Procedures are exhibited and analyzed for converting a sequence of i.i.d. Bernoulli variables with unknown mean $p$ into a Bernoulli variable with mean $\frac{1}{2}$. The efficiency of several procedures is studied.

In John Christopher's novel [2], the fate of the whole world is involved in the outcome of a game of Liar Dice. This paper presents a mathematical analysis of a class of zero-sum two-person games related to Liar Dice as described formally in Bell [1]. Karlin [4] provides the necessary background to the theory of games. Player $I$ receives a number $x$ chosen at random from a uniform distribution on the interval (0, 1). He then chooses a number $y$ in (0, 1) and claims that the number $x$ he received is at least $y$. Player $II$, not being informed of $x$, must accept or challenge $I$'s claim. If he accepts $I$'s claim, he loses an amount $b(x, y)$, a given function of $x$ and $y$. If he challenges $I$'s claim, he wins one if in fact $x < y$, and loses one if $x \geqq y$. The function $b(x, y)$ may be understood to represent player $II$'s expected loss in some other game that is played after $II$ accepts $I$'s claim. For example, if when $II$ accepts $I$'s claim, he must draw a number $z$ from the uniform distribution on (0, 1) and win one if $z > y$ and lose one if $z \leqq y$, then $II$'s expected loss is $b(x, y) = 2y - 1$. As another example, if when $II$ accepts $I$'s claim he must draw a number $z$ from the uniform distribution on $(x, 1)$, and win or lose one according as $z > y$ or $z \leqq y$, then $II$'s expected loss is \begin{align*}b(x, y) = -1\quad \text{if} \quad x \geqq y \\ = \frac{2y - x - 1}{1 - x}\quad \text{if} \quad x \leqq y.\end{align*} These two examples are taken later to illustrate the general theory. The use of a general $b(x, y)$ allows treatment of situations wherein the basic game is played again whenever $II$ accepts $I$'s claim, with the roles of the players reversed and with the distribution of the future $x$ dependent upon the past $x$ and $y$. Our solution to the general problem (Theorem 2) requires rather strong conditions on $b(x, y)$. The general problem is therefore not to be considered completely solved. However, when $b(x, y)$ is independent of the variable $x$, a complete solution is possible under the sole requirement that $b(y)$ (no longer a function of $x$) be nondecreasing in $y$. This is presented in Theorem 1 and the subsequent remarks.

Ehrenfeld and Zacks [2] presented randomized procedures for regular fractions and Zacks [3], [4] showed that an unbiased estimator of a given parameter vector in the saturated fractional replicate case exists only if one randomizes over all possible designs of a certain structure. In this paper, a similar method to the Randomized Procedure I in their paper is given for the saturated irregular main effect fractional replicates and an unbiased estimator of the main effect parameter vector is presented. Prior to this presentation, an invariant property of the information matrix is investigated. The explicit expression of the variance of the estimator is a remaining problem.

We show that a measure-theoretic extension of Cauchy's functional equation, namely, $g(x_1) + g(x_2) = h(f(x_1, x_2))$ a.e., for real-valued functions defined on measure spaces equipped with a "reasonably compatible" arcwise connected topology is equivalent to a theorem which characterizes one-parameter exponential families on such measure spaces in terms of a real-valued sufficient statistic.

Let $\mathbf{S}_i: p \times p(i = 1, 2)$ be independently distributed as Wishart $(n_i, p, \mathbf{\Sigma}_i)$. Let the characteristic roots of $\mathbf{S}_1 \mathbf{S}_2^{-1}$ and $\mathbf{\Sigma}_1 \mathbf{\Sigma}_2^{-1}$ be denoted by $l_i (i = 1,2, \cdots, p)$ and $\lambda_i (i = 1,2, \cdots, p)$ respectively such that $l_1 > l_2 > \cdots > l_p > 0$ and $\lambda_1 > \lambda_2 > \cdots > \lambda_p > 0$. Then the distribution of $l_1, \cdots, l_p$ can be expressed in the form (Khatri [8]) \begin{equation*} \tag{1.1} C|\mathbf{\Lambda}|^{-\frac{1}{2}n_1}|\mathbf{L}|^{\frac{1}{2}(n_1-p- 1)}\{\prod^p_{i<j}(l_j - l_i)\} \int_{O(p)}|\mathbf{I}_p + \mathbf{\Lambda}^{-1}\mathbf{HLH}'|^{-\frac{1}{2}(n_1+n_2)}(\mathbf{H}' d\mathbf{H})\end{equation*} where \begin{equation*}\begin{split}C = 2^{-p}\pi^{\frac{1}{4} p(p-1)}\{\prod^p_{i=1} \mathbf{\Gamma}(i/2)\} \mathbf{\Gamma}_p(\frac{1}{2}n_1 + frac{1}{2}n_2)\{\Gamma_ p(\frac{1}{2}p)\Gamma_p(\frac{1}{2}n_1)\Gamma_ p(\frac{1}{2}n_2)\}^{-1}, \\ \Gamma_p(t) = \pi^{\frac{1}{4}p(p-1)} \prod^p_{j=1} \Gamma(t - \frac{1}{2}j + \frac{1}{2}), \mathbf{L} = \operatorname{diag}(l_1, \cdots, l_p), \mathbf{\Lambda} = \operatorname{diag}(\lambda_1, \cdots, \lambda_p)\end{split}\end{equation*} and $(\mathbf{H}' d\mathbf{H})$ is the invariant measure on the group $O(p)$. However, this form is not convenient for further development. Also, since \begin{align*} \tag{1.2} I &= \int_{O(p)}|\mathbf{I}_p + \mathbf{\Lambda}^{-1} \mathbf{HLH}'|^{\frac{1}{2}(n_1 + n_2)}(\mathbf{H}' d\mathbf{H}) \\ &= C' \sum^\infty_{k=0} \frac{1}{k!} \sum_\kappa\frac{C_\kappa(-\mathbf{\Lambda}^{-1})C_\kappa(\mathbf{L})(n_1 + n_2)_\kappa}{C_\kappa(\mathbf{I}_p)}\end{align*} where $C' = 2^p\pi^{\frac{1}{4}p(p+1)}/\prod^p_{i=1} \Gamma(i/2)$ and the zonal polynomial $C_\kappa(\mathbf{T})$ of any $p \times p$ symmetric matrix $\mathbf{T}$ is defined in James [7], where $k$ is a partition of $k$ into not more than $p$ parts, the use of (1.2) in (1.1) gives a power series expansion, but the convergence of this series is very slow. In the one sample case G. A. Anderson [1] has obtained a gamma-type asymptotic expansion for the distribution of the characteristic roots of the estimated covariance matrix. In this paper we obtain a beta-type asymptotic representation of the roots distribution of $\mathbf{S}_1 \mathbf{S}_2^{-1}$ involving linkage factors between sample roots and corresponding population roots. If the roots are distinct the limiting distribution as $n_2$ tends to infinity has the same form as that of Anderson [1]. If, moreover, $n_1$ is assumed also large, then it agrees with Girshick's result [4], which was also discussed in Anderson [1].

This paper is concerned with the problem of finding reasonably explicit sufficient conditions for the almost admissibility of formal Bayes estimators (for definitions, see Section 2), where the underlying distribution is assumed known up to a single real parameter, and a real function of this parameter is being estimated with squared error as loss. The parameter space is assumed to be a possibly unbounded interval. These conditions are derived in Section 3. They are similar, in appearance, to results obtained by Karlin [3] when the underlying distribution is a member of the family of one parameter exponential distributions and the mean of this distribution is being estimated. The results of Section 3 should be viewed as a refinement of a heuristic argument given by Stein ([6] and [7] pages 233-240). In Section 2, some preliminary results and definitions are given. The results of Section 3 are applied in Section 4 to problems involving either the one dimensional exponential family or the estimation of a function of a single location parameter. Some of the results are known, in at least a similar form, while others are new. A counterexample based on a one dimensional location parameter problem is given in Section 5. It suggests that conditions of the type obtained here may even be necessary.

In this paper, we shall investigate a problem analogous to the one treated in Prakasa Rao [6]. We shall now suppose that there is a sample of $n$ independent observations from a distribution $F$ with monotone failure rate $r$ and the problem is to obtain a maximum likelihood estimator (MLE) of $r$ and its asymptotic distribution. Grenander [4] and Marshall and Proschan [5] have obtained the MLE of $r$ and the latter showed that these estimators are consistent. We obtain the asymptotic distribution of this estimator. The estimation problem is reduced at first to that of a stochastic process and the asymptotic distribution of MLE is obtained by means of theorems on convergence of distributions of stochastic processes. Methods used in this paper are similar to those in Prakasa Rao [6] and therefore, proofs are given only at places where they differ from the proofs of that paper. We shall consider the case of distributions with increasing failure rate (IFR) in detail. Results in the case of distributions with decreasing failure rate (DFR) are analogous to those of IFR and we shall state the main result in Section 7. Sections 2 and 3 deal with the definition and properties of distributions with monotone failure rate $r$. Some results related to the asymptotic properties of the MLE of $r$ are given in Section 4. The problem is reduced to that of a stochastic process in Section 5. The asymptotic distribution of the MLE is obtained in Section 6.

A random walk, $\{S_n\}^\infty_{n=0}$, having positive drift and starting at the origin, is stopped the first time $S_n > t \geqq 0$. The present paper studies the "excess," $S_n - t$, when the walk is stopped. The main result is an upper bound on the mean of the excess, uniform in $t$. Through Wald's equation, this gives an upper bound on the mean stopping time, as well as upper bounds on the average sample numbers of sequential probability ratio tests. The same elementary approach yields simple upper bounds on the moments and tail probabilities of residual and spent waiting times of renewal processes.

The coordinate free (geometric) approach to univariate linear models has added both insight and understanding to the problems of Gauss Markov (GM) estimation and hypothesis testing. One of the initial papers emphasizing the geometric aspects of univariate linear models is Kruskal's (1961). The coordinate free approach is used in this paper to treat GM estimation in a multivariate analysis context. In contrast to the univariate situation, a central question for multivariate linear models is the existence of GM estimates. Of course, it is the more complicated covariance structure in the multivariate case that creates the concern over the existence of GM estimates. As the emphasis is on GM estimation, first and second moment assumptions (as opposed to distributional assumptions) play the key role. Classical results for the univariate linear model are outlined in Section 1. In addition, a recent theorem due to Kruskal (1968) concerning the equality of GM and Least Squares (LS) estimates is discussed. A minor modification of Kruskal's result gives a very useful, necessary and sufficient condition for the existence of GM estimators for arbitrary covariance structures and a fixed regression manifold. In Section 2, the outer product of two vectors and the Kronecker product of linear transformations is discussed and applied to describe the covariance structure of a random matrix. This application includes the case of a random sample from a multivariate population with covariance matrix $\Sigma > 0 ("\Sigma > 0"$ means that $\Sigma$ is positive definite). The question of GM estimation in the standard multivariate linear model is taken up in Section 3. This model is described as follows: a random matrix $Y: n \times p$, whose rows are uncorrelated and each row has a covariance matrix $\Sigma > 0$, is observed. The mean matrix of $Y, \mu$, is assumed to have the form $\mu = ZB$ where $Z: n \times q$ is known and of rank $q$, and $B: q \times p$ is a matrix of regression coefficients. For this model, GM estimators for $\mu$ and $B$ exist and are well known (see Anderson (1958) chapter 8). The main result in Section 3 establishes a converse to this classical result. Explicitly, let $Y$ have the covariance structure as above and assume $\Omega$ is a fixed regression manifold. It is shown that if a GM estimator for $\mu\in\Omega$ exists, then each element $\mu\in\Omega$ can be written as $\mu = ZB$ where $Z: n \times q$ is fixed and $B: q \times p$ ranges over all $q \times p$ real matrices. The results in Section 4 and Section 5 are similar to the main result of Section 3. A complete description of all regression manifolds for which GM estimators exist is given for two different kinds of covariance assumptions concerning $\Sigma$ ($\Sigma$ as above). In Section 4, it is assumed that $\Sigma$ has a block diagonal form with two blocks. Section 5 is concerned with the case when $\Sigma$ has the so-called intra-class correlation form.

This paper is concerned with infinite-state probabilistic transition tables, representing the dynamical behavior of a probabilistic automaton and nonhomogeneous infinite-state Markov chains. Many known theorems are generalized from the finite-state case to the infinite-state case. A possible application to the problem of computing, approximately, products of infinite-state stochastic matrices is outlined.

A favorable game is one in which, loosely speaking, there are opportunities for favorable bets. We consider only games where there is a "double or nothing" return of the stake, the win probability $p$ being drawn independently each day from a distribution $F(p)$, and told to the player before he makes his bet. We are interested, not in maximizing the expected fortune over a number of plays, but in minimizing the probability of eventual ruin. In order that the gambler cannot stand pat, we use two devices. In Model I, he must pay a fee of $1 after each play. In Model II, he must bet a minimum of $1. His bet may never exceed his current fortune, $x$. Let $b$ denote an allowable betting strategy. We define the ruin function by $q(x) = \inf_bP$ (ruin $\mid b$, starting fortune $x$). In both models, if ever $x < 1$, ruin is inevitable. In Section 3, we introduce (following Ferguson, [3]) the idea of a loss function, $q_o(x)$, satisfying certain conditions. A "natural" choice for loss function is $q_0^\ast(x) = 1$ for $x < 1, = 0$ otherwise. We define the minimal expected loss after $n$ games by $q_n(x) = \inf_bEq_0(X_n)$ where $X_n$ is the fortune after the $n$th game. In Lemmas 3.1-3.5, we construct a suitable Borel-measurable betting function to show that $q_n(x)$ satisfies the dynamic programming relationship; if $x \rightarrow X_1(b_1)$ after bet $b_1$, then $q_n(x) = \inf_{b_1} Eq_{n-1} (X_1(b_1))$. The results of Section 3 do not depend on the assumption that the game is favorable, but from now on this assumption is required. Using the natural loss function $q_0^\ast(x)$, we show in Lemma 4.1 that $q_n^\ast(x) \rightarrow q(x)$ as $n \rightarrow \infty$, uniformly in $x$ (in passing, Theorem 5.1 shows that for any loss function, $q_n(x) \rightarrow q(x)$ without uniformity) and we employ this result in Section 6 to show that $q(x)$ is continuous, for Model I only. For both models, $q(x)$ is already known [1] to be lower semi-continuous. These results are sufficient to establish [3] that a particular stationary Markov betting function, $b^\ast(x, p)$, is optimal (although not necessarily unique). For large fortunes $x$, asymptotic forms for the ruin function $q(x)$, and as a consequence for $b^\ast(x, p)$, are already known [1], [3] under a slight restriction on $F(p)$, namely that it have no mass in a neighborhood of 1. On removing the restriction, some weaker results are obtained, and summarized in Section 8.

Let $X$ be a random variable with a family of possible distributions for $X$ indexed by $\lambda\in\Omega. \lambda$ is the realization of a random variable $\Lambda$ taking values in the space $\Omega$. For each $\lambda$, let $f_\lambda$ denote the conditional density of $X$ given $\Lambda = \lambda$ with respect to some $\sigma$-finite measure $\mu$. Let $\mathscr{G}$ be a family of possible a priori distributions $G$ for $\Lambda$. After observing $X$, we wish to test $H: \lambda\in\omega$ against $K: \lambda\in\omega'$ where $\omega$ is a subset of $\Omega$ and $\omega'$ its complement. To determine good tests for this problem, we use an analysis similar to the one of the Neyman-Pearson theory of hypothesis testing. Analogous to the type I and type II errors of the Neyman-Pearson theory are: type (i) error: $\Lambda\in\omega'$ decided and $\Lambda\in\omega$ occurs, type (ii) error: $\Lambda\in\omega$ decided and $\Lambda\in\omega'$ occurs. Analogous to the problem of finding uniformly most powerful level $\alpha$ tests is the problem: subject to: $P_G$(type (i) error) $\leqq \alpha$ for all $G\in\mathscr{G}$ minimize $P_G$(type (ii) error) uniformly for $G\in\mathscr{G}$. A test which achieves this is called a uniformly most powerful (UMP) level $\alpha$ test relative to $\mathscr{G}$. The existence of such UMP level $\alpha$ tests is proved for this hypotheses testing problem for various choices of the family of a priori distributions $\mathscr{G}$. As might be expected these results are closely related to the Neyman-Pearson theory of hypotheses testing. The second section gives four simple situations where the problem of finding UMP level $\alpha$ tests relative to a family of a priori distributions $\mathscr{G}$ reduces to an ordinary testing problem. In the third section, Theorem 1 gives for this testing problem an analogue of the concept of a least favorable distribution from the classical theory of hypotheses testing. Theorem 1 is used to prove Theorem 2 which gives the existence of a UMP level $\alpha$ test when $X$ is real-valued, $\Omega$ is a subset of the real numbers, the family of distributions indexed by $\lambda\in\Omega$ has a monotone likelihood ratio in $x$, and the family $\mathscr{G}$ satisfies a certain condition. The two theorems are applied to several examples. In the following, as always, a test (randomized) is a function $\delta$ defined on the range of $X$ which takes on values in the interval $\lbrack 0, 1\rbrack$. If $X = x$ is observed, $K$ is decided to be true with probability $\delta(x)$ and $H$ with probability $1 -\delta(x)$. For any test $\delta$ and $G\in\mathscr{G}$ we have \begin{equation*}\tag{1} P_G(\text{type (i) error of} \delta) = \int \int_\omega \delta(x)f_ \lambda(x) dG(\lambda) d \mu \quad\text{and}\end{equation*} \begin{equation*}\tag{2} P_G(\text{type (ii) error of} \delta) = \int \int_{\omega'} (1 - \delta(x))f_\lambda(x) dG(\lambda) d\mu\end{equation*} where the integral involving $X$ is over the entire space of $X$. It will often be convenient to think of $\lambda$ as a fixed but unknown parameter and the test $\delta$ as a test for the classical testing problem $H: \lambda\in\omega$ against $K: \lambda\in\omega'$. Changing the order of integration in (1) by Fubini's theorem, we have for the test $\delta$ the following relationship between the type I error of $\delta$, considered as a test for the classical problem, and the type (i) error of $\delta$, considered as a test for the problem of this paper: \begin{equation*}\tag{3} P_G(\text{type (i) error of} \delta) = \int_\omega P_\lambda (\text{type I error of} \delta) dG(\lambda).\end{equation*} In the same way, we have \begin{equation*}\tag{4} P_G(\text{type (ii) error of} \delta) = \int_{\omega'} P_\lambda (\text{type II error of} \delta) dG(\lambda).\end{equation*} We will now prove the existence of UMP level $\alpha$ tests for various families of a priori distributions.

Let $x_1, x_2, \cdots$ be a sequence of independent, observable, discrete, real random variables with respective density functions $f(x, \theta_1), f(x, \theta_2), \cdots$, where $\theta_i\in\Omega$ (a real interval), $i = 1,2, \cdots$. At any stage $s$, having observed $x_1, \cdots, x_s$, we wish to estimate $\theta_s$, relative to the loss function $L(d, \theta) \geqq 0$, under the restraint that $\theta_1 \leqq \cdots \leqq \theta_s$. Most of the literature concerned with estimating ordered parameters deals with maximum likelihood estimators (for example, Robertson and Waltman [3] and references contained therein). Recently, Blumenthal and Cohen [1] considered questions of minimaxity and admissibility of certain estimators of ordered translation parameters of continuous distributions. In Section 3 we obtain results concerning admissible and minimax estimators of $\theta_s$ for a large class of discrete distributions. Theorem 3.2 states that the minimax value $M_k$ of the problem of estimating $\theta_k$ at stage $k$ is the same for all $k = 1,2,\cdots$. Furthermore, the Corollary to Theorem 3.1 states that if $t(x_1, \cdots, x_s)$ is admissible for estimating $\theta_s$ at stage $s$, then $t(x_{k+1}, \cdots, x_{k+s})$ is admissible for estimating $\theta_{k+s}$ at stage $k + s$. This is not true, for example, when $x_k$ is normally distributed with mean $\theta_k$ and unit variance [4]. Hence in situations satisfying the conditions of Theorems 3.1 and 3.2, if there exists an admissible estimator $t(x_1)$ of $\theta_1$ having constant risk, then $t(x_s)$ will be the unique admissible minimax estimator of $\theta_s$, at stage $s$. It is this undesirable property (i.e. being based only on the last observation) which prompts us to look for other estimators which may be "better" in some sense. In Section 4 we use the early observations $(x_1, \cdots, x_{s-1}$, at stage $s$) to construct a sequence of estimators which is asymptotically subminimax $(s \rightarrow \infty)$ as well as having desirable properties for finite $s$. The results in Section 4 are obtained for general cumulative distribution functions as the methods are not peculiar to the discrete situation. However, it should be noted that the main results of Section 4 (Theorem 4.2 and Theorem 4.3) do not yield a worthwhile sequence of estimators unless we are in a situation similar to that of Section 3. That is, unless the existing admissible minimax estimators are undesirable as they are in the discrete case of Section 3.

A family of estimators, each of which dominates the "usual" one, is given for the problem of simultaneously estimating means of three or more independent normal random variables which have a common unknown variance. Charles Stein [4] established the existence of such estimators (for the case of a known variance) and later, with James [3], exhibited some, both for the case of unknown common variances considered here and for other cases as well. Alam and Thompson [1] have also obtained estimators which dominate the usual one. The class of estimators given in this paper contains those of James and Stein and also those of Alam and Thompson.

In [2], Lindeberg conditions of order $\nu \geqq 2$ are defined and shown to be NSC for convergence of $\nu$th absolute moments in the Central Limit Theorem when $\nu = 2k, k = 2,3, \cdots$. Section 4 contains the extension of that result to the case of all $\nu > 2$, the proof depending on some of the theorems, given in Section 2, relating the existence of moments to the integrability of the characteristic function near the origin. The proofs of the results of Section 2 are deferred to Section 3 and depend, in turn, on known results listed in Section 1. Throughout, we use the notations $\mathfrak{RI}x, \mathfrak{Im}x$ for the real, imaginary (respectively) parts of $x$, and $\lbrack x \rbrack$ to mean the largest integer strictly less than $x$.

Let $f(y)$ be a probability density function on the real line, and $F(y)$ the corresponding distribution function. It is assumed that \begin{equation*}\tag{1} F(y) = 0 \text{for} y \leqq 0,\quad F(y) > 0 \text{for} y > 0.\end{equation*} Let $\theta$ be a location parameter, and let $X = (x_1, \cdots, x_n)$ denote a sample of $n$ independent observations, with each $x_i$ distributed according to $F(x - \theta)$. In this paper, we study the minimum variance of unbiased estimators $t = t(X)$ of $\theta$, with special reference to the order, in $n$, of that variance. For example, if $F(y) = 1 - e^{-y}(y > 0)$, the minimum variance is $n^{-2}$ rather than of order $n^{-1}$ as in regular cases. We shall state conditions on $f(y)$ which determine this order. One of these is that $f(y)$ varies regularly at zero with exponent $c - 1 (c > 0)$ (cf. [3], chapter 8, sect. 8-9). Under the conditions imposed, the smallest attainable variance order is $n^{-1}$ if $c > 2$, but $(F^{-1}(n^{-1}))^2$ if $0 < c < 2$. The case $c = 2$ has special features. Since $F(y)$ varies regularly with exponent $c$, the minimum variance order will be $n^{-2/c}L(n^{-1})$ with slowly varying $L (0 < c < 2$; also true for $c = 2)$. When $c > \frac{1}{2}$, the Chapman and Robbins inequality [2] is used to obtain a lower bound for the minimum variance. For $0 < c \leqq \frac{1}{2}$, a new inequality is used; we then restrict slightly the class of unbiased estimators. The results carry over, of course, to distributions with $F(y) < 1$ for $y < 0, F(y) = 1$ for $y \geqq 0$. A generalization to biased estimators (or to mean square error) is straightforward, but some conditions on the bias function will be necessary. Some questions recently raised by Blischke et al. [1] are answered by the theorems. The conditions imposed here may probably be relaxed to some extent. Notation. $K$ and $K'$ denote positive, finite constants. If there exist $K$ and $K'$ such that $K < a(x)/b(x) < K'(|x| < x_0)$, we shall write $a(x) = \Omega(b(x)) (x \rightarrow 0)$. The qualification $(x \rightarrow 0)$ will often be omitted.

Let $f(y)$ be a probability density on the real line, with \begin{equation*}\tag{1} f(y) = ^+R(y) (y > 0),\quad f(y) = ^-R(|y|) (y < 0)\end{equation*} where $^+R$ and $^-R$ are normalized slowly varying functions as $y \rightarrow 0$ (cf. [6] chapter 8, sect. 8). Let $\theta$ be a location parameter. Denote by $X$ a sample $(x_1, \cdots, x_n)$ of $n$ independent observations from the distribution defined by $f(x - \theta)$. By $t = t(X)$ we denote an unbiased estimator of $\theta$. In this paper we study lower bounds for the variances $V_\theta(t)$, with special reference to their order, in $n$. Considering only densities with regular variation at $\theta$, (1) includes all cases where $\theta$ is the location either of a cusp or of a discontinuity with finite and positive values of $f(0 -)$ and $f(0 +)$. Under some conditions on $^+R$ and $^-R$, we calculate a function $\psi(h)$ such that \begin{equation*}\tag{2} V_\theta(t) \geqq K(\psi^{-1}(n^{-1}))^2\quad (\text{all} t) (0 < K < \infty).\end{equation*} It is surmised that this lower variance bound is of the best possible order of $n$. The bound is sometimes $o(n^{-1})$; hyperefficient estimators should then be possible. It is found that $\psi(h)$ depends heavily on the function $^+ \varepsilon(s)$ defined by $^+ R(y) = ^+ A \exp \{-\int^1_{y^+} \varepsilon(s)/s ds\}$, and on the corresponding function $^- \varepsilon(s)$. In view of [6] chapter 8, sect. 9 (or (10) below), this form of $^+R(y)$ is not a constraint on $f(y)$. The estimators $t_0$ constructed by Daniels [5] and Prakasa Rao [10] (for particular $^+R$ and $^-R$) have variances of the order of (2). This order is thus optimal with the densities considered: we have $V_\theta(t_0) \geqq \inf V_\theta(t) \geqq K(\psi^{-1}(n^{-1}))^2$. The Prakasa Rao estimators are hyperefficient. The calculations are based, partly, on ideas from the author's paper [9]. Since a cusp may be a mode, the results of this paper contribute to the discussion on the estimation of the mode (see [4], [11] and references therein). The generalization of (1) to regularly varying $f(y)$ as $y \downarrow 0$ and $y \uparrow 0$ will be treated elsewhere ([8]).--The generalization of (2) to biased estimators $t$ (or mean square error) is straightforward, but some conditions on the bias function will be necessary. Notation. $K$ and $K'$ denote positive, finite constants. If there exist $K, K'$ such that $K < a(x)/b(x) < K'$ for all $x, |x| < x_0$, we shall write $a(x) = \Omega(b(x)) (x \rightarrow 0)$; sometimes we omit $(x \rightarrow 0)$.

Equations are derived for updating estimates of a subset of the regression coefficients in a linear model when more data become available. It is assumed that estimates of the remaining coefficients in the model are of no interest. A simple, but non-trivial special case is the centering of the regression equation by removing an unknown constant.

Let $x: 1 \times n$ be a row vector of random variables such that each $x_i (i = 1,2, \cdots, n)$ is independently normally distributed with mean $\mu_i$ and variance one. Many authors have studied the distribution of what we shall call a univariate quadratic form $xAx'$, where $A$ is a positive definite $n \times n$ matrix. Three types of representation of the distribution have been developed, namely (i) power series about the origin, (ii) mixtures of chi-squares (we refer to these as Ruben-type representations), and (iii) series of Laguerre polynomials. Though all three types of expansion yield correct convergent representations, it has been found that the Laguerre series representation is computationally the most convenient and effective throughout the range of interesting values of the argument. Let $X: p \times n$ be a matrix whose column vectors are independently and identically distributed in multivariate normal distributions having zero mean vector and variance covariance matrix $\Sigma$. If $L$ is a positive definite $n \times n$ matrix, we refer to the $p \times p$ matrix $S = XLX'$ as a positive definite quadratic form with matrix argument. Khatri [11] has given a representation of the density function of the distribution of $S$, somewhat similar to the Ruben-type expansion for the univariate case. In this paper we express the density function of $S$ in terms of Laguerre polynomials with matrix argument. Our results can be easily extended to quadratic forms in a matrix argument when the common multivariate distribution of the column vectors is complex. In Section 2, we give definitions and notations. Section 3 gives some results on integration over orthogonal groups, and in Section 4 we derive the main results.

Several authors [1], [2], $\cdots$, [6], have derived characterizations of a conditional expectation operator. That is, if $T$ is a transformation which maps a particular set of functions into the same set, then necessary and sufficient conditions are specified so that $T$ is a conditional expectation operator. It is shown in the present paper that a similar sort of characterization can be found in the more general case when $T$ is a conditional expectation with respect to a $\sigma$-lattice operator even though $T$ need not be linear.

A limit theorem for random sums of independent random elements of $D\lbrack 0, 1\rbrack$ is proved. The theorem extends and simplifies the proof of a result obtained by R. Pyke in [3]. In a personal communication Professor Pyke remarked to me that the method employed is also the right approach to the problem studied in [4], by adequately defining sequences of random elements of $D\lbrack 0, 1 \rbrack$.

The topics of orthogonality and Fourier series occupy a central position in analysis. Nevertheless, there is surprisingly little statistical literature, with the exception of that of time series and regression, which involves Fourier analysis. In the last decade however, several papers have appeared which deal with the estimation of orthogonal expansions of distribution densities and cumulatives. Cencov [1] and Van Ryzin [6] considered general properties of orthogonal expansion based density estimators and the latter applied these properties to obtain classification procedures. Schwartz [3] and the authors [2] and [4] investigated respectively the Hermite and Trigonometric special cases. The authors also obtained certain general results which apply not only to estimators of the population density but also to estimators of the population cumulative [2], [4] and [5]. In this paper several results derived for the univariate case are extended to the multivariate case. Also a new relationship is obtained which involves general Fourier expansions and estimators. Although there is some reason for calling the Gram-Charlier estimation of distribution densities a Fourier method, one fundamental aspect of Fourier methods is not shared by Gram-Charlier estimation. Gram-Charlier techniques make no use of Parseval's Formula or related error relationships of Fourier analysis. The ease with which the mean integrated square error (MISE) is evaluated, when Fourier methods are applied, accounts for most of the recent interest in this area. Section 1 of this paper deals with an investigation of two general MISE relationships for multivariate estimates of Fourier expansions. The relationship given in Theorem 2 is particularly simple and yet includes the four MISE's which are involved in the estimation problem. In Section 2 the choice of orthogonal functions is restricted to the trigonometric polynomials. It is shown that the MISE of multidimensional trigonometric polynomial estimators are related in a simple way to the Fourier coefficients of the distribution which is being estimated. This result is of considerable utility since it yields a rule for deciding which terms should be included in the estimate of the multivariate density.

The purpose of this note is to indicate a direct and natural way of proving theorems stated in [4] by using an explicit expression for the sequence of normalizing constants belonging to a distribution function attracted to a stable law. This results in a remark concerning a counter example given by Tucker and a slightly sharpened version of his Lemma 5.

In [1], P. J. Bickel and the author presented a theorem giving sufficient conditions for the equivalence of invariant and almost-invariant functions. In the subsequent discussion, it was intimated without proof that the results extend to certain other classes of distributions that do not satisfy the hypotheses of the main theorem. A small lacuna was glossed over in the discussion; however, the conclusions are correct for the examples given there. In this note we justify those conclusions. We adopt the notation of [1] without further comment.

We shall use the notation of [1], Sections 2 and 3. Briefly, multiple decision problems (m.d.p.'s) of Type 1 have the following formulation. On the basis of the outcome of a random variable $(\operatorname{rv}) X$, which takes on values in the space $\mathscr{H}$, one element of the finite decision space $\mathscr{D} = \{d_0, d_1,\cdots, d_n\}$ is to be selected. The $\operatorname{rv} X$ has a density $p_\theta$ w.r.t. some $\sigma$-finite measure $\mu$ for some $\theta\in\Omega$. The parameter space $\Omega$ is partitioned into $(m + 1)$ disjoint subsets $\Omega = \Omega_0 \cup \cdots \cup\Omega_m$ and $\Omega_1 \cup \cdots \cup\Omega_m$ is denoted by $\Omega'$. The loss function $L: \Omega'x\mathscr{D} \rightarrow \lbrack 0, \infty)$ with $L(\theta, d) = w_{ij} \geqq 0$ for $d = d_j$ and $\theta\in\Omega_i (i = 1, \cdots, m; j = 0, \cdots, n)$ is constant over each $\Omega_i$ and is not defined for $\theta\in\Omega_0$. Type 1 m.d.p.'s satisfy the following assumptions: (i) $\Omega \subset R^s$, (ii) $\lbrack \Omega_1 \rbrack \cap \cdots \cap \lbrack \Omega_m \rbrack = \Omega_0' \neq \varnothing$, where $\lbrack \rbrack$ denotes closure w.r.t. the usual topology, (iii) for every test function $\varphi, E_\theta\lbrack\varphi(X) \rbrack$ is continuous in $\theta$. A m.d.p. of Type 1 will be called regular, if it satisfies the additional assumption (iv) for every test function $\varphi, E_{\theta_0}\lbrack \varphi(X) \rbrack = 0$ for all $\theta_0\in\Omega_0'$ implies that $E_\theta\lbrack \varphi(X) \rbrack = 0$ for all $\theta\in\Omega'$. A procedure $\delta = \delta(\varphi_0,\cdots, \varphi_n)$ is unbiased if and only if for $i = 1,\cdots, m$ the following inequalities hold \begin{equation*}\tag{1} \sum^n_{j=0} w_{hj} E_\theta \lbrack \varphi_j(X) \rbrack \geqq \sum^n_{j=0} w_{ij} E_\theta \lbrack \varphi_j(X) \rbrack,\quad\theta\in\Omega_i; h = 1, \cdots, m.\end{equation*} Let $W(M)$ denote the class of all unbiased (minimax risk) procedures for a m.d.p. of Type 1. Procedures with the same risk function will be identified. Let $w_{\cdot j}$ denote the point $(w_{lj},\cdots, w_{mj})$ of $R^m$ and let $S$ denote the convex hull of $w_{\cdot j}(j = 0, \cdots, n)$. Finally, let $E$ denote the set of points $e\in S$ with $e_1 = \cdots = e_m$. In [1] Theorem 4.1 it was proved that for problems of Type 1, a sufficient condition for $W \subset M$ is that there exists a point $e\in E$ that is both a minimax and a maximum point of $S$. This note establishes the following result. THEOREM. For regular problems of Type 1, necessary and sufficient for $W \subset M$ is that one of the following conditions holds: (i) $E = \varnothing$, (ii) $E$ consists of exactly one point $e$ and this is a minimax point of $S$. The following non-trivial example shows that for a regular m.d.p. of Type 1 the sufficient condition of Theorem 4.1 [1] is not necessary. Take $m = 3, n = 2, w_{\cdot 0} = (3, - 3, 0), w_{\cdot1} = (-3, 3, 0), w_{\cdot 2} = (6, -4, 6)$. Then $e = (0, 0, 0) = \frac{1}{2} w_{\cdot 0} + \frac{1}{2} w_{\cdot 1}$ is the unique convex representation with all coordinates equally large. Moreover, $e$ is a minimax point, so that condition (ii) of the Theorem is satisfied, but $e$ is not a maximin point, since the point $3w_{\cdot 1}/5 + 2w_{\cdot 2}/5 = (3/5, 1/5, 12/5)$ has all coordinates strictly larger than zero.

A combinatorial arrangement was proposed by T. G. Room (1955). Archbold and Johnson (1958) gave a method of construction for these arrangements (which they called Room's squares) and suggested an application to experimental design. The purpose of the present communication is to rectify an error in the analysis given by Archbold and Johnson in the above paper.