The concepts of significance, likelihood, and objective posterior have wide ranges of application in statistics. For certain very simple applications--single location variable--there has been recognition that the three concepts produce equivalent numerical results, specifically the equality of observed level of significance, integrated likelihood extreme values, and integrated objective posterior extreme values. The most general model permitting the use of the three concepts for the full parameter, and indeed for component parameters, is a structural model (or probability-space model). This paper examines the three concepts for a structural model and shows for both the full parameter and for component parameters the essential equivalence of observed level of significance, integrated likelihood extreme values, and integrated objective posterior extreme values.

It is well known that Wald's SPRT terminates with probability one and in fact the stopping time is exponentially bounded for every distribution $P$ except in the trivial case where the $\log$ likelihood ratio vanishes with probability one. The results in the literature for invariant SPRT's, however, have been considerably less complete. In all the parametric problems studied, moment conditions of certain random variables have been assumed to prove termination, and the finiteness of their moment generating function has also been assumed to prove the exponential boundedness of the stopping rule. In this paper, we try to remove or weaken these conditions for certain invariant SPRT's. In particular, we show that like the Wald SPRT, the sequential $t$- and $F$-tests always terminate with probability one for any distribution $P$ except in trivial cases. However, the stopping rules may fail to be exponentially bounded, and obstructive distributions are also exhibited. Sufficient conditions for exponential boundedness and finiteness of moments of the stopping rule are studied, and asymptotic expressions for the moments of the stopping rule are also given.

We present a unified large-sample theory of general chi-squared tests of fit under composite null hypotheses and Pitman alternatives. The statistics are quadratic forms in the standardized cell frequencies, and we allow random cells, $k$-variate observations from not necessarily continuous distributions, and quite general estimates of unknown parameters. Generalizations of published results on a number of specific chi-squared tests follow.

This paper deals with robustness studies of tests of two hypotheses (A) $\Sigma_1 = \Sigma_2$ in $N(\mu_i, \Sigma_i), i = 1, 2$, and (B) $\mu_1 = \cdots = \mu_l$ in $N(\mu_i, \Sigma), i = 1, 2, \cdots, l, \Sigma$ unknown, based on four test criteria (a) Hotelling's trace, (b) Pillai's trace, (c) Wilks' $\Lambda$ and (d) Roy's largest root. The robustness for (A) is against non-normality and for (B) against unequal covariance matrices and is studied in the exact case, unlike the results obtained earlier. In this connection, Pillai's density of the latent roots of $\mathbb{S}_1\mathbb{S}_2^{-1}$ under violations is used to derive the distributions or the moments of the criteria. Numerical studies of the tests of the two hypotheses based on the four criteria are made for the two-roots case.

Let $x_1, \cdots, x_n$ be a sample of size $n$ of an $\operatorname{rv}$ with $\operatorname{df} F(x - \theta)$, where $F$ is known but $\theta$ unknown. In this paper we make a Fourier approach to the problem of existence of a statistic $g(x_1, \cdots, x_n)$ which is a uniformly minimum variance (UMV) estimator of its own mean value. We mention only some of the results. If $n = 1$ we find an NASC for an estimator $g(x_1)$ to be, in a restricted sense, UMV. This condition is given in terms of the zeroes of the ch.f. of $F$ and the support of the Fourier transform of $g$. If $n \geqq 2$, it is shown that a statistic of the form $g(\bar{x})$, where $\bar{x}$ is the sample mean, cannot be UMV, unless $g$ is periodic or $F$ is a normal $\operatorname{df}$. We prove the non-existence of a UMV-estimator of $\theta$, provided that the tail of $F$ tends to zero rapidly enough. Finally, it is proved that no polynomial $P(x_1, \cdots, x_n)$ can be a UMV-estimator, unless $F$ is a normal $\operatorname{df}$.

We describe an approach to linear feature selection for $n$-dimensional normally distributed observation vectors which belong to one of $m$ populations. More specifically, we consider the problem of finding a rank $k k \times n$ matrix $B$ which minimizes the probability of misclassification with respect to the $k$-dimensional transformed density functions when a Bayes optimal (maximum likelihood) classification scheme is used. Theoretical results are presented which, for the case $k = 1$, give rise to a numerically tractable expression for the variation in the probability of misclassification with respect to $\mathbf{B}$. The use of this exression in a computational procedure for obtaining a $\mathbf{B}$ which minimizes the probability of misclassification in the case of two populations is discussed.

The problem of finding an optimum sampling strategy from the class of all linear unbiased strategies with a given expected sample size is considered. The paper deals with the properties of admissibility, completeness and strong admissibility of the Horvitz-Thompson strategies.

$X$ and $\Theta$ are random variables; for given $\Theta = \theta$, the conditional distribution of $X$ is binomial with parameters $N$ and $\theta$; the first $N$ moments of $\Theta$ are known. An estimate of $\Theta$ is made based on the observed value of $X$, the risk being defined in terms of squared error loss. It is shown that as conjectured by H. Robbins, the ratio of the Bayes risk to the minimax risk for all possible distributions of $\Theta$ uniformly tends to unity when $N \rightarrow \infty$.

A general continuous time distributed lag model is considered. The problem is that of estimating the parameters of the kernel when, as is often the case, the available data consist not of a continuous record but of discrete observations recorded at regular intervals of time. Fourier transformation of the model and insertion of the computable, discrete Fourier transforms of the variables produce an approximate model which is of non-linear regression type and is relatively easy to handle. Estimators are proposed and their asymptotic properties established, assuming principally that the variables are stationary and ergodic and that an "aliasing" condition on the independent variable is satisfied. The results of the paper imply a theory for the estimation of rather general continuous time systems, involving the operations of differentiation, integration and translation through time.

Let $X$ be an absolutely continuous real-valued random variable with additional restrictions to be imposed later. Venter (1967) ("On estimation of the mode," Ann. Math. Statist. 37 1446-1455) estimated the mode of $X$ by a point from the shortest interval containing a specified number $r = r(n)$ of observations. Venter demonstrated that such an estimator is strongly consistent under appropriate conditions on the distribution of $X$ and on $r(n)$. It is the purpose of this paper to show that strong consistency actually holds under very general conditions on the distribution of $X$. Convergence rates are also obtained which are, in some cases, much faster than those reported by Venter.

It is shown that complete sets of $(\nu - 1)/2$ by $\nu$ and $(\nu + 1)/2$ by $\nu$ multistage balanced Youden designs of type I and II can be constructed if $\nu$, the number of treatments, is a prime power of the form $4\lambda + 3$. It is also proved that the existence of a difference set with certain properties implies the existence of a 2-stage balanced Youden design. The usefulness of this latter result is demonstrated for those experiments where the number of treatments is not a prime power.

The main purpose of this paper is four-fold. First, to prove that the best upper bound on the number of mutually orthogonal $F(n, \lambda)$ squares is $(n - 1)^2/(m - 1)$, where $m = n/\lambda$. Second, to show that this upper bound is achievable if $m$ is a prime or prime power and $\lambda = m^h$. Third, to present a set of four mutually orthogonal $F(6; 2)$ squares design. This latter design is important because there are no orthogonal Latin squares of order 6 which could be used for this purpose. Fourth, to indicate a method of composing orthogonal $F$-squares designs. In addition, we have pointed out the way one may construct orthogonal fractional factorial designs and orthogonal arrays from these designs.

Let $\Omega_n$ denote a set of $n$ players, $p_{ij}$ the probability that player $i$ defeats player $j$ and $\Gamma$ the class of preference matrices $(p_{ij})$ with $p_{1j} \geqq \frac{1}{2}, j > 2$. Under the assumption that the outcomes of games are independent and distributed according to $(p_{ij}) \in \Gamma$, the effectiveness (relative to $(p_{ij})$) of a tournament plan, together with a rule to select a winner, is measured by the probability that player 1 (the "best" player) wins the tournament. A k.o. plan is a tournament plan in which a player is eliminated from the tournament if he loses one game. It is shown that there are no plans on $\Omega_n$ with $n - 1$ games that are more effective than k.o. plans relative to all matrices contained in certain reasonable subclasses of $\Gamma$. Among the k.o. plans for $2^m + k, 0 \leqq k < 2^m$, players, those which consist of a preliminary round of $k$ games followed by a "symmetric" k.o. tournament on the remaining $2^m$ players are more effective than all other plans relative to the preference matrices contained in two large subclasses of $\Gamma$. In order to prove these assertions, the tournament plans are interpreted as mappings with directed digraphs as domain and range.

Raghavarao (Ann. Statist. 1973) has given an $L_2$ design with $b = \nu = 49, r = k = 9, \lambda_1 = 3, \lambda_2 = 1$. It is shown that this design is isomorphic with a design given by Archbold and Johnson (Ann. Math. Statist. 1956).

A necessary and sufficient condition is established for doubly-connectedness in $b$-row and $k$-column designs in which all cells are filled. A necessary condition for doubly-connectedness in a generalized two-way elimination of heterogeneity designs is provided and a property of doubly-connected designs is given.

With respect to a general quadratic loss function, the Bayes rule for the mean of a probability distribution of unknown form is obtained, in the class of linear functions of the sample. The associated Bayes risk is also obtained. A number of recent results in the literature are shown to be direct corollaries of this result, and applications are given for the empirical distribution function of the sample.

In this paper a new method of approximating one distribution by another is introduced. The method is essentially a modification of the Edgeworth technique which eliminates the necessity of knowing the cumulants of the distributions involved.

It is shown that if $\mathbf{T}$ has an unknown exponential family distribution with natural parameter $\mathbf{\theta}$, then $\mathbf{G(\theta)} = \mathbf{ET}$ uniquely specifies the moment generating function. The converse is proved, namely, if $\{\mathbf{T_\theta}\}$ is a family of random variables with moment generating functions of a certain form, then it must be an exponential family. Moreover, several necessary and sufficient conditions are given so that a function can be the mean value function of an exponential family distribution.

It is known that the set $M = \{\mu_{r,n} : r = 1, 2, \cdots, n; n = 1, 2, 3, \cdots\}$ of expectations of order statistics of samples from a distribution $F$ completely determines $F$ (see, for example, Hoeffding (1953)). In addition Chan (1967), Konheim (1971) and Pollak (1973) have shown that certain subsets of $M$ determine $F$. In this note conditions are derived which are sufficient for a proper subset of $M$ to determine $F$.

Two nonparametric density estimators, based on Fourier series and the Fejer kernel, are presented. One of them $(\tilde{f}_N)$ is appropriate when the unknown density $f$ vanishes outside a known bounded interval; the other $(f_N^\sharp)$ is applicable without any assumptions about the support of $f$. The estimator $f_N^\sharp$ is of the type studied by Watson and Leadbetter (Sankhya A 26, 1964) and $\tilde{f}_N$ is almost of that type: both may be said to be of the "$\delta$-sequence type". If $f$ satisfies a certain Lipschitz condition at $x$ and the "number of harmonics" used in $\tilde{f}_N$ is asymptotically proportional to $N^\frac{1}{3}$, and $\rho_N/\log N \rightarrow \infty$, then $(N^{\frac{1}{3}}/\rho_N) \cdot |\tilde{f}_N(x) - f(x)| \rightarrow 0$ a.s.; a similar result holds for $f_N^\sharp$.

This paper shows the asymptotic equivalence of the classical and symmetric optimality criteria for the finite state, arbitrary action compound decision problem.

The paper deals with the density of the characteristic roots of $\mathbf{S}_1\mathbf{S}_2^{-1}$ where $\mathbf{S}_1$ has a noncentral Wishart distribution, $W(p, n_1, \mathbf{\Sigma}_1, \mathbf{\Omega})$, and $\mathbf{S}_2$ has an independently distributed central Wishart distribution $W(p, n_2 \mathbf{\Sigma}_2, \mathbf{0})$, under a condition. This density is basic for an exact study of robustness of tests of at least two multivariate hypotheses.

The characteristic polynomial is derived for the information matrix $M$ for those main-effect and second-order models based on fractions of the $3^n$ factorial which render the model symmetric under permutation of the factors. Explicit formulas for the determinant of $M$ and the trace of $M^{-1}$ result.