This paper studies the problem of how to space observations in regression so as to minimize the variance of an estimate of the regression function value at an arbitrary point in the domain of observations. Necessary and sufficient conditions are obtained for such a design, called a minimax design, in two dimensional polynomial regression of the type in which the regression function possesses a product structure. Such conditions are also obtained for minimax designs in one dimensional trigonometric and two dimensional spherical harmonics regression. Particular designs of the latter type are constructed.

Recently interest has arisen in statistical applications of the bispectrum of stationary random processes. (The bispectrum can be thought of as the Fourier transform of the third-order moment function of the process.) The principal area of statistical harmonic analysis to receive attention previous to this time has been second-order (i.e. spectral) theory on which there is a vast literature. However, the spectrum is most useful in problems of a "linear nature" (see discussion beginning on p. viii of Blackman and Tukey [2]) and provides insufficient information in nonlinear problems. A desire to study phenomena of a nonlinear character has attracted attention to the higher order theory. Such was the case, for example, in a recent study by Hasselmann, Munk and MacDonald [8] where the bispectrum is used in connection with oceanographic problems, among which, as the authors state, a number of interesting phenomena such as surf beats, wave breaking, and the energy transfer between wave components can be explained only by the nonlinearity of the wave motion. The bispectrum therefore provides a first glimpse at the nonlinear effects. It is the purpose of the present paper to discuss estimating the weighted and unweighted bispectral density given a set of observations of the process. The relevant properties (consistency and asymptotic unbiasedness) of the estimates are derived for certain general classes of processes.

In estimation of a real valued parameter $\theta$, using observations from the probability density $f(x \mid \theta)$, and using loss function $L(\theta, \phi)$, the prior density which minimizes asymptotic bias of the associated estimator is shown to be $J(\theta) = \varepsilon((\partial/\partial\theta) \log f)^2/\lbrack(\partial^2/\partial\phi^2)L(\theta, \phi)\rbrack^{\frac{1}{2}}_{\phi = \theta}$. Results are also given for estimation in higher dimensions.

The distribution of the latent roots depends on a definite integral over the group of orthogonal matrices. This integral defines a function of the latent roots of both the covariance matrix and the estimated covariance matrix. With an integration procedure involving first a substitution and then an expansion of the resulting integrand the first three terms of an expansion for the integral are found. This expansion is given in increasing powers of $n^{-1}$, where $n$ is the sample number less one. A numerical example is given for the distribution of the latent roots using the expansion for the definite integral given in this paper. Improved maximum likelihood estimates for the latent roots are found and the likelihood function is considered in detail.

The problem of classifying an observation into one of two multivariate normal populations with the same covariance matrix has been thoroughly discussed by Anderson [1] when the populations are completely known. Anderson and Bahadur [2] treated the case of different and known covariance matrices and obtained the minimal complete class restricting to linear classification rules. Wald [11], Anderson [1], and Rao [7] suggested some heuristic classification rules based on the sample estimates of the unknown parameters of the two normal distributions having the same covariance matrix. One of these heuristic rules is the maximum likelihood rule (ML rule) which classifies the observation into the population $\Pi_i$ if the maximum likelihood (likelihood maximized under the variation of the unknown parameters) obtained under the assumption that the observation to be classified comes from $\Pi_i$ is greater than the corresponding maximum likelihood assuming that the observation comes from $\Pi_j(i \neq j; i, j = 1, 2)$. Sitgreaves [8], [9], and John [5] obtained the explicit forms of the distributions of the classification statistics proposed by Anderson and Wald. Many other papers in this line are included in the book cited in the reference [9]. Ellison [4] derived a class of admissible rules which includes the ML rule for the problem of classification into more than two normal populations with different and known covariance matrices. Cacoullos [3] obtained an invariant Bayes rule and an admissible minimax rule for the problem of selecting one out of a finite number of completely specified normal populations which is "closest" to a given normal population whose covariance matrix is unknown. It will be shown in this paper that the ML rule is an unbiased admissible minimax rule when the common covariance matrix of the two normal populations is known; and, when the common covariance matrix is unknown, that the corresponding ML rule is unbiased and is an admissible minimax rule in an invariant class. The loss function in each problem is assumed to be a function (satisfying some mild restrictions) of the Mahalanobis distance between the two populations.

It is frequently of interest to an investigator to decide whether an unknown parameter lies in some interval and also to obtain a point estimate of the parameter. There are many instances in practice where one does this by first testing a hypothesis and then estimating. Sometimes the investigator will even allow the result of the hypothesis test to effect his method of estimation. Theoretical work on the procedure, where one follows a test of hypothesis by estimation, has been generally reviewed by Kitagawa [10]. Included in Kitagawa's review are references and discussions of other approaches to what we will call a hybrid problem. That is, we describe a hybrid problem as follows: Suppose we take $n$ observations on a random variable whose distribution is known to belong to a one-parameter family. Then, on the basis of these observations we decide whether the parameter (or some function of it) lies in a given interval of the parameter space (or the range space of the function of the parameter). Secondly, we estimate the parameter (or some function of it). In the next section we will explicitly state a decision theory formulation for a hybrid problem. Following the statement of the problem, we will give examples of situations where a hybrid problem is defined and for which the given formulation is appropriate. The formulation will be given for a hybrid problem when a single observation is made on a random variable whose distribution is of the exponential type, and when the decisions are made with regard to the expected value of the random variable. We decide whether the expected value, which is a function of the underlying parameter of the exponential distribution, lies in some given interval and we also estimate the expected value. For a large subclass of the exponential family, including the normal distribution with unknown mean, complete classes of procedures are found for the case when the given interval consists of a single point. These complete classes consist of procedures called interval-monotone procedures defined as follows: If the observation falls in a particular interval of the sample space, then decide that the expected value of the random variable lies in the given interval consisting of a single point and estimate it to be precisely that point. If the observation lies outside that particular interval of the sample space, decide that the expected value lies outside the given interval and estimate it by some monotone analytic function of the observation. Another result for the given formulation is concerned with the case when the distribution is normal with unknown mean and the given fixed interval is any symmetric interval about the origin. Here, a class of admissible procedures is found, where each procedure is a symmetric interval-monotone procedure, the symmetry pertaining both to the interval and the estimate. Furthermore, included in this class is the following intuitively appealing procedure: If the observation falls in some particular symmetric interval about the origin, then decide that the parameter lies in the given fixed interval of the parameter space and estimate the parameter by zero; while if the observation falls outside the particular interval of the sample space, then decide that the parameter lies outside the given interval and estimate it by the value of the observation. It is interesting to note that the estimate resulting from the above procedure was proved to be an inadmissible estimate for the squared error loss function. (See [5]). The analogue of the admissibility result for the normal case is next found for the cases where the probability distribution of the random variable is binomial, Poisson and gamma, and the left end point of the given interval of the parameter space is zero. In the next section then, we state the problem and give examples. In Section 3 we give complete class theorems. The admissibility results for the normal case and binomial, Poisson, and gamma cases follow in Sections 4 and 5, respectively.

Presented here is a new $k$-sample model in order statistics, in which $k$ random samples of equal size are first ordered within themselves in the usual manner, and are then ordered among themselves by considering the size of the maximum value in each sample. The distribution functions and several probabilities relevant to the model are derived. Consider $k$ random samples of size $n (X_{11}, X_{21}, \cdots, X_{n1}), (X_{12}, X_{22}, \cdots, X_{n2}), \cdots, (X_{1k}, X_{2k}, \cdots, X_{nk})$, where the $X_{ir}$ are independent and identically distributed according to the absolutely continuous distribution function $F(x)$. Arrange each sample in decreasing order, and let $Z_{1r}$ be the greatest random variable in the sample $(X_{1r}, X_{2r}, \cdots, X_{nr})$, let $Z_{2r}$ be the second greatest random variable in the same sample, and so on. The $k$ samples, after ordering, are then $(Z_{11}, Z_{21}, \cdots, Z_{n1}), (Z_{12}, Z_{22}, \cdots, Z_{n2}), \cdots, (Z_{1k}, Z_{2k}, \cdots, Z_{nk})$. Form the set $S = \{Z_{11}, Z_{12}, \cdots, Z_{1k}\}$, so that the elements of $S$ are the greatest random variables in each of the $k$ samples. Order the random variables in $S$, and let $Y_{11}$ denote the greatest, $Y_{12}$ the second greatest, and so on to $Y_{1k}$. Then for each point in the sample space, $Z_{1r}$ corresponds to some $Y_{1j}$. Define $Y_{ij}$ as the $i$th ranked random variable from the same sample as the $Z_{1r}$ mentioned above. In other words, for each point in the sample space where $Z_{1r}$ corresponds to $Y_{1j}$, the sample $(Z_{1r}, Z_{2r}, \cdots, Z_{nr})$ will be denoted by $(Y_{1j}, Y_{2j}, \cdots, Y_{nj})$. Since $Z_{1r} > Z_{2r} > \cdots > Z_{nr}$, it follows that $Y_{1j} > Y_{2j} > \cdots > Y_{nj}$. $Y_{ij}$ is called the $i$th order statistic in the $j$th sample. The number $i$ is called the rank of $Y_{ij}$ within the sample, and the number $j$ is called the rank of the sample. The above model is useful in flood frequency analysis, when it is desired to combine flood records at several independent stations in an attempt to study rare floods. Also the above model can be applied in the following situation. A large number $(nk)$ of displays have been entered in a science fair, at which $k$ prizes are to be awarded. The judge feels it is impractical to consider all $nk$ displays at once, and so he randomly divides the displays into $k$ groups of $n$ elements in each group. He then judges each group separately, ordering the $n$ displays in each group according to excellence. When each group has thus been ordered, he then considers the $k$ best displays, one from each group, and awards the 1st prize, 2nd prize, $\cdots, k$th prize to these $k$ displays according to their relative excellence. $Y_{ij}$ represents the display that was $i$th best in its group where the best display in that group won $j$th prize. The distribution function of $Y_{ij}$ is found in Section 2. A method of comparing $Y_{ij}$ with $X$, an additional random variable with the same distribution function $F(x)$, is given by the equation for $P(X > Y_{ij})$, derived in Section 3. Section 4 contains $P(Y_{i_1j_1} > Y_{i_2j_2})$ where $Y_{i_1j_1}$ and $Y_{i_2j_2}$ are two ordered random variables from the same array of $nk$ random variables. The results of Section 4 permit the comparison of any two of the ordered random variables. The comparison of $Y_{1,k}$ with $Y_{2,1}$ is related to the probability of $Y_{1,k}$ exceeding $\max_j Y_{2,j}$ which was considered by Cohn, Mosteller, Pratt, and Tatsuoka (1960). In Section 5 the expected value of $Y_{ij}$ is discussed. The fairness of the above method of awarding prizes is examined in Section 6.

Although the theory of asymptotically most powerful rank tests is fairly well developed, only little attention (see [4]) has been paid to censored data. In this paper we derive the a.m.p.r.t. for censored data from the a.m.p.r.t. based on the complete sample. For simplicity, we consider the two-sample location problem for data censored on the right. The methods used, however, can be applied to find the a.m.p.r.t. for multi-censored data and also the a.m.p.r.t. for the two-sample dispersion problem when the data are censored.

This paper extends the use of finite fields for the construction of confounding plans, to include "asymmetrical" or "mixed" factorials. The technique employed is to define addition and multiplication of elements from distinct finite fields, by mapping these elements into a finite commutative ring containing sub-rings isomorphic to each of the fields in question. The standard and relatively simple techniques for symmetrical factorials are then applied in a straightforward manner to the asymmetrical case. The paper is concluded with a numerical example for the case of a $3^2 \times 5$ factorial design, and an outline of some possible confounding plans. Fractional factorials are discussed briefly.

The distribution of the number of successes in independent trials is shown to be bell-shaped of every order. The most likely number of successes is "almost uniquely" determined from the mean number and from the mean plus the largest and smallest probability of success on any trial. Bounds on the distribution function of the number of successes are obtained and extended to an infinite number of trials, including the Poisson distribution.

We consider a system with a finite number of states, $1, 2, \cdots, S$. Periodically we observe the current state of the system and perform an action, $a$, from a finite set $A$ of possible actions. As a joint result of $s$, the current state, and $a$, the action performed, two things occur: (1) we receive an immediate return $r(s, a)$; and (2) the system moves to a new state $s'$ with probability $q(s' \mid s, a)$. (For several interesting and occasionally amusing concrete examples of this setup, the reader is referred to Howard's excellent book, [4].) Let $F$ be the (finite) set of functions from $S$ to $A$. A policy $\pi$ is a sequence $(\cdots, f_n, \cdots, f_1)$ of members of $F$. Using $n$ steps of the policy $\pi$ means observing the system, and upon finding $s_0$, performing action $f_n(s_0)$, observing the next state $s_1$ and performing $f_{n-1}(s_1)$ and so on until after $n - 1$ of these steps have been completed one observes $s_{n-1}$ and performs action $f_1(s_{n-1})$ and then stops. The expected (total) return using $n$ steps of $\pi$ given that the initial state is $s_0$ is denoted $v_n(s_0, \pi)$. A policy $\pi$ is optimal if $v_n(s, \pi) \geqq v_n(s, \pi')$ for any policy $\pi'$ and all $n$ and $s$. In other words, $\pi$ is optimal if the return using $n$ steps of $\pi$ cannot be exceeded by using $n$ steps of any other policy regardless of $n$ and the initial state of the system. Obviously, $v_n(s, \pi)$ for optimal $\pi$ may be calculated by value iteration, that is, by the use of the recursion $v_{n+1}(s, \pi) = \max_a \lbrack r(s, a) + \sum_{s'} q(s' \mid s, a)v_n(s', \pi)\rbrack.$ Similarly an optimal $\pi$ may be generated by letting $f_1(s)$ be any $a$ with $r(s, a) \geqq r(s, a')$ for all $a'$, and letting $f_{n+1}(s)$ be any $a$ for which the expression on the right side of the above equation attains its maximum. This paper is concerned with the return of an optimal policy $\pi$. The principal results are as follows: A policy is eventually stationary if for $m, n$ sufficiently large, $f_m = f_n$, that is, if the sequence which is the policy consists of one member of $F$ repeated infinitely often followed by a finite arbitrary sequence. In Section 3, an example is given where there is only one optimal policy and it has the form $(\cdots, f, g, f, g, f, g)$, an oscillating sequence of two members of $F$ which shows that there may not be an eventually stationary optimal policy. The author has not determined whether the example represents the worst possible behavior of optimal policies or whether there are cases where there is no periodic optimal policy. For an optimal policy $\pi$, the gain of the policy, defined by $\lim_n n^{-1}v_n(s, \pi)$, exists; it will be denoted by $v^\ast(s, \pi)$. It is shown in Lemma 3.1 that if $\pi$ is optimal then there is an $N$ such that $\lim_n \lbrack v_{nN+r}(s, \pi) - (nN + r)v^\ast(s, \pi)\rbrack$ exists for any $r$, that is, asymptotically, the return oscillates periodically around the long range average return. One of the major results of the paper is that there is a stationary policy $\sigma$, i.e., one for which $f_m = f_n$ for all $m, n$, which has the same gain as the optimal policy $\pi$, symbolically, $v^\ast(s, \sigma) = v^\ast(s, \pi)$ for all $s$. In fact a stronger result is true (Theorem 4.2), namely that there is some constant $C$ such that $v_n(s, \pi) - v_n(s, \sigma) < C$ for all $n$ and $s$. Although it is not known whether all optimal policies are eventually periodic, it is shown (Corollary 4.9 of Theorem 4.8) that there is an eventually periodic policy whose return is an arbitrarily small amount less than that of the optimal policy. Formally, for any $\epsilon > 0$, there is a policy $\sigma$ with the property that for some $N, f_{n+N} = f_n$ for $n$ sufficiently large such that $v_n(s, \pi) - v_n(s, \sigma) < \epsilon$ for all $n$ and any $s$. The final section contains further results for the special case that $q(s' \mid s, a) > 0$ for all $s', s, a$.

Brunk [3] discusses conditional expectations given $\sigma$-lattices. This note is concerned with the observation that the reciprocal of the conditional expectation given a $\sigma$-lattice with respect to some measure $\mu$ of a positive random variable $X$ is the conditional expectation of $1/X$ given the complementary $\sigma$-lattice with respect to another measure. (For the case of a $\sigma$-field this result is equivalent to the familiar result about the reciprocal of a Radon-Nikodym derivative.) Associated with this property of conditional expectations is a mapping of the set of all convex functions on $(0, \infty)$ into itself which leads to alternative ways of formulating certain extremum problems. Let $(\Omega, \mathfrak{A}, \mu)$ be a measure space with $\mu(\Omega) < \infty$. Let $I_A$ denote the indicator function of a set $A$. We shall let $\mathscr{L}$ denote a $\sigma$-lattice of subsets of $\Omega(\mathscr{L} \subset \mathfrak{A})$. A $\sigma$-lattice is, by definition, closed under countable unions and intersections and contains both $\Omega$ and the null set $\varnothing$. If $\mathscr{L}$ is such a $\sigma$-lattice then $\mathscr{L}^c$ will denote the $\sigma$-lattice of all subsets of $\Omega$ which are complements of members of $\mathscr{L}$. We say that a random variable $X$ is $\mathscr{L}$-measurable if $\{X > a\} \varepsilon \mathscr{L}$ for each real number $a$. Let $L_2$ denote the class of square integrable random variables and $L_2(\mathscr{L})$ the class of $\mathscr{L}$-measurable, square integrable random variables. Later we shall want to restrict our attention to strictly positive random variables, so for any set $S$ of random variables we let $S^+$ denote the set of all those members of $S$ which are strictly positive. Let $\mathscr{B}$ denote the class of Borel subsets of the real line. The following is one of several available definitions for the conditional expectation, $E_\mu(X \mid \mathscr{L})$, of $X$ given $\mathscr{L}$ (see Brunk [3]). Definition. If $X \varepsilon L_2$ then $Y \varepsilon L_2(\mathscr{L})$ is equal to $E_\mu(X \mid \mathscr{L})$ if and only if $Y$ has both of the following properties: \begin{equation*}\tag{1}\int(X - Y)Z d\mu \leqq 0 \quad\text{for each} Z \varepsilon L_2(\mathscr{L})\end{equation*} \begin{equation*}\tag{2}\int_B (X - Y) d\mu = 0 \quad\text{for each} B \varepsilon Y^{-1}(\mathscr{B})\end{equation*} (Brunk [3] shows the existence of such a $Y$ and that it is unique in the sense that if $W$ is any other member of $L_2(\mathscr{L})$ having these properties then $W = Y\lbrack\mu\rbrack$.)

Let $Y_1$ and $Y_2$ be two real valued, independently and identically distributed random variables with variance $\sigma^2$. Then $\sigma^2 = \varepsilon(Y_1 - \varepsilon Y_1)^2 = 2^{-1} \cdot \varepsilon(Y_1 - Y_2)^2$. Note that \begin{equation*}\tag{0.1}(Y_1 - Y_2)^2 = \begin{vmatrix}1 & 1 \\ Y_1 & Y_2\end{vmatrix}^2\end{equation*} is the square of the length of the interval $\lbrack Y_1, Y_2\rbrack$. Given a sample of size $n$, a well known unbiased estimator for $\sigma^2$ is given by \begin{equation*}\tag{0.2}\lbrack n(n - 1)\rbrack^{-1} \sum_{1\leqq i_1<i_2\leqq n} (Y_{i_1} - Y_{i_2})^2 = (n - 1)^{-1} \sum^n_{i=1}(Y_i - \bar Y)^2.\end{equation*} The present note will discuss a $k$-dimensional generalization of this situation. Throughout our discussion we will assume that the following condition is satisfied. Condition $mathscr{I} X_1, X_2, \cdots, X_n$ are $n(>k)$ independently and identically distributed, $k$-vector valued random variables with (unknown) expected vector $\mu$ and covariance matrix $\not\sum$. One natural generalization of the above parameter $\sigma^2$ then is (the $X_i$ in the following formulae being understood as one-column matrices with $k$ components) the parameter $\theta$ defined as follows: \begin{equation*} \tag{0.3} \theta = \lbrack(k + 1)!\rbrack^{-1}\varepsilon \big(\begin{vmatrix}1 & 1& \cdots & 1 & 1 \\ X_1 & X_2 & \cdots & X_k & X_{k+1}\end{vmatrix}^2\big).\end{equation*} Theorem 1 will show that $\theta = \det \not\sum$. Note that the absolute value of the determinant in the second member of (0.3) is known to be $k$! times the $k$-dimensional content of the simplex (for a definition e.g. see [4], p. 10) which has the $(k + 1)$ points (i.e., the $(k + 1) k$-tuples) $X_1, X_2, \cdots, X_{k+1}$ for its vertices (an enlightening discussion of determinants as connected with volumes can be found in [3], pp. 152-162). This furnishes a geometric interpretation to the parameter $\theta = \det \not\sum$. By an argument well known from the theory of $U$-statistics and using the equality $\binom{n}{k+1} \cdot (k + 1)! = n(n - 1) \cdots (n - k) \equiv n_{k+1}$, say, one finds (see corollary 1.1) that an unbiased estimator for $\theta = \det \not\sum$ is given by $\hat{\theta}$, where $\hat{\theta}$ is defined by: \begin{equation*}\tag{0.4}\hat{\theta} = n^{-1}_{k+1} \cdot \sum \begin{vmatrix}1 & 1 & \cdots & 1 \\ X_{i_1} & X_{i_2} & \cdots & X_{i_{k+1}}\end{vmatrix}^2.\end{equation*} where summation is over all $(i_1,i_2, \cdots, i_{k+1})$ with $1 \leqq i_1 < i_2 < \cdots < i_{k+1} \leqq n$. Theorem 2 will point out a simple relation between $\hat{\theta}$ and Wilks' internal scatter $S_{k,\bar{X},n} = \det U$ (cf. Wilks [5], equations (4.8), (4.2) and (4.3); Wilks [6], equation (18.1.23); and Section 2 below): \begin{equation*}\tag{0.5} (n - 1)(n - 2) \cdots (n - k) \hat{\theta} = S_{k,\bar{X},n} = \det U.\end{equation*} This solves a question, left open by Wilks [5], p. 493, namely: under which conditions is it true that \begin{equation*}\tag{0.6} \varepsilon S_{k,\bar{X},n} = (n - 1)(n - 2) \cdots (n - k) \det \not\sum?\end{equation*} Corollary 2.1 shows that equality (0.6) is true as soon as the trivial Condition $\mathscr{I}$ is satisfied. This clearly constitutes a much stronger result than Wilks' preliminary statement. In fact, equation (0.6) is a full generalization of the equality liminary statement. In fact, equation (0.6) is a full generalization of the equality $\varepsilon\sum_i (Y_i - \bar{Y})^2 = (n - 1)\sigma^2$, and valid under conditions of the same scope.