Let $y = \theta + e$ where $\theta$ and $e$ are independent random variables so that the regression of $y$ on $\theta$ is linear and the conditional distribution of $y$ given $\theta$ is homoscedastic. We find prior distributions of $\theta$ which induce a linear regression of $\theta$ on $y$. If in addition, the conditional distribution of $\theta$ given $y$ is homoscedastic (or weakly so), then $\theta$ has a normal distribution. The result is generalized to the Gauss-Markoff model $\mathbf{Y} = \mathbf{X\theta} + \mathbf{\varepsilon}$ where $\mathbf{\theta}$ and $\mathbf{\varepsilon}$ are independent vector random variables. Suppose $\bar{y}_i$ is the average of $p$ observations drawn from the $i$th normal population with mean $\theta_i$ and variance $\sigma_0^2$ for $i = 1,\cdots, k$, and the problem is the simultaneous estimation of $\theta_1,\cdots, \theta_k$. An estimator alternative to that of James and Stein is obtained and shown to have some advantage.

A formal structure for conditional confidence (cc) procedures is investigated. Underlying principles are a (conditional) frequentist interpretation of the cc coefficient $\Gamma$, and highly variable $\Gamma$. The latter allows the stated measure of conclusiveness to reflect how intuitively clear-cut the outcome of the experiment is. The methodology may thus answer some criticisms of the Neyman-Pearson-Wald approach, but is in the spirit of the latter and includes it. Example: $X$ has one of $k$ densities $f_\omega \operatorname{wrt} \nu$. A nonempty set of decisions $D_\omega \subset D$ is "correct" for state $\omega$. A nonrandomized cc procedure consists of a pair $(\delta, Z)$ where $\delta$ is a nonrandomized decision function and $Z$ is a conditioning rv. The cc coefficient is $\Gamma_\omega = P_\omega\{\delta^{-1}(D_\omega)\mid Z\}$. If $X = x_0$, we make decision $\delta(x_0)$ with "cc $\Gamma_\omega(x_0)$ of being correct if $\omega$ is true"; it is unnecessary, but often a practical convenience (as for un-cc intervals), to have $\Gamma_\omega$ independent of $\omega$. Possible notions of "goodness" are discussed; e.g., $(\bar{\bar{\delta}}, \bar{\bar{Z}})$ at least as good as $(\bar{\delta}, \bar{Z})$ if $P_\omega\{\bar{\bar{\delta}}(X)\in D_\omega$ and $\bar{\bar{Gamma}}_\omega > t\} \geqq p_\omega\{\bar{delta}(X) \in D_\omega$ and $\bar{\Gamma}_\omega > t\}^\forall t, \omega,$ and $P_\omega\{\bar{\bar{\Gamma}}_\omega = 0\} \geqq P_\omega\{\bar{Gamma}_\omega = 0\}^\forall\omega$. It is proved that cc procedure $(\delta, Z)$ is admissible if the non-cc $\delta$ is admissible. For 2-hypothesis problems the converse is true; otherwise, "star-shaped" partitions of the likelihood ratio space are needed. Other loss structures are also treated.

A measurable cross-section for orbits of a sample space under a free (exact) transformation group is shown to exist under topological regularity conditions. This is used to represent the sample space as essentially the product of a maximal invariant and an equivariant part, which implies Stein's representation for the density of the maximal invariant.

For a normally distributed mixed model with two unknown variance components $\theta_1$ and $\theta_2$, a tractable characterization is given for the admissible estimators within the class $\tilde{\mathscr{N}}_\delta$ of invariant quadratic unbiased estimators for $\delta_1\theta_1 + \delta_2\theta_2$. Here the term admissible is used with reference only to the class $\tilde{\mathscr{N}}_\delta$. This characterization is based on a result for general linear models which characterizes the admissible estimators within the class of linear unbiased estimators. The admissibility of MINQUE estimators and the usual analysis of variance estimators is considered.

Some concepts of relative and absolute efficiency for sequential tests are considered. These are sequential analogs of Hodges-Lehmann and Chernoff efficiencies. Using these criteria, several sequential tests for the mean of a normal distribution are evaluated. Among them are Wald's SPRT, Anderson's triangular boundary, Bayes and APO tests and a repeated significance test. Truncated versions of these tests are also considered. Asymptotic expressions for the (expected) stopping times and error rates are given.

By means of a general weak convergence theorem some invariance principles are proven for the multivariate sequential empirical process and for the multivariate rank order process w.r.t. stronger metrics than the generalized Skorohod metric. The underlying random variables are neither assumed to be independent nor to be stationary. These results are then applied to derive convergence of the weighted empirical cumulatives and for the weighted rank order process. Finally by a new representation asymptotic normality is proven for a general class of linear multivariate rank order statistics.

Lower and upper bounds on the power of a Smirnov test for symmetry $H_0: \bar{F} = F$ versus $H_1: \bar{F} \geqq F, \sup_x\lbrack\bar{F}(x) - F(x)\rbrack = \Delta > 0$ are obtained exactly or estimated for selected values of sample size $N$, level $\alpha$, and asymmetry $\Delta$. Furthermore the asymptotic power of the test as $N^{\frac{1}{2}}\Delta_N \rightarrow c$ is shown to be bounded by $\Phi(c - k_\alpha)$ and 1 if $c \geqq k_\alpha$ and by $\alpha$ and $2\Phi(c - k_\alpha)$ if $c < k_\alpha$, where $k_\alpha$ is the critical point. These bounds compare favorably in some respects with those of the Wilcoxon and other monotone rank tests studied in "Power bounds and asymptotic minimax results for one-sample rank tests," Ann. Math. Statist. 42 12-35.

This paper examines the relationships between optimality criteria which are commonly used for undiscounted, discrete-time, countable state Markovian decision models. One approach, due to Blackwell, is to maximize the expected discounted total return as the discount factor approaches 1. Another, due to Veinott, is to maximize the Cesaro means of the finite horizon expected returns as the horizon tends to infinity. Derman's is to maximize the long-run average gain. Denardo, Miller and Lippman showed that Blackwell's and Veinott's approaches are equivalent for finite state and action spaces. As shown here, that equivalence breaks down when the state space is countable. Also, policies optimal according to Blackwell's or Veinott's approach need not be optimal according to Derman's. On the positive side, fairly weak conditions are given under which Blackwell's and Veinott's criteria imply Derman's, and somewhat stronger conditions under which Blackwell's and Veinott's criteria are equivalent.

A new method of construction of affine resolvable balanced incomplete block design from a given affine resolvable balanced incomplete block design is given.

Let $s_{ju}$ be the number of varieties in common to the $j$th and $u$th blocks of a symmetric semiregular group divisible design. Connor (1952) and Saraf (1961) have given inequalities for $s_{ju}$. Both these inequalities lead to the same stronger inequality $\lambda_1 \leqq s_{ju} \leqq 2\lambda_2 - 1$. Both the upper and lower bounds are attained by a series of designs derived from lattices.

When a variety is lost or omitted from a balanced incomplete block design, the residual design is not, in general, balanced. A lower bound is obtained for the overall efficiency in this case and it is shown that the overall loss of efficiency even in the unbalanced case is small.

Conditional properties of the usual confidence intervals for the situations referred to in the title are investigated. It is shown that there can be no negatively biased relevant selections in a sense which implies that there can be no negatively biased relevant subsets in the sense of Buehler (1959). The intuitive meaning of these results is that there is no way of betting that the quoted confidence levels are too high which yields positive expected return for all parameter values. In addition it is reported that the coverage probabilities for the Behrens-Fisher intervals are always larger than the nominal significance level would suggest. Thus the Behrens-Fisher and Student's $t$ procedures can be considered to be conservative.

It is shown that under certain regularity conditions the Bernstein-v. Mises approximation of the posterior distributions is of order $O(n^{-\frac{1}{2}})$ with high probability.

Let $\theta$ be the natural parameter of a continuous one-parameter exponential family. An empirical Bayes test is constructed for testing $\theta \leqq 0$ against $\theta > 0$ with a piecewise linear loss function. Since the problem is monotone, Bayes tests for a given prior distribution can be characterized by a single parameter, e.g., the size of the test under $\theta = 0$. Therefore the construction of an empirical Bayes test can be reduced to the construction of an estimator of this parameter. Such an estimator is constructed and the convergence rate of its mean squared error is investigated. The empirical Bayes test constructed in this way has not only nice asymptotic properties, but it can also be applied to small samples because of its (weak) admissibility.

A generalization of a Bernoulli process which incorporates a dependence structure was given by Klotz (1972, 1973), in which he considered $X_1, X_2, \cdots, X_n$ as a stationary two-state Markov chain with state space $\{0, 1\}$. The parameters of the process are $p = P(X_i = 1)$ and $\lambda$, which measures the degree of persistence in the chain. Klotz was unable to solve the equations arising from the full likelihood for the M.L.E.'s of $p$ and $\lambda$, so proposed and investigated an ad hoc procedure. Here explicit solutions are obtained for M.L.E.'s based on a modified likelihood function, where the modification consists of neglecting the first term of the full likelihood. In addition it is observed that Klotz's equations can in fact be solved explicitly.

In the literature one finds (at least) two approaches towards proving that the sample mean is uniformly minimum variance (UMV), among unbiased estimates that "ignore the labels," for the finite population mean: The "traditional approach" and the "scale-load approach." The identity of results under the two approaches extends to a more general setting, as shown in this paper: The Horvitz-Thompson estimate is UMV unbiased for any given fixed effective size design.

It is often stated that the variance of an unbiased estimator of a function of a real parameter can attain the Cramer-Rao lower bound only if the family of distributions is a one-parameter exponential family. A rigorous proof of this statement, subject to certain regularity conditions, has been given by Wijsman. However, in general, the statement is not true. Assuming a revised set of regularity conditions it is shown here that there exists a more general class of distributions for which the Cramer-Rao lower bound for the variance is attained for almost all or even all values of the parameter in an interval. The class reduces to the exponential class only by imposing a restriction requiring the absolute continuity in the parameter of a function involving the logarithm of the probability density.

The approximate Bahadur efficiency and the Pitman efficiency for hypothesis testing problems are considered. A theorem is stated and proved which gives a condition under which the existence of the limiting (as the alternative approaches the hypothesis) approximate Bahadur efficiency implies the existence of the limiting (as the significance level approaches 0) Pitman efficiency and the equality of the two limits. Several examples are then given to show how the theorem may be used in computing previously unknown limiting Pitman efficiencies using the Bahadur approach.

Fixed cost sequential sampling schemes are introduced in this article. In these schemes units are observed sequentially according to a given sampling method until the total cost reaches a preassigned value; it is assumed that the cost of examining each unit is unknown in advance. It is shown how the notion of sufficiency in sampling can be used to construct unbiased estimators of population parameters under these schemes.

A stochastic approximation process for estimating an unknown parameter in nonlinear regression is discussed. The process was suggested by Albert and Gardner [Stochastic Approximation and Nonlinear Regression. Research Monograph No. 42. M.I.T. Press, Cambridge, Massachusetts]. An almost sure convergence of the process is proved. The proof is an application of a theorem of Robbins and Siegmund on the almost sure convergence of nonnegative almost supermatingales. The conditions given here are weaker than those given by Albert and Gardner.