Suppose that we observe random variables $X_1, \cdots, X_n$ which are identically and independently distributed according to some distribution $F$ where $F$ ranges over a family $\mathscr{F}$. The following question was posed by J. Steffensen in this abstract context in [7]. (1) If a functional $q$ is given on $\mathscr{F}$, when does there exist a statistic $\delta(X_1, \cdots, X_n)$ such that, \begin{equation*} \tag{1.1} E_F(\delta(X_1, \cdots, X_n)) = q(F),\end{equation*} for all $F \varepsilon \mathscr{F}$? (As usual $E_F$ denotes the expectation under the assumption that $F$ is the common distribution of the $X_i$.) In other words, when does there exist an unbiased estimate of $q(F)$ based on $n$ observations? A question which naturally follows from question 1 was raised and considered by Halmos in [2]. (2) Let the degree of $q$ be the smallest $n \leqq \infty$ such that (1.1) holds for all $F \varepsilon \mathscr{F}$ and some $\delta$. Characterize the degree of estimable functionals $q$ for various families $\mathscr{F}$. (By estimable we mean merely that (1.1) should hold for some finite $n$ and $\delta$.) In his paper [2] Halmos dealt with question 2 as well as the more important issue of existence of best unbiased estimates in the context of families $\mathscr{F}$ which are very large, and asked whether anything can be said about more interesting families $\mathscr{F}$. The next two sections of our paper deal exclusively with question 2 for families $\mathscr{F}$ which are closed under finite mixtures (convex combinations). In Section 2, some examples lead to our main result, Theorem 2.1, which gives an elementary characterization of degree for functionals defined on families $\mathscr{F}$ as above. The theorem is followed by further examples which illustrate its utility. (However, the required convexity property excludes the usual parametric families.) Theorem 2.2, an extension of a theorem of Halmos [2], characterizes the degree of a product of estimable functionals. It is applied to give the degree of the $r$th cumulant of a distribution when $\mathscr{F}$ is the family of all distributions with moments of all orders. The third section deals with the problem of degree when we observe samples from two or more populations with different distributions. Theorem 3.1 shows that the notion of degree can be successfully extended in this situation. The theorem is followed by several examples in which the degree of various functionals such as two-sample distance criteria is computed. Finally, in Section 4 we address ourselves to question 1. Theorem 4.1 gives a simple characterization of estimable functionals when the family $\mathscr{F}$ is $\mathscr{F}(\mu)$ where, \begin{equation*} \tag{1.2} \mathscr{F}(\mu) = \{F:F \text{is absolutely continuous with respect to} \mu\}\end{equation*} for a fixed $\sigma$ finite measure $\mu$. The characterization is suggested by Theorem 2.1. After some examples we state without proof a further theorem (4.2) along these lines and illustrate its applicability.

In this paper we develop some theorems for an extended compound decision problem which has as the component a general statistical decision problem. The results generalize and strengthen some results previously reported by Swain (1965) and Johns (1967). From the outset it is assumed that the reader is familiar with the compound decision problem.

This paper consists of new results continuing the series of papers on optimal design theory by Kiefer (1959), (1960), (1961), Kiefer and Wolfowitz (1959), (1960), Farrell, Kiefer and Walbran (1965) and Karlin and Studden (1966a). After disposing of the necessary preliminaries in Section 1, we show in Section 2 that in several classes of problems an optimal design for estimating all the parameters is supported only on certain points of symmetry. This is applied to the problem (introduced by Scheffe (1958)) of multilinear regression on the simplex. In Section 3 we consider optimality when nuisance parameters are present. A new sufficient condition for optimality is given. A corrected version is given to the condition which Karlin and Studden (1966a) state as equivalent to optimality, and we prove the natural invariance theorem involving this condition. These results are applied to the problem of multilinear regression on the simplex when estimating only some of the parameters. Section 4 consists primarily of a number of bounds on the efficiency of designs; these are summarized at the beginning of that section.

For the regression model $Y_{\nu i} = \alpha + \beta C_{\nu i} + \epsilon_{\nu i}, i = 1, \cdots, N_\nu$, where the $\epsilon_{\nu i}$ are independent and identically distributed random variables (iidrv), optimum rank order tests for the hypothesis that $\beta = 0$ are due to Hoeffding (1950), Terry (1952) and Hajek (1962), among others. In the present paper, the theory is extended to the problem of testing the homogeneity of the regression coefficients from $k(\geqq 2)$ independent samples. Allied efficiency results are also presented.

The paper studies Wald's minimax risk (M.R.) criterion and Lehmann's unbiasedness condition for a very general class of Type I problems (Section 2) which contains nearly all nonsequential multiple decision problems (m.d.p.'s) from parametric statistics (at least when no reasonable a priori knowledge is available--apart from knowledge restricting the parameter space in, for example, one-sided problems or in trend situations--and provided that the problems can be formulated by means of a loss function which is constant for the various kinds of error.) Type I problems turn out to behave in a degenerate way. Generally the M.R. procedure is not unique (Situation 1, see Theorem 3.1 and the Sections 5, 11, 12); when a unique M.R. procedure exists, then this is trivial and useless (Situation 2, see Corollary 3.1 and Sections $6, \cdots, 10$). We try to remedy this by applying Lehmann's unbiasedness condition (Section 4). This has to be done cautiously for the unbiasedness condition might be too restrictive: sometimes the class $W$ of all unbiased procedures is so small (Corollary 4.1) that $W = \varnothing$ (Theorem 8.1(i)) or such that $W$ contains only poor and useless procedures (Theorem 8.1 (iii) and Section 12). Fortunately we can show for some problems in Situation 1 that $W \subset M$ where $M$ denotes the class of all M. R. procedures (cf. Theorem 4.1; the unbiasedness restriction seems to be very attractive when the sufficient conditions of Lemma 4.2 are satisfied, see Theorem 5.1 where $W = M,$ Lemma 7.1 (ii) and Theorem 11.1; in Section 12 we also have $W \subset M$ but nevertheless the unbiasedness restriction is not attractive). For problems in Situation 2 the unique M.R. procedure $\delta^\ast$ is trivial and useless. Nevertheless we regard it as an advantage of the unbiasedness restriction when $\delta^\ast \varepsilon W$ (Sections 6, $\cdots$, 10), whereas $\delta^\ast \not\in W$ is regarded as an indication that $W$ might be too small (Theorem 8.1 (iv), Remark 8.1 and Lemma 10.1). We always try to obtain the procedure with "the most attractive appropriate optimum property." As shown by Lehmann ordinary two-decision testing problems (Section 5) and products of such problems (Section 11) do not present extensive difficulties because the unbiasedness restriction is attractive (Theorems 5.1 and 11.1) and reduces our m.d.p.'s to problems in the Neyman-Pearson formulation. Many three-decision two-sided problems are solved (Section 6) though we have to be content with criterions which are not very compelling. Similar results are obtained for some slippage problems (Sections 7, $\cdots$, 9) where the "optimum" procedure turns out to be not of the "natural" form considered in literature. For many m.d.p.'s of Type I (except for those in Sections 5, 6 and 11) the actual construction of the "optimum" procedure is forbidding and we have to content ourselves with results relating the classes $W$ and $M$ (cf. Sections 7, $\cdots$, 10, 12; the problems can be solved only in very simple special cases).

Henry Teicher [10] has initiated a systematic study called "identifiability of finite mixtures" (these terms to be defined in Section 1) which has significance in several areas of statistics. [10] gives a sufficiency condition that a family $\mathscr{F}$ of cdf's (cumulative distribution functions) generate identifiable finite mixtures, and consequently establishes that finite mixtures of the one-dimensional Gaussian or gamma families are identifiable. From [9] it is known that the Poisson family generates identifiable finite mixtures, and the binomial and uniform families do not. In [11], Teicher proves that the class of mixtures of $n$ products of any identifiable one-dimensional family is likewise identifiable (and that the analogous statement for finite mixtures is valid). Spragins and I have shown [13] that the finite mixtures on a family of cdf's is identifiable if and only if $\mathscr{F}$ is linearly independent in its span over the real numbers, and that $\mathscr{F}$ generates identifiable finite mixtures if $\mathscr{F}$ is any of the following: the $n$-dimensional normal family, the union of the $n$-dimensional normal family and the family of $n$ products of one dimensional exponential distributions, the Cauchy family, the negative binomial family, and the translation parameter family generated by any one dimensional cdf. (In this last case, our proof directly generalizes to any $n$-dimensional translation parameter family.) In view of the fact that many of the important distribution families have been seen to give identifiable finite mixtures, it would seem appropriate to seek methods for performing this identification, and therefore the intention of this paper is to reveal (Section 2) a general algorithm for construction of a consistent estimator. In Section 3 we demonstrate that the algorithm is effective for all the identifiable families mentioned above. Our results, in addition to having application to an interesting problem in communication theory [6], can be used to extend the empiric Bayes approach to a certain type of decision problem. Section 4 will discuss the details of this application.

In testing a hypothesis concerning the correlation coefficient in a bivariate normal distribution where all the parameters are unknown, the Pearson product moment statistic is appropriate. It may happen, however, that there are relations among the parameters in the distribution, in which case the Pearson statistic would not utilize this information. In particular if the variances of the two marginal distributions are equal, it is possible to test the correlation coefficient by means of a simpler statistic which makes use of this information. In this paper we explain how this statistic arises and present some properties of its distribution. This statistic as well as its properties developed here are utilized in the latter part of this paper where we consider the problem of estimating the difference of the means when some of the observations corresponding to one of the variables are missing.

Let $X_1, X_2, \cdots, X_n$, be a sample of $n$ independent observations of a random variable $X$ with distribution $F(x) = F(x_1, \cdots, x_m)$ or $R^m$ and Lebesgue density $f(x) = f(x_1, \cdots, x_m)$. To estimate the density $f(x)$ consider estimates of the form \begin{equation*} \tag{1} f_n(x) = n^{-1} \sum^n_{j=1} K_n(x, X_j),\quad K_n(x, X_j) = h_n^{-m}K(h_n^{-1}(x - X_j));\end{equation*} where $K(u) = K(u_1, \cdots, u_m)$ is a real-valued Borel-measurable function on $R^m$ such that \begin{equation*} \tag{2} K(u) \text{is a density on}\quad R^m\end{equation*} \begin{equation*} \tag{3} \sup_{u\varepsilon R^m} K(u) < \infty\end{equation*} \begin{equation*} \tag{4} \|u\|^m K(u) \rightarrow 0 \text{as} \|u\|^2 = \sum^m_{i=1} u_i^2 \rightarrow \infty\end{equation*} and $\{h_n\}$ is a sequence of numbers such that \begin{equation*} \tag{5} h_n > 0,\quad n = 1, 2, \cdots;\quad \lim_{n\rightarrow\infty} h_n = 0 \text{and} \lim_{n\rightarrow\infty} nh_n^m = \infty.\end{equation*} Such density estimates have been shown to be weakly consistent (that is, $f_n(x) \rightarrow f(x)$ in probability as $n \rightarrow \infty$) on the continuity set, $C(f),$ of the density $f(x)$ by Parzen [4] for $m = 1$ and by Cacoullos [1] for $m > 1$. In Theorem 1, we state conditions under which strong consistency (that is, $f_n(x) \rightarrow f(x)$ with probability one as $n \rightarrow \infty$) of such estimates obtains. Theorem 2 gives conditions under which uniform (in $x$) strong consistency of the estimates (1) is valid. In this respect, our results are very similar in the case $m = 1$ to those of Nadaraya [4], although the method of proof and conditions imposed are different. Theorem 3 concerns the estimation of the unique mode of the density $f(x)$ when it exists.

Recent results by Birnbaum, Esary and Marshall (1966), Barlow and Proschan (1967) and others suggest that the exponential models used be Epstein and Sobel (1953) and others for life testing problems should be extended to models in which the lifetimes have increasing failure rate average (IFRA) distributions. In this paper, IFRA scale models are dealt with. Consider two independent random samples $X_1, \cdots, X_m$ and $Y_1, \cdots, Y_n$ from populations with distributions $F(x)$ and $G(y) = F(\Delta y)$ respectively, where $F$ is a continuous, unknown, IFRA distribution. The null hypothesis $H_0:\Delta \leqq 1$ is to be tested against the alternative $H_1:\Delta > 1$. Since $F$ is unknown, it is not possible to maximize the power $E\lbrack \phi\mid F(\cdot), F(\Delta\cdot) \rbrack, \Delta > 1$. However, it is shown (Theorems 2.1, 2.2, 3.1, 3.2) that the tests that maximize the power for the exponential alternative $F(\Delta y) = 1 - \exp (- \Delta y)$ actually maximize the minimum power $\inf_{F(\cdot)} E\lbrack \phi\mid F(\cdot), F(\Delta\cdot) \rbrack$. Thus these tests are minimax. They have been computed by Lehmann (1953), Savage (1956), and Rao, Savage and Sobel (1960). The results indicate that in the case of uncensored samples, one should use one of the statistics \begin{align*}L = {\prod}^n_{i=1} (N + i - s_{n+1-i}), \text{or} \\ S = \sum^m_{i=1} J_0(r_i), \text{with} J_0(k) = \sum^N_{j=N+1-k} 1/j,\end{align*} where $N = m + n$ and $r_1, \cdots, r_m (s_1, \cdots, s_n)$ are the ordered ranks of the $X$'s ($Y$'s) in the combined sample of $X$'s and $Y$'s. $L$. is minimax for $\Delta$ in an interval about two, and $S$ is minimax for $\Delta$ in an interval $(1, \delta)$ to the right of one. The minimax statistics in the case of censored samples are more complicated (see (3.1) and (3.2)) and one might use one of the approximations suggested by Gastwirth (1965) or Basu (1967) (see (3.3)). Only finite sample size properties are dealt with. Asymptotic results are given in [6].

The objective of the present paper is to define a robust point estimator of the parameter $\beta$ in the model \begin{equation*} \tag{1.1} y_j = \alpha + g_j(\beta) + z_j,\quad j = 1, 2, \cdots, n,\end{equation*} where $\alpha$ and $\beta$ are unknown parameters, $g_1, g_2, \cdots, g_n$ are real-valued functions of real variable satisfying suitable conditions and $z_1, z_2, \cdots, z_n$ are independent identically distributed random variables having a distribution function belonging to a specified class. An important special case of (1.1) is the regression model obtained by taking $g_j(\beta) = \beta x_j, j = 1,2, \cdots, n$, where the $x$'s are known constants. Robust point estimators of $\beta$ in this case have already been given by Adichie [1], who followed the method of Hodges and Lehmann and by Brown and Mood [7], who considered the so-called "median" estimators. The estimator presented in this paper, which is also a Hodges-Lehmann type estimator, therefore, provides a third alternative for the regression model. In Section 2 of this paper, a robust point estimator for $\beta$ in the model (1.1) under some suitable regularity conditions on $g_j$ and $z_j$ is defined. In Section 3, a simple computational technique for the calculation of this estimator is given. A small sample property of the estimator is given in Section 4, and in Section 5, asymptotic normality is established under some regularity conditions. In Section 6, some special cases of the model are considered.

Suppose that independent normally distributed random vectors, $X^{n\times1}$ and $Y^{k\times1}$, are observed with $E(X) = 0, E(Y) = \mu, \operatorname{Cov}(X) = \sigma^2I$, and $\operatorname{Cov}(Y) = \sigma^2I$. It is known [2, 5] that the best invariant estimator of $\mu$ is admissible if $k \leqq 2$ and inadmissible if $k > 2$. It is also known [1, 7] that the best invariant estimator of $\sigma$ is inadmissible. In this paper, these results are extended to show that the best invariant estimator of $\theta = A\mu + \eta\sigma$, for a given matrix $A$ and a given vector $\eta$, is inadmissible if $|\eta|$ is sufficiently large (when $k = 1, A = 1, \theta$ is a quantile).

Common estimates of multivariate location parameters have the property that each component of the parameter is estimated using only the corresponding component of the observations. This is true of the sample mean, sample median and the vector of medians of averages (studied in [1]), as well as of the rank-order statistics often applied to testing for location. In some cases, particularly the multivariate normal, such estimators achieve asymptotic efficiency, but in general information is lost. This paper presents three methods of estimating multivariate location parameters which use more information than is available in the marginal distributions. These classes of estimators are asymptotically nearly efficient (ANE), in the sense that for every $\epsilon > 0$ there is an estimator in the class with asymptotic efficiency $> 1 - \epsilon$ (if efficiency is measured by a comparison of the asymptotic covariance matrix to the inverse of the information matrix). Our ANE estimators are motivated by those of Ogawa [6] for univariate location parameters. Ogawa obtained the asymptotically minimum-variance asymptotically unbiased estimator (ABLUE) for location or scale from a chosen set of sample quantiles. It was soon observed (Tischendorf [10]) that the reciprocal of the asymptotic variance of Ogawa's estimator (properly normalized) is essentially a Riemann sum for the information integral for the parameter being estimated. Thus under mild regularity conditions the ABLUE approaches asymptotic efficiency as larger sets of more closely spaced quantiles are chosen for use. Ogawa's estimators are therefore ANE for univariate location parameters. In the present paper we describe three classes of ANE estimators for multivariate location parameters. The first two consist of linear estimators, and represent multivariate generalizations of Ogawa's ANE class. Our three classes are as follows: (1) Choose a set of marginal sample quantiles in each direction from a continuous $r$-variate location parameter distribution. These quantiles generate a random partition of Euclidean $r$-space $R_r$, and for $r > 1$ the observed cell frequencies contain additional information. We obtain the ABLUE's in terms of the sample quantiles and the observed cell frequencies for $r = 2$ and show that they are ANE. (2) For all $r > 1$, linear ANE estimators are obtained by choosing a single sample quantile in each direction and partitioning $R_r$ by marking off fixed distances from these. The ABLUE's in terms of the $r$ chosen quantiles and the observed cell frequencies are ANE. These estimators have much simpler coefficients than do those of class (1). (3) Finally, ANE estimators can be obtained by exploiting analytic properties of RBAN estimators (Neyman [5]) for a sequence of multinomial problems related to the given location parameter family. These estimators are usually not expressed in closed form. Ogawa proceeded by applying least squares theory to the asymptotic distribution of his chosen set of sample quantiles. The ABLUE's of classes (1) and (2) are here derived by the same method, but establishing the joint asymptotic distribution of the marginal sample quantiles and the observed cell frequencies is non-trivial. Our method is to reduce the problem to one involving the multinomial distribution. A similar idea was used by Weiss [11] to obtain the joint asymptotic distribution of the quantiles alone, but the present problem requires more elaborate arguments. Section 2 contains a preliminary result for the multinomial distribution. The ANE classes (1) and (2) are discussed in Section 3, while Section 4 presents the third class. Estimators of all three classes will require use of a computer if the distribution function cannot be expressed in closed form, and may therefore be of limited practical usefulness. Section 5, however, contains an example for which estimators of classes (1) and (2) can be computed with relative ease. For this example, a bivariate logistic distribution, the performance of our estimators is compared with that of the sample mean and median and of Ogawa's univariate ANE estimators. Throughout, $K$ denotes a generic positive constant, $\mathscr{L}\{X\}$ is the probability law of the random variable $X$, and $\mathscr{L}\{X_n\} \rightarrow \mathscr{L}\{X\}$ designates convergence in law. $N(\mu, \Sigma)$ is the normal law with mean $\mu$ and covariance matrix $\Sigma$ (which may be $1 \times 1$).

Let $X_1, X_2, \cdots$ be a sequence of independent identically distributed random vectors in $R^n$ (Euclidean $n$-space). Let the $X_i$'s have a distribution which is a product of $n$ Borel probability measures along an orthogonal set of axes. After sampling $m$ times let $H_m$ be the convex hull of $\{X_1, \cdots, X_m\}$. All possible limiting shapes for $H_m$ are found along with necessary and sufficient conditions that the limit be obtained.

Draper and Mitchell (1968) gave the complete set of even 512-run designs of resolution $\geqq 6$ and the complete set of 256-run designs of resolution $\geqq 5$. The authors showed that each of these designs can be obtained from one or other of five Reference Designs, all of resolution 6. The present note gives near-cyclic representations of the complete sets of interactions in the identity relationships of the Reference Designs. Some of these sets are shown to be related to certain incomplete block designs, including a resolvable balanced incomplete block design of a type that seems to have eluded attention hitherto.

A vector variable $\mathbf{X}$ is said to have a linear structure if it can be written as $\mathbf{X} = \mathbf{AY}$ where $\mathbf{A}$ is a matrix and $\mathbf{Y}$ is a vector of independent random variables called structural variables. In earlier papers the conditions under which a vector random variable admits different structural representations have been studied. It is shown, among other results, that complete non-uniqueness, in some sense, of the linear structure characterizes a multivariate normal variable. In the present paper we prove a general decomposition theorem which states that any vector variable $\mathbf{X}$ with a linear structure can be expressed as the sum $(\mathbf{X}_1 + \mathbf{X}_2)$ of two independent vector variables $\mathbf{X}_1, \mathbf{X}_2$ of which $\mathbf{X}_1$ is non-normal and has a unique linear structure, and $\mathbf{X}_2$ is multivariate normal variable with a nonunique linear structure.