For general optimality criteria $\Phi$, criteria equivalent to $\Phi$-optimality are obtained under various conditions on $\Phi$. Such equivalent criteria are useful for analytic or machine computation of $\Phi$-optimum designs. The theory includes that previously developed in the case of $D$-optimality (Kiefer-Wolfowitz) and $L$-optimality (Karlin-Studden-Fedorov), as well as $E$-optimality and criteria arising in response surface fitting and minimax extrapolation. Multiresponse settings and models with variable covariance and cost structure are included. Methods for verifying the conditions required on $\Phi$, and for computing the equivalent criteria, are illustrated.

It is proved that the jackknife estimate $\tilde{\theta} = n\hat{\theta} - (n - 1)(\sum \hat{\theta}_{-i}/n)$ of a function $\theta = f(\beta)$ of the regression parameters in a general linear model $\mathbf{Y} = \mathbf{X\beta} + \mathbf{e}$ is asymptotically normally distributed under conditions that do not require $\mathbf{e}$ to be normally distributed. The jackknife is applied by deleting in succession each row of the $\mathbf{X}$ matrix and $\mathbf{Y}$ vector in order to compute $\hat{\mathbf{\beta}}_{-i}$, which is the least squares estimate with the $i$th row deleted, and $\hat{\theta}_{-i} = f(\hat\mathbf{\beta}_{-i})$. The standard error of the pseudo-values $\tilde{\theta}_i = n\hat{\theta} - (n - 1)\hat{\theta}_{-i}$ is a consistent estimate of the asymptotic standard deviation of $\tilde{\theta}$ under similar conditions. Generalizations and applications are discussed.

Asymptotic normality of linear rank statistics for testing the hypothesis of independence is established both under fixed alternatives (or the null hypothesis) and under converging alternatives. The results of Ruymgaart, Shorack and van Zwet [14] are used to obtain a further weakening of the smoothness conditions on the score functions. In the present case the score functions are allowed to have a finite number of discontinuities of the first kind. The results of the present paper and of the paper [14] will be summarized in the author's thesis [13].

Frequency tables are examined in which some cells are not distinguishable. Log-linear models are proposed for these tables which lead to likelihood equations closely related to those associated with log-linear models for conventional frequency tables. Just as in conventional tables, the maximum likelihood equations are shown to be the same under Poisson or multinomial sampling. Applications are made to the problem of estimation of gene frequencies from observed phenotype frequencies.

Earlier results on asymptotically optimal sequences of regression designs for autoregressive stochastic processes are extended (nearly) to the equivalence classes of such processes.

A method is derived to place an approximate bound on the mean-square error incurred by using an incorrect covariance matrix in the Gauss-Markov estimator of the coefficient vector in the full-rank general linear model. The bound thus obtained is a function of the incorrect covariance matrix $\tilde{S}$ actually used, the Frobenius norm of $S - \tilde{S}$, where $S$ is the correct covariance matrix, and the basis matrix $\phi$. This estimate can therefore be computed from known or easily-approximated data in the usual regression problem. All mathematics related to the method is derived, and numerical examples are presented.

Statistics are derived for testing a sequence of observations from an exponential-type distribution for no change in parameter against possible two-sided alternatives involving parameter changes at unknown points. The test statistic can be chosen to have high power against certain of a variety of alternatives. Conditions on functionals on $C\lbrack 0,1\rbrack$ are given under which one can assert that the large sample distribution of the test statistic under the null-hypothesis or an alternative from a range of interesting hypotheses is that of a functional on Brownian Motion. We compute and tabulate distributions for functionals defined by nonnegative weight functions of the form $\psi(s) = as^k, k > -2$. The functionals for $-1 \geqq k > -2$ are not continuous in the uniform topology on $C\lbrack 0, 1\rbrack$.

Consider the problem of estimating a common mean of two independent normal distributions, each with unknown variances. Note that the problem of recovery of interblock information in balanced incomplete blocks designs is such a problem. Suppose a random sample of size $m$ is drawn from the first population and a random sample of size $n$ is drawn from the second population. We first show that the sample mean of the first population can be improved on (with an unbiased estimator having smaller variance), provided $m \geqq 2$ and $n \geqq 3$. The method of proof is applicable to the recovery of information problem. For that problem, it is shown that interblock information could be used provided $b \geqq 4$. Furthermore for the case $b = t = 3$, or in the common mean problem, where $n = 2$, it is shown that the prescribed estimator does not offer improvement. Some of the results for the common mean problem are extended to the case of $K$ means. Results similar to some of those obtained for point estimation, are also obtained for confidence estimation.

The following class of games is considered: a sequence chooser produces an infinite sequence of 0's and 1's, and a predictor observes the sequence for a finite time, stopping when he pleases and choosing an action from a finite set. The predictor wins an amount depending only on the action chosen and on the first two unobserved terms of the sequence. The value of such games is determined and the formula obtained is used to give a derivation of the well-known values of the 1-0 game and the two-move lag bomber-battleship game. Values for some generalizations of the latter game are given. Optimal strategies for the sequence chooser are discussed.

Four problems associated with the use of Zelen's calculus of factorials in the statistical analysis of nonorthogonal $n$-way classification data are solved. These are for the situations for which (i) some effect parameters are equated to zero, (ii) some combinations (subclasses) contain no observations, (iii) expected values of mean squares under fixed, mixed, and random models are desired, and (iv) expected values of single degree of freedom sums of squares are wanted. A unified approach to these problems was developed. Relationships to previous work, to blocked experiments, to fractional replication, and to "messy data" situations are discussed. The various analyses are first described for a nonorthogonal two-way classification and then generalized to an $n$-way classification in the final section of the paper. Numerical examples are presented to illustrate the various procedures.

The main result of the current research describes a monotonicity property of certain invariant tests for the multivariate analysis of variance problem. Suppose $X: r \times p$ has a normal distribution, $EX = \Theta$ and the rows of $X$ are independent, each with unknown covariance matrix $\Sigma: p \times p$. Let $S = p \times p$ have a Wishart distribution $W(\Sigma, p, n), S$ independent of $X$. If $K$ is the acceptance region of an invariant test for the null hypothesis $\Theta = 0$, let $\rho_K(\delta)$ denote the power function of $K$, where $\delta = (\delta_1, \cdots, \delta_t), t \equiv \min (r, p)$ and $\delta_1^2,\cdots,\delta_t^2$ are the $t$ largest characteristic roots of $\Theta\sigma^{-1}\Theta'$. A main result is THEOREM. If $K$ is a convex set (in $(X, S))$, then $\rho_K(\delta)$ is a Schur-convex function of $\delta$. Standard tests to which the above theorem can be applied include the Roy maximum root test and the Lawley-Hotelling trace test.

It is shown that among all cumulative distribution functions passing through $k \geqq 2$ given points there is a unique one with minimal Fisher information; it is obtained by a curious type of spline interpolation. This answers some questions raisd by D. G. Kendall and J. W. Tukey.

The main result in this paper concerns the limiting behavior of normalized cumulative sums of induced order statistics obtained from $n$ independent two-dimensional random vectors, as $n$ increases indefinitely. By means of a Skorokhod-type embedding of these cumulative sums on Brownian Motion paths, it is shown that under certain conditions the sample paths of these normalized sums converge in a certain sense to a process obtained from the Brownian Motion by a transformation of the time-axis. This yields an invariance principle similar to Donsker's. In particular, the asymptotic distribution of the supremum of the absolute values of these normalized cumulative sums is obtained from a well-known result for the Brownian Motion. Large sample tests of a specifieds regression function are obtained from these results.

It is shown that in every repeated zero-sum game of incomplete information in which the information to both players is identical, $\lim \nu_n$ and $\nu_\infty$ exist and are equal.

Some generalizations of Dupac's dynamic stochastic approximation have been worked out to the more general cases of time variation. Sufficient conditions for convergence in the mean square and with probability one are given in case of deterministic trend and convergence with a bound is proved for the random trend case, using the estimation scheme $x_{n+1} = g_n(x_n) + a_n(\alpha - y_{n+1}(g_n(x_n)))$. This estimation procedure seems to be of practical use to a variety of problems in estimation, prediction and control.

Let $(X, S)$ be a measurable space and $M$ a collection of probability measures on S. Pitcher (Pacific J. Math. (1965), pages 597-611) introduced a condition called compactness on the statistical structure $(X, S, M)$, more general than domination by a fixed $\sigma$-finite measure. Under this condition he gave a construction of a minimal sufficient subfield. In this paper a counterexample invalidating this construction is presented. We give a nonconstructive proof of the existence of a minimal sufficient subfield under a condition slightly weaker than compactness. The proof proceeds by considering the intersection of an uncountable collection of sufficient subfields, and relies on a martingale convergence theorem with directed index set, due to Krickeberg.

Suppose the arms of a two-armed bandit generate i.i.d. Bernoulli random variables with success probabilities $\rho$ and $\lambda$ respectively. It is desired to maximize the expected sum of $N$ trials where $N$ is fixed. If the prior distribution of $(\rho, \lambda)$ is concentrated at two points $(a, b)$ and $(c, d)$ in the unit square, a characterization of the optimal policy is given. In terms of $a, b, c$, and $d$, necessary and sufficient conditions are given for the optimality of the myopic policy.