A rotationally invariant exponential model, which includes the Fisher-von Mises and Bingham distributions as special cases, is proposed for directional data in $R^p(p \geqslant 2)$. A new regression estimator for the model parameters is developed as a competitor to the maximum likelihood estimator. Both the new estimator and the MLE are asymptotically efficient at the postulated model and are robust under small departures from that model. Computationally, the regression estimator is much simpler since it requires no iterations or numerical integrations. Goodness-of-fit can be assessed by fitting nested special cases of the general model to the data.

This paper considers Robbins-Monro stochastic approximation when the regression function changes with time. At time $n$, one can select $X_n$ and observe an unbiased estimator of the regression function evaluated at $X_n$. Let $\theta_n$ be the root of the regression function at time $n$. Our goal is to select the sequence $X_n$ so that $X_n - \theta_n$ converges to 0. It is assumed that $\theta_n = f(s_n)$ for $s_n$ known at time $n$ and $f$ an unknown element of a class of functions. Under certain conditions on this class and on the sequence of regression functions, we obtain a random sequence $X_n$ such that $|X_n - \theta_n|$ converges to 0 in Cesaro mean with probability 1. Under more stringent conditions, $X_n - \theta_n$ converges to 0 with probability 1.

When $y = M(x) + \varepsilon$, where $M$ may be nonlinear, adaptive stochastic approximation schemes for the choice of the levels $x_1, x_2, \cdots$ at which $y_1, y_2, \cdots$ are observed lead to asymptotically efficient estimates of the value $\theta$ of $x$ for which $M(\theta)$ is equal to some desired value. More importantly, these schemes make the "cost" of the observations, defined at the $n$th stage to be $\sum^n_1(x_i - \theta)^2$, to be of the order of $\log n$ instead of $n$, an obvious advantage in many applications. A general asymptotic theory is developed which includes these adaptive designs and the classical stochastic approximation schemes as special cases. Motivated by the cost considerations, some improvements are made in the pairwise sampling stochastic approximation scheme of Venter.

Consider the problem of sequential testing of a one sided hypothesis when the risk function is a linear combination of a probability of an error component and an expected cost component. Sobel's results on monotonicity of Bayes procedures and essentially complete classes are extended. Sufficient conditions are given for every Bayes test to be monotone. The conditions are satisfied when the observations are from an exponential family. They are also satisfied for orthogonally invariant tests of a mean vector of a multivariate normal distribution and for scale invariant tests of two normal variances. Essentially complete classes of tests are the monotone tests for all situations where these sufficient conditions are satisfied.

The notion of Bahadur efficiency is used to compare multivariate linear hypothesis tests based on six criteria: (1) Roy's largest root, (2) the likelihood ratio test, (3) the Lawley-Hotelling trace, (4) Pillai's trace, (5) Wilks' $U$, and (6) Olson's statistic. Bahadur exact slope is computed for each statistic as a function of noncentrality parameters using results for probabilities of large deviations. The likelihood ratio test is shown to be asymptotically optimal in the sense that its slope attains the optimal information value, and the remaining tests are shown not to be asymptotically optimal. Inequalities are derived for the slopes showing order of preference.

For linear rank tests in the two-sample case the concept of approximate Bahadur efficiency (BE) is developed, and as the main result of this paper the equality of the approximate and exact local BE is shown. According to a result of Wieand, local approximate BE equals Pitman efficiency under rather general conditions and as a consequence these three approaches to efficiency generally coincide for the class of linear rank tests.

Let $P_\eta, \eta \in \Theta \times \Gamma \subset \mathbb{R} \times \mathbb{R}^k$, be an exponential family. It is shown that the sequence of tests $(\varphi^\ast_n)_{n\in\mathbb{N}}$, where $\varphi^\ast_n, n \in \mathbb{N}$, is u.m.p. in the class of all tests similar with respect to the nuisance-parameter $\gamma$ for the hypothesis $\{P^n_{(\theta, \gamma)}: \gamma \in \Gamma\}$ against alternatives $P^n_{(\theta_1, \eta_1)}, \theta_1 > \theta, \eta_1 \in \Gamma$, is asymptotically efficient in the class $\Phi^\ast_\alpha$ of test-sequences which are asymptotically of level $\alpha$ (continuously in the nuisance-parameter). Here, asymptotic efficiency of $(\varphi^\ast_n)_{n\in\mathbb{N}}$ means that for all $\gamma \in \Gamma, t > 0$, the power of $\varphi^\ast_n$ evaluated at local alternatives $P^n_{(\theta+tn^{-1/2},\gamma)}$ asymptotically attains the upper bound given for test-sequences in $\Phi^\ast_\alpha$.

Let $S_{p \times p}$ have a nonsingular Wishart distribution with unknown matrix $\Sigma$ and $k$ degrees of freedom. For two different loss functions, estimators of $\Sigma^{-1}$ are given which dominate the obvious estimators $aS^{-1}, 0 < a \leqslant k - p - 1$. Our class of estimators $\mathscr{C}$ includes random mixtures of $S^{-1}$ and $I$. A subclass $\mathscr{C}_0 \subset \mathscr{C}$ was given by Haff. Here, we show that any member of $\mathscr{C}_0$ is dominated in $\mathscr{C}$. Some troublesome aspects of the estimation problem are discussed, and the theory is supplemented by simulation results.

A balanced incomplete block (BIB) design with $b$ blocks is said to have the support size $b^\ast$ when exactly $b^\ast$ of the $b$ blocks are distinct. BIB designs with $b^\ast < b$ have interesting applications in design of experiments and controlled sampling as explained in details in Foody and Hedayat (1977) and Wynn (1977). A method called "trade-off" is introduced for the construction of BIB designs with repeated blocks. This method is utilized to study BIB designs with arbitrary $v$ treatments in blocks of size $k = 3$ in general and with $v = 7$ and $k = 3$ in particular. It is shown that BIB designs with $v = 7, k = 3$, any $b$, and any $b^\ast$ exist if and only if (i) $b$ is divisible by 7, (ii) $7 \leqslant b^\ast \leqslant \min(b, 35)$, (iii) $b^\ast \neq 8, 9, 10, 12$, or $16$, (iv) $(b, b^\ast) \neq (28, 24), (28, 27), (35, 30), (35, 32), (35, 33), (35, 34)$ or (42, 34).

A decision is a mapping from states of nature to consequences. Given a utility $u$ on the set of consequences and a measure $\nu$ on the set of states, the expected utility of a decision $f$ is $\int u(f(e)) d\nu(e)$. By the "expected utility hypothesis" on a set of choices made by an individual we mean that there exists a utility and a measure such that the individual always chooses the decision of highest expected utility. We present a set of necessary and sufficient conditions that a set of choices between two decisions be consistent with the expected utility hypothesis. We assume the set of states and the set of consequences to be finite and we do not assume the ordering, given by the choices, to be complete. Our conditions require the individual to make new choices, between decisions which involve repetitions of states, in a consistent way. There are finitely many new choices and they do not involve utility.

It is supposed that an observable time series $\{x_t\}$ is representable as the sum of a "signal" $s_t$ and a "noise" $n_t$, and that it is desired to extract the signal $s_t$, i.e., to obtain an estimate $\hat{s}_t$ of $s_t$. Corresponding to any such estimate is a signal extraction error, $\delta_t = s_t - \hat{s}_t$, which for nondeterministic stochastic processes possesses a nonzero mean square. For the class of homogeneously nonstationary processes, characterizations of the extraction error process are given, and it is shown that the mean square of the error does not exist unless the nonstationary autoregressive operators in the $s$- and $n$-processes have distinct roots. The MSE, autocorrelations and spectrum of the error, when it is stationary, are illustrated for some special cases, including two stochastic-model approximations to the Census $X$-11 seasonal adjustment procedure.

Consider samples of size $n$ from each of $k$ symmetric populations, differing only in their location parameters. The decision problem is to select the best population--the one with the largest location parameter--with control on the probability of correct selection (PCS) whenever the largest parameter is at least $\Delta$ units larger than all others, and whenever the common error distribution belongs to a specified neighborhood of the standard normal. It is shown that, if the sample size $n$ is chosen according to a formula given herein, and Huber's $M$-estimate is applied to each of the $k$ samples with the population having the largest estimate being selected as best, that the PCS goal is achieved asymptotically (as $\Delta\downarrow 0$)--the procedure is robust. Moreover, no other selection procedure can achieve this goal asymptotically with a smaller sample size--the procedure is most economical. Comparisons with other procedures are given. These results are based on a uniform asymptotic normality theorem for Huber's $M$-estimate, contained herein.

In the regression design problem with observations which are second order processes the estimation of the mean function involves function space valued random variables. The best unbiased linear estimator of the mean function is found and an exact analogue of the Kiefer-Wolfowitz theorem in design theory is proved.

A subset selection problem is formulated as a multiple decision problem. Then, restricting attention to rules which attain a certain minimum probability of correct selection, the minimax value is computed, under general conditions, for loss measured by subset size and number of non-best populations selected. Applying this to location and scale problems, previously proposed rules are found to be minimax. But for problems involving binomial, multinomial and multivariate noncentrality parameters, such as $\chi^2$ and $F$, previously proposed rules are found to be not minimax.

A result due to Ray is extended to cover hypothesis testing problems in which samples are taken sequentially in pairs from two independent distributions of a one-dimensional univariate exponential family. In particular, for certain reasonable restrictions on losses and costs, an upper bound to the exact stage of truncation is obtained for a truncated Bayes procedure.