The two-sample problem for directional data with dimension greater than one is considered. A large family of permutation tests is proposed and studied. The statistics upon which the tests are based are related to those introduced by Gine in the context of tests for uniformity, and are defined in terms of Sobolev norms. Examples treated include the unit spheres $S^p$ and hemispheres $H^p$ for directions in $(p + 1)$-dimensional Euclidean space, and the torus $T^2 = S^1 \times S^1$ for pairs of directions in two dimensions. Computable forms of the statistics with specified consistency properties are obtained for each of these examples. Sampling from the permutation distribution is proposed as a means of implementing the tests in practice. Several tests for uniformly on the torus $T^2$ are also obtained.

In this paper we consider adaptive one-sample rank tests of the following type: the score function $J$ of the test is estimated from the sample under the restriction that $J \in \mathscr{J}$, for some given one-parameter family $\mathscr{J} = \{J_r, r \in I \subset R^1\}$. Using deficiencies, we compare the performance of such tests to that of rank tests with fixed scores. Conditions on the estimator $S$ of the parameter $r$ and on $J_r$ are given, under which the deficiency tends to a finite limit, which is obtained. For a particular class of estimators which are related to the sample kurtosis, explicit results are obtained.

The functional $T(F) = \int F^{-1}(t)J(t) dt$ associated with linear combinations of order statistics is shown to have a Frechet-type differential. As a corollary, the statistic $T(F_n)$ obtained by evaluating $T(\cdot)$ at the sample df $F_n$ is seen to be asymptotically normal and to obey a law of the iterated logarithm.

Questions of admissibility of statistical estimators are reduced to considerations involving differential inequalities. The coefficients of these inequalities involve moments of the underlying distributions; and so are, in principle, not difficult to derive. The methods are "heuristic" because it is necessary to verify on an ad-hoc basis that error terms are small. Some conditions on the structure of the problem are given which we believe will guarantee that these error terms are small. Several different statistical estimation problems are discussed. Each problem is transformed (if necessary) so as to meet the above mentioned structure conditions. Then the heuristic method is applied in order to generate conjectures concerning the admissibility of certain generalized Bayes procedures in these problems.

When samples are taken independently from different populations, the sample means are jointly admissible for the population means with the squared error as loss function. The result supplements a previous result that when there are many variate values associated with the population units, the sample means of the variate values are jointly admissible for the population means for general loss functions.

A method is presented for the construction of all families of smallest simultaneous confidence sets (SCS) in a given class, for a family $\{\psi_i(\gamma)\}$ of parametric functions of the parameter of interest $\gamma = \gamma(\theta)$. The method is applied to the MANOVA problem (in its canonical form) of inference about $M = EX$, where $X$ is $q \times p$ and has rows that are independently multivariate normal with common covariance matrix $\Sigma$. Let $S$ be the usual estimate of $\Sigma$ and put $W = (M - X)S^{-\frac{1}{2}}$. It is shown that smallest equivariant SCS for all $a'M, a \in R^q$, are necessarily those that are exact with respect to the confidence set for $M$ determined by $\lambda_1(WW') \leqslant \operatorname{const} (\lambda_1 = \text{maximum characteristic root})$, i.e., derived from the acceptance region of Roy's maximum root test (this is strictly true for $p < q$, and true for $p \geqslant q$ under a weak additional restriction). It is also shown that smallest equivariant SCS for all tr $NM$, with rank $(N) \leqslant r$, are necessarily those that are exact with respect to $\|W\|_{\varphi_r} \leqslant 1$, where $\varphi_r$ is a symmetric gauge function that, on the ordered positive cone, depends only on the first $r$ arguments. Taking $r = 1$, the simultaneous confidence intervals for all $a'Mb$ of Roy and Bose emerge, and $r = \min(p, q)$ results in the simultaneous confidence intervals for all tr $NM$ of Mudholkar.

Along with the asymptotic distribution, expressions for the asymptotic bias and asymptotic dispersion matrix of the preliminary test maximum likelihood estimator for a general multi-sample parametric model (when the null hypothesis relating to the restraints on the parameters may not hold) are derived and compared with the parallel expressions for the unrestricted and restricted maximum likelihood estimators. This study reveals the robustness property of the preliminary test estimator when the assumed restraints may not hold.

We give a decision-theoretic justification of a heuristic principle suggested by Robbins. Suppose a sequence $X_1, X_2, \cdots$ of independent random variables can be observed; they are known to be identically distributed with unknown mean and variance $u, \sigma^2(u)$ respectively. We assume further that the common distribution is of exponential type, so that the sample mean $\bar{X}$ is a sufficient statistic. The problem is to estimate $u$ taking into account the cost of sampling, which is assumed to be linear in the sample size, $n$. For the case of a fixed-size sample, with squared-error loss function, the minimum risk (expected loss) is obtained, after rescaling if necessary, as $\min_nE(n + A(\bar{X} - u)^2) = \min_n(n + A\sigma^2(u)/n) = 2A^{\frac{1}{2}}\sigma(u)$, and is attained at $n_0 = A^{\frac{1}{2}}\sigma(u)$. Since $u$ is unknown, this optimum value is not available; and for any fixed $n$, if $\sigma(u)$ is unbounded, the risk $n + A\sigma^2(u)/n$ can be much larger than $2A^{\frac{1}{2}}\sigma(u)$. For calibration of the performance of sequential stopping rules, Robbins advocated consideration of the "regret" $E(n + A(\bar{X} - u)^2) - A^{\frac{1}{2}}\sigma(u),$ and several authors have constructed procedures with uniformly bounded regret. Two questions arise, which we settle here (in a certain asymptotic sense, and after making certain smoothness assumptions). First, can any procedure have strictly negative regret, for all $u$? Second, if a procedure has uniformly bounded regret, is it necessarily close to being optimum, in the sense that for each (suitably smooth) sequence of prior distributions on $u$, is it only boundedly worse than the corresponding sequence of Bayes procedures? Our answers are no, and yes, respectively. Several examples are discussed, and analogies pointed out with the fixed-sample-size concepts of the asymptotic optimality of maximum-likelihood estimates, and the super-efficiency phenomenon.

The problems of estimating sequentially the intensity parameter of a homogeneous Poisson process and of estimating sequentially the mean of a sequence of i.i.d. Poisson rv's, are considered. The procedures suggested are shown to perform well for large values of the parameter and/or for small sampling cost. Having bounded regret, the procedure for estimating the mean of the Poisson sequence is asymptotically Bayes w.r.t. any sequence of a priori densities, which spread mass in a suitably smooth manner.

Let $X_1, \cdots, X_n$ be a random sample from a full-rank multivariate normal distribution $N(\mu, \Sigma)$. The two cases (i) $\mu$ unknown and $\Sigma = \sigma^2\Sigma_0, \Sigma_0$ known, and (ii) $\mu$ and $\Sigma$ completely unknown are considered here. Transformations are given that transform the observation vectors to a (smaller) set of i.i.d. uniform rv's. These transformations can be used to construct goodness-of-fit tests for these multivariate normal distributions. Two examples are given to illustrate the application of these tests to numerical problems.

Let $X_1, \cdots, X_n$ be an ordered sample from a distribution $A_n$ on [0, 1]. The $k$-spacings $D_1(N, k), \cdots, D_N(N, k)$ are defined and the weak convergence of their empirical distribution function under a sequence of alternatives $A_n$ approaching the uniform distribution is established. This is then applied to find the limiting distribution of $W_n(g, k) = N^{-\frac{1}{2}}\Sigma^N_{i=l}(g(NkD_i(N, k)) - a)$ where $g$ is a smooth function and $k$ is fixed. The statistics $W_n(g, k)$ can be used to test the hypothesis that the observations are uniformly distributed in [0, 1]. The asymptotic relative efficiency of $W_n(g, k)$ with respect to $W_n(g, 1)$ is shown to increase without limit for several functions $g$. The test with $g(x) = x^2$ is shown to be asymptotically optimal within the class $W_n(g, k)$ for any fixed $k$. The paper extends results of Rao and Sethuraman.

Let $\mathscr{E}_x = \{X, S_X; P_\theta, \theta\in\Theta\}$ and $\mathscr{E}_Y = \{Y, S_Y; Q_\theta, \theta\in\Theta\}$ be two statistical experiments with the same parameter space $\Theta$. Some implications of the sufficiency of $\mathscr{E}_X$ for $\mathscr{E}_Y$, according to Blackwell's definition, are given in terms of Kullback-Leibler information and Fisher information matrices. For a scale parameter $\theta$, and $k_1 > k_2 > 0$, the experiment with parameter $\theta^{k_1}$ is proved to be sufficient for the experiment with parameter $\theta^{k_2}$ for a class of distributions including the gamma distribution and the normal distribution with known mean. Some results of Stone are generalized to the class of experiments with both location and scale parameters. A concept of sufficiency is proposed in which $\mathscr{E}_X$ is more informative than $\mathscr{E}_Y$ for a fixed prior distribution of $\theta$ if the expected Bayes risk from $\mathscr{E}_X$ is not greater than that from $\mathscr{E}_Y$ for every decision problem involving $\theta$. This concept is then used to develop a definition of marginal Bayesian sufficiency in the presence of nuisance parameters.

Consider the realization of the process $y(t) = \Sigma^n_{k=1}\theta_kf_k(t) + \xi(t)$ on the interval $T = \lbrack 0, 1\rbrack$ for functions $f_1(t), f_2(t), \cdots, f_n(t)$ in $H(R)$, the reproducing kernel Hilbert space with reproducing kernel $R(s, t)$ on $T \times T$, where $R(s, t) = E\xi(s)\xi(t)$ is assumed to be continuous and known. Problems of the selection of functions $\{f_k(t)\}^n_{k=1}$ are discussed for $D$-optimal, $A$-optimal and other criteria of optimal designs.

Each of two arms generate an infinite sequence of Bernoulli random variables. At each stage we choose which arm to observe based on past observations. The parameter of the left arm is known; that of the right arm is a random variable. There are two conflicting desiderata: to observe a success at the present stage and to obtain information useful for making future decisions. The payoff is $\alpha_m$ for a success at stage $m$. The objective is to maximize the expected total payoff. If the sequence $(\alpha_1, \alpha_2, \cdots)$ is regular an observation of the left arm should always be followed by another of the left arm. A rather explicit characterization of optimal strategies for regular sequences follows from this result. This characterization generalizes results of Bradt, Johnson, and Karlin (1956) who considered $\alpha_m$ equal to 1 for $m \leqslant n$ and 0 for $m > n$ and of Bellman (1956) who considered $\alpha_m = \alpha^{m-1}$ for $0 \leqslant \alpha < 1$.

Let $Y_t$ satisfy the stochastic difference equation $Y_t = \sum^p_{j = 1}\eta_jY_{t - j} + e_t$ for $t = 1, 2, \cdots$, where the $e_t$ are independent identically distributed $(0, \sigma^2)$ random variables and the initial conditions $(Y_{-p + 1}, Y_{-p + 2}, \cdots, Y_0)$ are fixed constants. It is assumed the true, but unknown, roots $m_1, m_2, \cdots, m_p$ of $m^p - \sum^p_{j = 1}\eta_jm^{p - j} = 0$ satisfy $m_1 = m_2 = 1$ and $|m_j| < 1$ for $j = 3, 4, \cdots, p$. Let $\hat{\mathbf{\eta}}$ denote the least squares estimator of $\mathbf{\eta} = (\eta_1, \eta_2, \cdots, \eta_p)'$ obtained by the least squares regression of $Y_t$ on $Y_{t - 1}, Y_{t - 2}, \cdots, Y_{t - p}$ for $t = 1, 2, \cdots, n$. The asymptotic distributions of $\hat{\mathbf{\eta}}$ and of a test statistic designed to test the hypothesis that $m_1 = m_2 = 1$ are characterized. Analogous distributional results are obtained for models containing time trend and intercept terms. Estimated percentiles for these distributions are obtained by the Monte Carlo method.

The monotone regression of a variable $X$ on another variable $Y$ is of particular interest when $Y$ cannot be directly observed. The correlation of $X$ and $Y$ can be tested if at least high and low values of $Y$ can be recognized. If all the components of a random vector have monotone regression on a variable $Y$, and if they are all uncorrelated given $Y$, then an inequality due to Chebyshev shows that marginal zero covariances imply that all but at most one of the components are uncorrelated with $Y$. Cases are examined where marginal uncorrelatedness of attributes implies their independence. Applications to contaminated experiments and to discriminant analysis are noted.

For the two-action two-state compound decision problem, a class of decision rules is found that are both asymptotically optimal and admissible at each stage $n$. These rules are Bayes rules with respect to appropriate exchangeable prior distributions.

Let $X_1, \cdots, X_n$ be a sequence of independent random vectors taking values in $\mathbf{\mathbb{R}}^d$ with a common probability density $f$. If $f_n(x) = (1/h) h^{-d}_n\sum^n_{i = 1}K((x - X_i)/h_n)$ is the kernel estimate of $f$ from $X_1, \cdots, X_n$ then conditions on $K$ and $\{h_n\}$ are given which insure that $\int|f_n(x) - f(x)|dx \rightarrow_n 0$ in probability or with probability one. No continuity conditions are imposed on $f$.

Fisher considered the problem of constructing sequentially a "better" finite population from a given infinite one. The purpose of this paper is to prove the optimality of Fisher's procedure.

A family of estimators which dominate the maximum likelihood estimators of regression coefficients is given when the dependent variable and (3 or more) independent variables have a joint normal distribution.