a real parameter and a density function satisfying Cramer-Rao regularity conditions it has the form \begin{align*}I_F(\theta) &= \int \lbrack(\partial/\partial\theta) \ln f(x \mid \theta)\rbrack^2f(x \mid \theta) dx \\ &= \int - (\partial^2/\partial\theta^2) \ln f(x \mid \theta)f(x \mid \theta) dx;\\ \end{align*} for a vector parameter $\mathbf{\theta}$ it becomes a matrix \begin{align*}\mathbf{I}_F(\mathbf{\theta}) = \operatorname{cov} \{(\partial/\partial\mathbf{\theta}) \ln f(x \mid \mathbf{\theta})\mid \mathbf{\theta}\} \\ &= \int - (\partial/\partial\mathbf{\theta})(\partial/\partial\mathbf{\theta}') \ln f(x \mid \mathbf{\theta})f(x \mid \mathbf{\theta}) dx\\ \end{align*} where $\operatorname{cov}$ stands for covariance matrix and where the integrand being averaged in the second expression is the matrix of partial derivatives of the log-likelihood. Shannon (1948) proposed a definition of information for communication theory. In its primary form it measures variation in a distribution; with a change of sign it measures concentration and is thereby more appropriate for statistics: $I_S(\theta) = \int \ln f(x \mid \theta)f(x \mid \theta) dx.$ Kullback (1959) considers a definition of information for "discriminating in favor of $H_1(\theta_1)$ against $H_2(\theta_2)$": $I_K(\theta_1, \theta_2) = \int \ln \lbrack f(x \mid \theta_1)/f(x \mid \theta_2)\rbrack f(x \mid \theta_1) dx.$ These information functions are additive with independent observation; in fact, additivity is taken as an essential property in most developments of information. The three information functions are defined for quite general statistical models, (mild regularity required for Fisher's definition). And the Fisher and the Kullback definitions are tied closely to large sample theory and to Bayes' theory respectively. The emphasis in this paper is not on information in a general model based on general principles. Rather it is on information in an important and somewhat special model--the location model and, more generally, the transformation-parameter model.

This note discusses the asymptotic independence of the "$(d)$-structured" order statistics $F(X^{(i)})$ and some of the more usual statistics of structure $(d)$. (For a discussion of structure $(d)$ and related concepts, see references [2], [3] and [4].) Asymptotic independence holds, in particular, in the case of the Kolmogoroff-Smirnoff statistic, and thus provides approximate significance levels for the simultaneous test of the hypothesis that the population c.d.f. has specified form and the hypothesis that the sample contains no outlying observations. If the Kolmogoroff-Smirnoff statistic is used in conjunction with the largest order statistic $X^{(n)}$, the acceptance region of the resulting test can be characterized as follows: For acceptance (of the hypothesis that the population c.d.f. has specified shape and that the sample contains no outliers), the $n$ "risers" of the sample c.d.f. must fall within a "three-sided" region. This region is the intersection of the usual Kolmogoroff-Smirnoff region with the half-plane to the left of the critical value for $X^{(n)}$. The example of the last section deals with this case. Section 1 contains the theory on which the subsequent development is based. It involves the multivariate probability integral transformation $T_G$ discussed in [10], which has the property that $T_G(X), X$ distributed according to $G$, is uniform over the unit cube, so that $T_H^{-1}T_G(X)$ has distribution $H$. The transformation enters the argument in essentially this way: If $Y$ is a vector distributed according to $H,f$ and $g$ are two functions of $Y$, and $H(t)$ is the conditional distribution of $Y$, given $g(Y) \leqq t$, then $f(Y)$ and $g(Y)$ are independent if and only, for essentially all $t, f(Y)$ and $f(T_H^{-1}T_{H(t)}(Y))$ have the same distribution conditionally on $g(Y) \leqq t$. In the present application, $Y$ is a random sample from the uniform distribution, $g$ is an order statistic of this sample, and $f$ is the function of $Y$ corresponding to a statistic of structure $(d)$ such as the Kolmogoroff-Smirnoff statistic or Sherman's statistic [12]. It may appear that the above method recommends itself primarily on grounds of novelty, rather than suitability. This is borne out by the fact that the relatively weak requirement that the distributions of $f(Y)$ and $f(T_H^{-1}T_{H(t)}, Y))$ agree asymptotically is verified below by showing that in fact $f(Y)$ and $f(T_H^{-1}T_{H(t)}(Y))$ themselves agree asymptotically, i.e., that their difference converges in probability to zero. This last is made to follow in turn from the very rapid covergence of $|Y - T_H^{-1}T_{H(t)}(Y)|$ to zero. One has the option of having $H$ or $g$ perform the ordering of the sample $Y$. By this is meant that $H$ can be the uniform distribution over the unit cube, and $g$ the function "$k$th largest coordinate of $Y$". Or $H$ can be the joint distribution of the order statistics of $Y$ and $g$ the function "$k$th coordinate of $Y$". The second of these two approaches yields a form for $T^{-1}_HT_{H(t)}$ especially suitable (due to the common multiplier $C_n(x, t, l)$ appearing in Equations (1)) for establishing the convergence to zero of $f(Y) - f(T^{-1}_HT_{H(t)}(Y))$ for a fairly large class of statistics $f$. These include the statistics of Kolmogoroff-Smirnoff type, whose essential combining operation is "sup", and also certain other statistics with less tractable combining operations, such as Sherman's statistic, where the combining operation is addition. Section 2 is devoted to this second approach, applied to Sherman's statistic. The first approach, though seemingly unnatural for statistics with combining operations more demanding than "sup", has the advantage of yielding expressions as easy to manipulate for bivariate (e.g., two order statistics) as for univariate $g$. Section 3 is devoted to this first approach, applied to statistics of Kolmogoroff-Smirnoff type, with $g$ bivariate. The last section contains an application, together with a computation indicating that independence sets in quite rapidly.

The logistic curve $y = k/(1 + \alpha e^{-\beta t})$ has been used in studies pertaining to population growth by Verhulst [17] and by Pearl and Reed [14] and by several later authors. The logistic function $P = 1/(1 + e^{-(\alpha+\beta x)})$ has been very widely used by Berkson [1], Berkson and Hodges [2] as a model for analyzing bioassay and other experiments involving quantal response. Gumbel [8] has shown that the asymptotic distribution of the midrange of exponential type initial distributions is logistic. In connection with problems involving censored data, Plackett [15], [16] has considered the use of the logistic distribution. In this paper a random variable $Y$ is said to follow a logistic distribution (denoted by $L(\mu, \sigma^2))$ if its cumulative distribution function (c.d.f.) is \begin{equation*}\tag{1.1}F(y; \mu, \sigma) = 1/\lbrack 1 + e^{-\lbrack(y-\mu)/\sigma\rbrack\cdot(\pi/3\frac{1}{2})}\rbrack. \end{equation*} The probability density function (p.d.f.) corresponding to (1.1) is \begin{equation*}\tag{1.2}f(y; \mu, \sigma) = (\pi/\sigma 3^{\frac{1}{2}})e^{-\pi(y-\mu)/3\frac{1}{2}\sigma}/\lbrack 1 + e^ {-\pi(y-\mu)/3\frac{1}{2}\sigma}\rbrack^2,\end{equation*} where $-\infty < y < \infty, -\infty < \mu < \infty$ and $\sigma > 0$. It should be noted that the distribution (1.2) is symmetrical with mean $\mu$ and variance $\sigma^2$. The moment generating function of $X = (Y - \mu)/\sigma$ is easy to derive (see for example, Gumbel [8]) and is \begin{equation*}\tag{1.3}M_X(t) = \Gamma(1 + t/g)\Gamma(1 - t/g),\quad g = \pi/3^{\frac{1}{2}}.\end{equation*} In this paper order statistics from the standard logistic distribution $L(0, 1)$ are studied. If $X_1, X_2, \cdots, X_n$ are $n$ independent and identically distributed logistic random variables with density function, \begin{equation*}\tag{1.4}f(x) = (\pi/3^{\frac{1}{2}})(e^{-x\pi/3\frac{1}{2}})/(1 + e^{-x\pi/3\frac{1}{2}})^2,\quad -\infty < x < \infty,\end{equation*} then we are concerned with the moments, the distribution and some estimation problems using the ordered random variables $X_{(1)}, X_{(2)}, \cdots, X_{(n)}$ where \begin{equation*}\tag{1.5}X_{(1)} \leqq X_{(2)} \leqq \cdots \leqq X_{(k)} \leqq \cdots \leqq X_{(n)}.\end{equation*} In the sequel, we shall call $X_{(k)}$, the $k$th order statistic in a sample of size $n$ from the logistic distribution $L(0, 1)$. In this paper the exact expressions for the moments of $X_{(k)}$ have been derived. The values of the first four exact moments for all sample sizes $n$ from 1 to 10 have been tabulated (Table I). More generally, the moments of $X_{(k)}$ have been expressed in terms of expressions involving Bernoulli and Stirling numbers of 1st kind. These derivations are obtained from the moment generating function which has been derived. The cumulants of $X_{(k)}$ are expressed in terms of polygamma functions, as was pointed out by Plackett [15]. Birnbaum and Dudman [3] have tabulated expected values and standard deviations from the logistic distribution using tabulated values of the digamma and trigamma functions. Table III of the present paper gives the percentage points of $X_{(k)}$ (i) for all $k(k \leqq n)$ and all $n$ from 1 to 10 (ii) for $k = 1, n$ and $\frac{1}{2}n$ and $\frac{1}{2}(n + 2)$ ($n$ even) or $\frac{1}{2}(n + 1)$ ($n$ odd) for $n = 11(1)25$. In Section 3, we obtain series expansions for the joint moment generating function and covariance of the two order statistics. In Section 5, the use of one and two order statistics for estimating $\mu$ and $\sigma$ in $L(\mu, \sigma^2)$ is shown. In Section 6, expressions (closed form) are derived for the cumulative distribution function and the density function of the sample range. Using the results of Section 6, a short table (Table II) of the sample range of the logistic is given for $n = 2$ and 3. Section 7 gives a description of the tables in this paper.

We define the incomplete gamma function ratio for positive $N$ by \begin{align*}\tag{1}P(N, b, X) &= \int^X_0 t^{N-1}e^{-bt} dt/ \int^\infty_0 t^{N-1}e^{-bt} dt \\ &\equiv \int^X_0 D(N, b, t) dt, \text{say},\\ \end{align*} where $N$ and $b$ are positive real numbers and $0 < X \leqq \infty$. In what follows, we generally use the notation $P_N$ to denote the same function, unless $b$ or $X$ are given some specific values. It should be noted that the notation $P(N, b, X)$ does not mean that the function is one of three variables, since we have $P(N, b, X) = P(N, 1, bX)$. The positive real number $b$ is only a scale factor. The function $P_N$ is of appreciable importance in probability and statistics and it is known as the gamma distribution and for $b = \frac{1}{2}$ as the chi-square distribution. It is also of importance in many other branches of applied mathematics. Like the incomplete beta distribution, it is also a special case of the more general confluent hypergeometric function. During the last decade, interest has been revived in the development, use and application of these functions and a number of publications, such as Erdelyi et al. (1953), Tricomi (1954) and Slater (1960), appeared. In these publications, the function treated is defined as the numerator in the middle part of (1) with $b = 1$, and the definition is generally extended for all complex values of $N$ such that $R(N)$ is not a non-positive integer. Many of the properties of $P_N$ are derived in these publications. Bancroft (1949) derived some new properties of the incomplete beta function (particularly, recurrence relations) from the more general properties of the parent hypergeometric function. Similar methods may also be used to derive new properties of $P_N$ of use in statistical work. In this paper, however, we derive such properties directly from the well-known difference-differential properties of $P_N$ given by \begin{align*}\tag{2}(\partial^r/\partial X^r) P_N &= (-b)^r\Delta^rP_{N-r}, \\ &= (\partial^{r-1}/\partial X^{r-1})D(N, b, X),\quad N > r,\\ \end{align*} where $r = 0, 1, 2, 3, \cdots$ and where \begin{equation*}\tag{3}\Delta^{r+1} P_T = \Delta^r(\Delta P_T) = \Delta^r(P_{T+1} - P_T)\end{equation*} are the advancing finite differences of $P_N$ with respect to $N$ and with a unit difference interval. Property (2) for $r = 1$ may be used, in fact, to define the function $P_N$ itself, as in Milne-Thomson (1933). This property was also used in 1947 by Khamis in an unpublished Ph.D thesis (1950) to derive, inter alia, a series expansion in terms of chi-square integrals to approximate to statistical distribution functions and also to formulate a computational method for the tabulation of the chi-square distribution itself. This method was later used by Hartley and Pearson (1950) in computing a five decimal table for this distribution. The chi-square series expansion was later generalized by Khamis (1960a) into an expansion in terms of incomplete gamma function ratios and the computational method was employed by Khamis, (1964a) and (1965) to prepare a six decimal table of the chi-square integral and a master ten decimal table, for very fine $N$ and $X$ intervals, of the incomplete gamma function (1) for $b = \frac{1}{2}$ (a description of this table is given by Khamis (1964b). In this paper we make further use of the property (2) to derive other new and useful properties of the function (1). In Section 2 we derive a simple recurrence relation for $P(T, b, X)$ for $T = N, N + 1$ and $N + 2$. In Section 3 we derive new $N$-wise sum formulae as consequences of this recurrence relation. We then give in Section 4 an $X$-wise sum formula based on the Taylor expansion of (2) in the neighbourhood of $X$. This is made possible by extending the definition of $P(N, b, X)$ for $N \leqq 0$, and thus enabling the removal of the condition $N > r$ in (2) above. Other examples of the use of property (2) are given in Section 5 where in particular, the Laguerre polynomials are expressed in terms of the differences of the function $P(N, b, X)$. Section 5 contains also examples of some numerical applications of the previous results. The importance of the properties derived here and the simplicity inherent in such derivations due to the nature of property (2) are further enhanced by work recently carried out by Wise (1950) and developed further by H. O. Hartley and E. J. Hughes (in process of publication) where the incomplete gamma function ratio is shown to provide quite satisfactory approximations to the incomplete beta function ratio and to many other statistical functions. We note finally, that the notation used in (1) is only one of many notations used by different authors. The references given above include most of the different notations used for the function $P_N$ and other related functions.

This paper presents some generalizations of the elementary renewal theorem (Feller [9]) and the deeper renewal theorem of Blackwell [1], [2] to planar walks. Let $U\lbrack A\rbrack$ denote the expected number of visits of a transient 2-dimensional nonarithmetic random walk to a Borel set $A$ in $R^2$. Let $S(\mathbf{y}, a)$ denote the sphere of radius $a$ about the point $\mathbf{y}$ for a given norm $\| \cdot \|$ of the Euclidean topology. Then, the elementary renewal theorem for the plane, given in Section 2, states that $\lim_{a\rightarrow\infty} U\lbrack S(\mathbf{O}, a)\rbrack/a = 1/ \|E\lbrack\mathbf{X}_1\rbrack\|$, where $\mathbf{X}_1 = (X_{11}, X_{21})$ is the first step of the walk, if $E\lbrack\mathbf{X}_1\rbrack$ exists. Farrell has obtained similar results for nonnegative walks in [7]. Section 4 contains the main result of the paper, the generalization of the Blackwell renewal theorem in the case of polygonal norms for random walks which have both $E\lbrack X^2_{11}\rbrack$ and $E\lbrack X^2_{21}\rbrack$ finite and one of $E\lbrack X_{11}\rbrack, E\lbrack X_{21}\rbrack$ different from 0. The theorem states that $\lim_{a\rightarrow\infty} \{U\lbrack S(\mathbf{0}, a + \Delta,)\rbrack - U\lbrack S(\mathbf{0}, a)\rbrack\} = \Delta/\|E\lbrack\mathbf{X}_1\rbrack\|$ for every $\Delta \geqq 0$ and $\| \cdot \|$ specified above. This result is also established with no restrictions on $E\lbrack X^2_{11}\rbrack, E\lbrack X^2_{21}\rbrack$ under different regularity conditions, in particular, for the $L_\infty$ norm if both $E\lbrack X_{11}\rbrack$ and $E\lbrack X_{21}\rbrack$ are different from 0, and correspondingly for the $L_1$ norm if $\|E\lbrack X_{11}\rbrack| \neq \|E\lbrack X_{21}\rbrack|$. Farrell in [8] has obtained more general results for nonnegative walks under somewhat more restrictive regularity conditions and by a different method. The next section gives the Blackwell theorem for totally symmetric transient walks with finite step expectations, both of whose marginal walks are recurrent. We conclude with a discussion of extensions of these results to higher dimensions and some open questions.

The case $\beta = 0$ is the famous Polya (1931) Urn; a discussion of its elementary properties can be found in (Feller, 1960, Chapter IV) and (Frechet, 1943). These facts about the Polya Urn are a classical part of the oral tradition, although some have yet to appear in print (see Blackwell and Kendall, 1964). The fractions $(W_n + B_n)^{-1}W_n$ converge with probability 1 to a limiting random variable $Z$, which has a beta distribution with parameters $W_0/\alpha, B_0/\alpha$. Given $Z$, the successive differences $W_{n + 1} - W_n :n \geqq 0$ are conditionally independent and identically distributed, being $\alpha$ with probability $Z$ and 0 with probability $1 - Z$. Proofs are in Section 2. If $\beta > 0$, the situation is radically different. No matter how large $\alpha$ is in comparison with $\beta$, the fractions $(W_n + B_n)^{-1}W_n$ converge to $\frac{1}{2}$ with probability 1. This seemingly paradoxical result can be sharpened in several ways. Abbreviate $\rho$ for $(\alpha + \beta)^{-1}(\alpha - \beta)$. If $\rho > \frac{1}{2}$, it is proved in Section 3 that $(W_n + B_n)^{-\rho}. (W_n - B_n)$ converges with probability 1 to a nondegenerate limiting random variable. This result in turn fails for $\rho \leqq \frac{1}{2}$. If $0 < \rho \leqq \frac{1}{2}$, the sequence $(W_n + B_n)^{-\rho}(W_n - B_n)$ has plus infinity for superior limit and minus infinity for inferior limit, with probability 1. If $\rho < 0$, the sequence $(W_n - B_n)$ has plus infinity for superior limit and minus infinity for inferior limit, with probability 1. In both cases, the tail $\sigma$-field of $(W_n, B_n) :n \geqq 0$ is trivial. If $\rho < \frac{1}{2}$, it is proved in Section 5 that the distribution of $n^{-\frac{1}{2}}(W_n - B_n)$ converges to normal with mean 0 and variance $(\alpha - \beta)^2/(1 - 2\rho)$. When $\rho = \frac{1}{2}$, the last fraction is not defined; but the distribution of $(n \log n)^{-\frac{1}{2}}(W_n - B_n)$ converges to normal with mean 0 and variance $(\alpha - \beta)^2$. The asymptotic normality of $W_n - B_n$ for $\rho \leqq \frac{1}{2}$ was observed by Bernstein (1940). I am grateful to J. A. McFadden for calling this paper to my attention. Consider taking $\alpha = 0$ and $\beta = 1$, so $\rho < 0$. Since $(W_n + B_n)^{-1}W_n$ converges to $\frac{1}{2}$, therefore $W_n - B_n$ is asymptotically like the sum of $n$ independent random variables, each equal to $+1$ with probability $\frac{1}{2}$ and $-1$ with probability $\frac{1}{2}$. It is tempting to conclude that the distribution of $n^{-\frac{1}{2}}(W_n - B_n)$ converges to normal with mean 0 and variance 1. From the preceding paragraph, however, the asymptotic variance is $\frac{1}{3}$. There is an even more startling difference between the asymptotic behavior of $(W_n - B_n) : n \geqq 0$ and that of a coin-tossing game. Let $X_n :n \geqq 1$ be independent and $\pm 1$ with probability $\frac{1}{2}$ each. Let $S_n = X_1 + \cdots + X_n, S_{j/n,n} = n^{-\frac{1}{2}}S_j$. Define $S_{t,n}$ for $0 \leqq t \leqq 1$ and $nt$ not integral by linear interpolation. By the celebrated Invariance Principle of Donsker (1951), the law of $\{S_{t,n} :0 \leqq t \leqq 1\}$ converges in a strong way to the law of a Brownian motion. However, for $\rho < \frac{1}{2}$ suppose we define $Z_{j/n,n} = n^{-\frac{1}{2}}(W_j - B_j)$ and $\{Z_{t,n} :0 \leqq t \leqq 1\}$ by linear interpolation. The law of $\{Z_{t,n} :0 \leqq t \leqq 1\}$ converges in the sense of the Invariance Principle to the law of a process $\{Z_t :0 \leqq t \leqq 1\}$. Now $Z_t$ is normal with mean 0 and variance $(1 - 2\rho)^{-1}(\alpha - \beta)^2t$. But $\{Z_t :0 \leqq t \leqq 1\}$, though Gaussian, does not have independent increments. On the other hand, $\{t^{-\rho}Z_t :0 \leqq t \leqq 1\}$ is a nonhomogeneous Brownian motion. If $\rho = \frac{1}{2}$, it is necessary to put $Z_{j/n,n} = (n \log n)^{-\frac{1}{2}}(W_j - B_j)$. In the limit, $Z_t = t^{\frac{1}{2}}Z_1$, where $Z_1$ is normal with mean 0 and variance $(\alpha - \beta)^2$. These results were obtained independently by K. Ito and myself. Details will be given in a future joint paper. D. Ornstein has obtained this very intuitive proof that $(W_n + B_n)^{-1}W_n$ converges to $\frac{1}{2}$ with probability 1 for $\beta > 0$. Suppose first $\alpha > \beta$. If $0 \leqq x \leqq 1$ and $P\lbrack\lim \sup (W_n + B_n)^{-1}W_n \leqq x\rbrack = 1$, by an easy variation of the Strong Law, with probability 1, in $N$ trials there will be at most $Nx + o(N)$ drawings of a white ball; so at least $N(1 - x) - o(N)$ drawings of black. Therefore, with probability 1, $\lim \sup (W_n + B_n)^{-1}B_n$ is bounded above by $\lim_{N\rightarrow\infty}\{\alpha\lbrack Nx + o(N)\rbrack + \beta\lbrack N(1 - x) - o(N)\rbrack\}/N(\alpha + \beta)$ or $(\alpha + \beta)^{-1}\lbrack\beta + (\alpha - \beta)x\rbrack$. Starting with $x = 1$ and iterating, $P\lbrack\lim \sup (W_n + B_n)^{-1} \leqq \frac{1}{2}\rbrack = 1$ follows. Interchange white and black to complete the proof for $\alpha > \beta$. If $\alpha < \beta$, and $P\lbrack\lim \sup (W_n + B_n)^{-1}W_n \leqq x\rbrack = 1$, then a similar argument shows $P\lbrack\lim \sup (W_n + B_n)^{-1}B_n \leqq (\alpha + \beta)^{-1}. (\alpha + (\beta - \alpha)x)\rbrack = 1$. The argument proceeds as before, except both colors must be considered simultaneously.

Estimation of the spectral density of a discrete stationary process is considered under the assumption that some of the observations are missing due to some binomially distributed random mechanism. The asymptotic variance of the estimate is derived for normally distributed random variables. As in the author's dissertation [7], the extension to processes which are stationary of fourth order is fairly standard. It is hoped that one will be able to extend these results to more complicated random mechanisms.

Let $(\mathscr{X}, \mathscr{a})$ be a measurable space and $\Theta$ be an open subset of a $k$-dimensional Euclidean space. For each $\theta \varepsilon \Theta$ let $P_\theta$ be a probability measure on $\mathscr{a}$. We assume that for every $\theta \varepsilon \Theta, \{X_n, n \geqq 0\}$ is a Markov process defined on $(\mathscr{X}, \mathscr{a})$ into $(R, \mathscr{B})$, where $(R, \mathscr{B})$ denotes the Borel real line. Furthermore, we denote by $\mathscr{a}_n$ the $\sigma$-fields induced by the random variables $\{X_0, X_1, \cdots, X_n\}$, and by $P_{n, \theta}$ the restriction of $P_\theta$ to $\mathscr{a}_n$. In the present paper we give conditions under which the sequence of families of probability measures $\{P_{n,\theta}, \theta \varepsilon \Theta\}$ has the desirable property of being differentially asymptotically normal. This implies that in some neighborhood of $\theta \varepsilon \Theta, \{P_{n,\theta}, \theta \varepsilon \Theta\}, n \geqq 0$ can be, for certain problems, approximately treated as if they were normal. For a detailed account of the notions involved in this paper the reader is referred to [5], in particular, Section 5. Also in the Appendix of the present paper one can find the definitions of the concepts most frequently used in this work, including that of the differentially asymptotically normal families of distributions. In Section 2 the required notation is introduced and also the assumptions being made throughout the paper are listed. In Section 3 we state the main result, and in the following subsections we give the proof of it in several steps.

Let $X_1, X_2, \cdots$ be a sequence of independent, identically distributed random variables, and write $Z_n$ for the maximum of $X_1, X_2, \cdots X_n$. Then there are two well known theorems concerning the limiting behaviour of the distribution of $Z_n$. (See, for example, Gumbel [7].) Firstly, if for some sequence of pairs of numbers $a_n, b_n$, the quantities $a^{-1}_n(Z_n - b_n)$ have a non-degenerate limiting distribution as $n \rightarrow \infty$, then this limit must take one of three forms. Secondly, if $c_n = c_n(\xi)$ is defined by $P\lbrack X > c_n\rbrack \leqq \xi/n \leqq P\lbrack X \geqq c_n\rbrack$, then $P\lbrack Z_n \leqq c_n\rbrack$ tends to $e^{-\xi}$ as $n \rightarrow \infty$. Suppose now that we drop the assumption of independence of the $X_j$, and require instead that the sequence $\{X_n\}$ be a stationary stochastic process: then it might be expected that similar results will hold, at least if $X_i$ and $X_j$ are nearly independent when $|i - j|$ is large. In Section 2 it will be shown that, if the process $\{X_n\}$ is uniformly mixing, then the only possible non-degenerate limit laws of $a^{-1}_n(Z_n - b_n)$ are just those that occur in the case of independence, and that the only possible limit laws of $P\lbrack Z_n \leqq c_n\rbrack$ are of the form $e^{-k\xi}, k$ being some positive constant not greater than one. The uniform mixing property is rather strong at first sight. It is however clear that some restriction is necessary, at least for example to ergodic processes, and the parallel which exists to a certain extent with normed sums of random variables suggests that uniform mixing hypotheses may be appropriate. (Cf. Rosenblatt [9]). In the independent case converse results hold, as we have already observed for the second problem, giving necessary and sufficient conditions for the existence of the limits. We give some results concerning this problem in Section 3, but they are not altogether satisfactory. Berman ([2] and especially [3]) has investigated the same problem, under somewhat different conditions. His results for the Gaussian case in [3], however, include those which can be obtained from Lemmas 1 and 2 of the present paper, and in consequence we do not reproduce them here. The second problem mentioned above was considered by Watson [10] for $m$-dependent stationary processes, and his paper did in fact suggest the present investigations. His results are contained in Section 3. Certain results were announced without proof by Chibisov [4], but these appear not to overlap our results.

Assuming that the distribution being sampled is absolutely continous, Parzen [3] has established the consistency and asymptotic normality of a class of estimators $\{f_n(x)\}$ based on a random sample of size $n$, for estimating the probability density. In this paper, we relax the assumption of absolute continuity of the distribution $F(x)$ and show that the class of estimators $\{f_n(x)\}$ still consistently estimate the density at all points of continuity of the distribution $F(x)$ where the density $f(x)$ is also continuous. It is further shown that the sequence of estimators $\{f_n(x)\}$ are asymptotically normally distributed. The extension of these results to the bi-variate and essentially the multi-variate case with applications and a discussion on the construction of higher dimensional windows will be presented at the International Symposium in Multivariate Analysis to be held in Dayton, Ohio during June 1965.

Let $F(x)$ be a probability distribution function. Assuming the singular part to be identically zero, it is well known (see e.g. Cramer [1] pp. 52, 53) that $F(x)$ can be decomposed into $F(x) = F_1(x) + F_2(x)$ where $F_1(x)$ is an everywhere continuous function and $F_2(x)$ is a pure step function with steps of magnitude, say, $S_\nu$ at the points $x = x_\nu, \nu = 1, 2, \cdots, \infty$ and that finally both $F_1(x)$ and $F_2(x)$ are non-decreasing and are uniquely determined. In this paper the problem of estimating the jump $S_i$ corresponding to the saltus $x = x_i$ is considered. Also considered are the problems of estimation of reliability and hazard rate. Based on a random sample $X_1, X_2, \cdots X_n$ of size $n$ from the distribution $F(x)$, consistent and asymptotically normal classes of estimators are obtained for estimating the jump $S_i$ corresponding to the saltus $x = x_i$. Based on the earlier work of the author [2] on estimation of probability density, consistent and asymptotically normal estimates are obtained for the reliability and hazard rate.

We consider a sequence of independent and identically distributed positive stochastic variables $x_1, x_2, x_3, \cdots$ with the distribution function $F(x)$. Let $y_n$ be the smallest of the values taken by the $n$ first of these variables and $S_n = y_1 + y_2 + \cdots + y_n$. It is then shown that $S_n/\log n$ tends in probability to the value $F = \lim_{t\downarrow 0} t/F(t)$ assumed to exist as a finite or infinite number.

Let $x_1, \cdots, x_n$ be independent observations on a $p$-dimensional random variable $X = (X_1, \cdots, X_p)$ with absolutely continuous distribution function $F(x_1, \cdots, x_p)$. An observation $x_i$ on $X$ is $x_i = (x_{1i}, \cdots, x_{pi})$. The problem considered here is the estimation of the probability density function $f(x_1, \cdots, x_p)$ at a point $z = (z_1, \cdots, z_p)$ where $f$ is positive and continuous. An estimator is proposed and consistency is shown. The problem of estimating a probability density function has only recently begun to receive attention in the literature. Several authors [Rosenblatt (1956), Whittle (1958), Parzen (1962), and Watson and Leadbetter (1963)] have considered estimating a univariate density function. In addition, Fix and Hodges (1951) were concerned with density estimation in connection with nonparametric discrimination. Cacoullos (1964) generalized Parzen's work to the multivariate case. The work in this paper arose out of work on the nonparametric discrimination problem.

Let the function $F(x)$ be a distribution function for a continuous symmetric distribution, and let $X_{(i)}$ represent the $i$th order statistics from a sample of size $n$. It is shown in this paper that for $i \geqq (n + 1)/2$ $E(X_{(i)}) \geqq G(i/(n + 1)) \quad\text{if} F \text{is unimodal}$ and $E(X_{(i)}) \leqq G(i/(n + 1)) \quad\text{if} F is U\text{shaped},$ where $x = G(u)$ is the inverse function of $F(x) = u$. The definitions of unimodal and $U$-shaped distributions are given in Section 3. The above inequalities are of interest, since it is known (Blom (1958), Chapters 5 and 6) that for sufficiently large $n$ the bound $G(i/(n + 1))$ approaches $E(X_{(i)})$.

Let the random vector $X = (X_1 \cdots X_p)'$ have a multivariate normal distribution with unknown mean $\xi$ and unknown nonsingular covariance matrix $\Sigma$. Write $\bar\Gamma = \Sigma^{-1}\xi = (\Gamma_1, \Gamma_2, \Gamma_3)'$, where $\Gamma_1, \Gamma_2$ and $\Gamma_3$ are subvectors of $\bar\Gamma$ containing first $q$, next $p' - q$ and last $p - p'$ components of $\bar\Gamma$ respectively. We will consider here, the problem of testing the hypothesis $H_0 : \Gamma_3 = \Gamma_2 = 0$ against the alternative $H_1 : \Gamma_2 = 0, \Gamma_2 \neq 0$ when $p > p' > q$ and $\xi, \Sigma$ are both unknown. The origin of the problem and its likelihood ratio test have been discussed in Giri (1964a, 1964b). It has also been shown there that for $p = p' > q$, the likelihood ratio test of $H_0$ against $H_1$ is uniformly most powerful invariant similar. In this paper we will prove that the likelihood ratio test of $H_0$ against $H_1$ is uniformly most powerful invariant similar for the general case, i.e. $p > p' > q$. A corollary that the likelihood ratio test is uniformly most powerful similar among the group of tests with power depending only on $\Delta(H_1)$ (defined in Section 1) will follow from this. The problem of testing $H_0$ against $H_1$ remains invariant under the group $G$ of $p \times p$ nonsingular matrices $g = \begin{pmatrix}g_{11} & 0 & 0 \\ g_{12} & g_{22} & 0 \\ g_{13} & g_{23} & g_{33}\end{pmatrix}$ operating as $(X, \xi, \Sigma) \rightarrow (gX, g\xi, g\Sigma g')$, where $g_{11}, g_{22}$ and $g_{33}$ are $q \times q, (p' - q) \times (p' - q)$ and $(p - p') \times (p - p')$ submatrices of $g$ respectively. We may restrict our attention to the space of minimal sufficient statistic $(\bar X, S)$ of $(\xi, \Sigma)$. A maximal invariant in $(\bar X, S)$ under $G$ is $R = (R_1, R_2, R_3)'$ and a corresponding maximal invariant in $(\xi, \Sigma)$ under $G$ is $\Delta = (\delta_1, \delta_2, \delta_3)'$ where $R_i \geqq 0, \delta_i \geqq 0$ are defined in Section 1. In terms of maximal invariant the above problem reduces to that of testing $H_0 :\delta_3 = \delta_2 = 0, \delta_1 > 0$ against $H_1 :\delta_3 = 0, \delta_2 > 0, \delta_1 > 0$ when $\xi, \Sigma$ are both unknown. It has been shown in Giri (1964a) that on the basis of $N$ random observations the likelihood ratio test of $H_0$ against $H_1$ is given by reject $H_0$, if $Z = (1 - R_1 - R_2)/(1 - R_1) \leqq Z_0$, where the constant $Z_0$ is determined in such a way that the test has size $\alpha$ and under $H_0, Z$ has beta-distribution with parameters $(N - p')/2, (p' - q)/2$. In Section 1 we will find the maximal invariants $R$ and $\Delta$ along with the distribution of $R$. Actually, we will first find here the maximal invariant in $(\bar X, S)$ under a more general group $G_k$ (to be defined in Section 1) and its distribution. The maximal invariant $R$ under $G$ and its distribution will follow from this as a special case. In Section 2 we will prove the theorem that the likelihood ratio test is uniformly most powerful invariant similar.

This note deals with a two-parameter family of discrete distributions which has interesting moment properties. We are indebted to Arthur Schleifer, Jr. for having informed us that the distributions derived below by formal methods are integer parameter cases of what are called Beta-Pascal in Raiffa and Schlaifer (1961), p. 238. We were led to the present family by searching for discrete distributions to be used in numerical work with inventory problems. In particular, we were interested in enlarging our discrete repertoire (binomial, Poisson, geometric, negative binomial, etc.) in two directions. First, we sought convenient closed form expressions for various associated probabilities and expectations. This would of course simplify computation of penalty functions and optimal inventory levels. Second, we sought representation of rather extreme behavior so as to subject our theoretical formulations to stresses such as large dispersions about average future demand. It turns out that the present family meets both criteria. Not only is it highly tractable but corresponding to each value $r = 0, 1, 2, \cdots$ there is a member distribution possessing its $r$th moment but whose $(r + 1)$st moment fails to exist. Our approach is entirely analogous to the following formal "development" of geometric distributions. We start with the observation that $\sum^\infty_{i = 1} q^i = 1/(1 - q)$ is a convergent series of positive terms for $0 < q < 1$. Then we normalize to unit sum and consider the series to represent a probability distribution: $\sum^\infty_{i = 1} pq^i = 1$ where $p = 1 - q$. If this is done, we obtain the one-parameter geometric distribution whose mode is zero, whose mean equals $q/p$ and whose variance-to-mean ratio equals ($1 + mean$). Our present effort similarly begins with a convergent series (of factorial power functions) and we find a two-parameter family where, interestingly enough, not only is the mode zero but the variance-to-mean ratio, when it exists, approaches the quantity $(1 + \text{mean})$.