In this paper we consider estimation of the location parameter $\theta \in R^d$ based on a random sample from $(\theta + X, Y),$ where $X$ is a $d$-dimensional random vector, $Y$ is a random element of some measure space $\mathscr{Y},$ and $(X, Y)$ has a known distribution. We first define the Fisher information $\mathscr{J}(\theta + X, Y)$ and the inverse information $\mathscr{J}^-(\theta + X, Y)$ under no regularity conditions. The properties of these quantities are investigated. Supposing that $E|X|^\delta < \infty$ for some $\delta > 0$ we show that for $n$ sufficiently large the Pitman estimator $\hat{\theta}_n$ of $\theta$ based on a random sample of size $n$ is well defined, unbiased, and its covariance, which is independent of $\theta$, satisfies the inequality $n \operatorname{Cov} \hat{\theta}_n \geqq \mathscr{J}^-(\theta + X, Y)$. Moreover, $\lim_{n\rightarrow \infty} n \operatorname{Cov} \hat{\theta}_n = \mathscr{J}^-(\theta + X, Y)$ and $n^\frac{1}{2}(\hat{\theta}_n - \theta)$ is asymptotically normal with mean zero and covariance $\mathscr{J}^-(\theta + X, Y)$.

Conditions are given for admissibility of procedures invariant under two-dimensional translation. These conditions may be applied to obtain admissibility of (i) the best invariant procedure for estimating a two-dimensional location parameter and (ii) procedures which are Bayes in the class of procedures invariant under two-dimensional translation, the prior distribution being over values of some other parameter.

Let $X$ and $Y$ be two random variables with continuous distribution functions $F$ and $G$ and means $\mu$ and $\xi$. In a linear model, the crucial property of the contrast $\Delta = \xi - \mu$ is that $X + \Delta =_\mathscr{L} Y$, where $= _\mathscr{L}$ denotes equality in law. When the linear model does not hold, there is no real number $\Delta$ such that $X + \Delta = _\mathscr{L} Y$. However, it is shown that if parameters are allowed to be function valued, there is essentially only one function $\Delta(\bullet)$ such that $X + \Delta(X) = _\mathscr{L} Y$, and this function can be defined by $\Delta(x) = G^{-1}(F(x)) - x$. The estimate $\hat{\Delta}_N(x) = G_n^{-1}(F_m(x)) - x$ of $\Delta(x)$ is considered, where $G_n$ and $F_m$ are the empirical distribution functions. Confidence bands based on this estimate are given and the asymptotic distribution of $\hat{\Delta}_N(\bullet)$ is derived. For general models in analysis of variance, contrasts that can be expressed as sums of differences of means can be replaced by sums of functions of the above kind.

In this paper the asymptotic behaviors of the Bayes test and Bayes risk are studied in both the one-sided and two-sided testing problems, where the independent observations are taken from one member of a one-parameter exponential family. Precise asymptotic expressions are found which show the Bayes procedure to be relatively insensitive to the prior distribution as the sample size increases to infinity.

This paper provides a means for testing the hypothesis that two simple regression lines are parallel, when the two sets of error terms have two arbitrary unknown continuous distributions. The non-parametric test which is developed here is analogous to the two-sample Wilcoxon test. At the end of the paper, two additional problems in non-parametric regression analysis are briefly referred to.

This study is concerned with the joint distribution of Gerig's (1969) statistics when applied to tests for shift in various marginal distributions pertaining to complete two-way multivariate data. The exact small-sample distribution can be found using conditional permutation arguments, and the limiting permutation distribution is shown to belong to a known class of multivariate chi-square distributions. A special case yields the limiting joint distribution of Friedman's (1937) $_{\chi r^2}$ statistics for the one-dimensional marginal distributions. Berry-Esseen bounds are given for the rate of convergence of the joint distribution to its limiting form when the underlying distributions are identical over replications.

In recent years, a theory has been emerging concerning the statistical power of small computers. In the present paper it is proven that in the sense peculiar to this literature, small computers (mathematically equivalent to finite automata) can in general be designed to solve multiple simple hypothesis testing problems. In many cases, only one state for each hypothesis is needed. In a more conventional sense, we reveal the construction of finite automata which implement sequential decision procedures having the capacity to distinguish between any given finite set of probabilities with any desired accuracy. Finally, some results on the ability of finite automata to track time--changing hypotheses are outlined.

A theory of matrix differentiation is presented which uses the concept of a matrix of derivative operators. This theory allows matrix techniques to be used in both the derivation and the description of results. Several new operations and identities are presented which facilitate the process of matrix differentiation. The derivative theorems and new operations are then applied to the problem of determining optimal policies in a linear decision model with unknown coefficients, a problem which would be cumbersome if not impossible to solve without these theorems and operations.

We study the probability of error asymptotically for testing one Gaussian stochastic process against another when the mean vectors are zero and we have the choice between two given covariance matrices. It is shown that under certain conditions the probabilities of error form asymptotically a geometric progression with a ratio that is derived. The approach employs Laplace's method of approximating integrals and does not appeal to Fourier analysis; in this sense it can be said to be elementary.

Let $(X(t), Y(t))$ be a complex vector process stationary of order $k$ for any $k, k = 1,2,\cdots$, such that $Y(t)$ is expressed as a polynomial functional of degree 2 operating on $X(t)$. Then $Y(t)$ can be rewritten as a sum of orthogonal projections $G_j(K_j, Y(t)), j = 0, 1, 2$. It is shown that there is a set of functionals which approximate in mean square the projection $G_2(K_2, Y(t))$. Moreover, it is possible to determine the kernels associated with these functionals.

Estimation of the $k$th derivative of a df by means of the $k$th-order difference quotients of the empiric df is investigated. In particular, consistency conditions are given, the asymptotic bias, variance, and mean-squared error of the estimator are computed, and means of minimizing the latter are discussed.

Sanov's statement of first-order asymptotic behaviour of probabilities of large deviations of an empirical distribution function is here established for empirical probability measures, with attendant simplification of conditions. For the case of distribution functions, our theorem is strictly more general than a specialisation of results of Hoadley.

Consider independent and normally distributed random variables $X_1,\cdots, X_n$ such that $0 < \operatorname{Var} X_i = \sigma^2; i = 1,\cdots, n$ and $E(X_1,\cdots, X_n)' = A'\beta$ where $A'$ is a known $n \times k$ matrix and $\beta = (\beta_1,\cdots, \beta_k)'$ is an unknown column matrix. (The prime denotes transposition.) The cases of known and totally unknown $\sigma^2$ are considered simultaneously. Denote the experiment obtained by observing $X_1,\cdots, X_n$ by $\mathscr{E}_A$. Let $A$ and $B$ be matrices of, respectively, dimensions $n_A \times k$ and $n_B \times k$. Then, if $\sigma^2$ is known, (if $\sigma^2$ is unknown) $\mathscr{E}_A$ is more informative than $\mathscr{E}_B$ if and only if $AA' - BB'$ is nonnegative definite (and $n_A \geqq n_B + \operatorname{rank} (AA' - BB'))$.

Two classes of rank tests are considered for ordered alternatives in a randomized block design with $k$ treatments and $n$ blocks: tests based on among-blocks rankings ($A$-tests) and tests based on within-blocks rankings ($W$-tests). Previous efficiency comparisons for fixed $k, n \rightarrow \infty$ under the normal distribution have indicated that $A$-tests are more sensitive. In the present paper it is shown that this behavior is not typical under other distributions. Further, analysis of efficiencies for fixed $n, k \rightarrow \infty$ indicates greater sensitivity for $W$-tests. Considering these results and certain other desirable properties of the $W$-tests, the latter are recommended for most applications.

A recent paper by Villegas (1969) uses a fiducial type argument to lend support to the use of Jeffrey's invariant prior for the variance-covariance matrix in the non-informative situation. Sampling is assumed to be on a multi-normal random variable. Villegas proceeds by selecting a pivotal quantity which has a fixed distribution. Since there is nothing unique about this pivotal quantity, we note in this paper, that the Villegas fiducial approach could lead to other priors, unless more restrictions are imposed. In Section 2, we construct an example involving a "lower triangular" decomposition of a Wishart matrix variable that has the distribution of the disguised Wishart variable of Tan and Guttman (1971). Interestingly, an "upper triangular" decomposition of the same Wishart matrix leads to yet another prior.

For $U$-Statistics and von Mises' differentiable statistical functions, when the regular functional is stationary of order zero, almost sure convergence to appropriate Wiener processes is studied. A second almost sure invariance principle, particularly useful in the context of the law of iterated logarithm and the probability of moderate deviations, is also established.

For testing $\beta_i = \beta, i = 1,\cdots, k$, in the model $Y_{ij} = \alpha + \beta_iX_{ij} + Z_{ij} j = 1,\cdots, n_i$ a class of rank score tests is presented. The test statistic is based on the simultaneous ranking of all the observations in the different $k$ samples. Its asymptotic distribution is proved to be chi square under the hypothesis and noncentral chi square under an appropriate sequence of alternatives. The asymptotic efficiency of the given procedure relative to the least squares procedure is shown to be the same as the efficiency of rank score tests relative to the $t$-test in the two sample problem. The proposed criterion would be an asymptotically most powerful rank score test for the hypothesis if the distribution function of the observations is known.

When classifying an observation $\mathbf{z}$ which has arisen (with known prior probabilities) from one of two $p$-variate nonsingular normal populations with known parameters, the discriminant, say $U$, which minimizes the total probability of misclassification is based on the logarithm of the ratio of the densities of the two populations. When the parameters are unknown, the "classical" procedure has been to substitute sample estimates for the unknown parameters in $U$ and use the resulting sample discriminant, say $V$, as the basis for classifying future observations. This procedure need not enjoy the property of minimizing the probability of misclassification and has been justified, from the classical point of view, almost entirely on the grounds that it seems intuitively reasonable. When the covariance matrices of the two normal populations are equal, $U$ is a linear function of the observation vector $\mathbf{z}$. The fact that $U$ minimizes the probability of misclassification does not imply that $V$ will. Further, although $U$ is linear, the sample discriminant which minimizes the probability of misclassification will, in general, not be linear. Here, using the Bayesian notion of a predictive distribution, we obtain from amongst the class of linear sample discriminants that one which minimizes the predictive probability of misclassification.

We consider countable state, finite action dynamic programming problems with bounded rewards. Under Blackwell's optimality criterion, a policy is optimal if it maximizes the expected discounted total return for all values of the discount factor sufficiently close to 1. We give an example where a policy meets that optimality criterion, but is not optimal with respect to Derman's average cost criterion. We also give conditions under which this pathology cannot occur.