Consider estimators $\hat{\theta}_n$ of the location parameter $\theta$ based on a sample of size $n$ from $\theta + X$, where the random variable $X$ has an unknown distribution $F$ which is symmetric about the origin but otherwise arbitrary. Let $\mathscr{F}$ denote the Fisher information on $\theta$ contained in $\theta + X$. We show that there is a nonrandomized translation and scale invariant adaptive maximum likelihood estimator $\hat{\theta}_n$ of $\theta$ which doe not depend on $F$ such that $\mathscr{L}(n^{\frac{1}{2}}(\hat{\theta}_n - \theta)) \rightarrow N(0, 1/\mathscr{J})$ as $n \rightarrow \infty$ for all symmetric $F$.

An asymptotically efficient sequence of estimators for the location parameter of an (unknown) symmetric density $f$ is given under weak conditions on the tails of $f$. The estimators are related to the usual linear combination of order statistics except that the unknown $f$ has to be estimated and only some of the order statistics are used.

The weak convergence of empirical distribution functions subject to random perturbations and scale factors to a Gaussian process is established. This result is used to study the efficiencies of tests based on spacings in goodness-of-fit problems.

Direct products are used in some problems involving balanced factorial models for analysis of variance. Problems considered include demonstration that least squares estimators are minimum variance unbiased linear estimators when mixed models are considered, determination of the covariance structure of lest squares estimators, calculation of expected values of sums of squares and mean squares, and determination of the distribution of sums of squares and least squares estimators under the normality assumption.

Let $\pi_1,\cdots, \pi_k$ be $k$ populations with $\pi_i$ characterized by a scalar $\lambda_i \in \Lambda$, a specified interval on the real line. The object of the problem is to make some inference about $\pi_{(k)}$, the population with largest $\lambda_i$. The present work studies rules which select a random number of populations between one and $m$ where the upper bound, $m$, is imposed by inherent setup restrictions of the subset selection and indifference zone approaches. A selection procedure is defined in terms of a set of consistent sequences of estimators for the $\lambda_i$'s. It is proved the infimum of the probability of a correct selection occurs at a point in the preference zone for which the parameters are as close together as possible. Conditions are given which allow evaluation of this last infimum. The number of non-best populations selected, the total number of populations selected, and their expectations are studied both asymptotically and for fixed $n$. Other desirable properties of the rule are also studied.

Suppose $T$ is a partially ordered set and that associated with each $t$ in $T$ we have a distribution with "location parameter" $m(t)$. In this paper we discuss consistency properties of estimates of $m(\cdot)$ which are isotonic with respect to a partial order. The results extend results in the literature, some of which are contained in Brunk (1970) (Estimation of isotonic regression in Nonparametric Techniques in Statistical Inference, Cambridge University Press, 177-195), Cryer, et al. (1972) (Monotone median regression in Ann. Math. Statist. 43), Hanson, et al. (1973) (On consistency in monotone regression in Ann. Statist. 1), and Robertson and Wright (1973) (Multiple istonic median regression in Ann. Statist. 1).

Maximum likelihood estimation of the parameters $\lambda$ and $\mu$ of a simple (linear) birth-and-death process observed continuously over a fixed time interval is studied. Asymptotic distributions for large initial populations and for large periods of observation are derived and some nonstandard results appear. The result problem of estimation from the discrete skeleton of the process is also discussed.

Sequential tests that are LMP for certain one-sided testing problems are discussed. In all cases considered, the stopping rule is the first time a certain random walk leaves a bounded interval. (Thus various inequalities and approximations due to Wald can be utilized in obtaining properties of these tests.) For models in a one-parameter exponential family, each LMP sequential test is shown to be a Wald SPRT for a family of paired (conjugate) simple hypotheses.

A sequential procedure for testing the hypothesis that the distribution of a sequence of i.i.d. random variables is symmetric about zero is given, where the test statistic is a function of the signs and the rank of the absolute values of the observations. Necessary and sufficient conditions that the individual signed ranks be independent are given. The critical region, power, and expected sample size of the test are determined approximately by using the fact that the test statistic behaves asymptotically like a Brownian motion process.

A general class of quadratic rank tests for randomness versus trend is introduced and studied in this paper. Included within this class are the Cramer-von Mises two sample test, the Watson two-sample test, and their extensions to trend alternatives. Analytical study is made of the asymptotic power and efficiency of such tests in a neighborhood of the null hypothesis.

Let $C$ be the cone of functions $\phi$ that are concave-convex about the origin, continuous at the origin, and have $\phi(0) = 0$, and $\phi'(t) \leqq \phi' (-t)$ for $t \geqq 0$. Necessary and sufficient conditions are given for $\phi(\int x(t) dH(t)) \leqq \int \phi(x(t)) dH(t)$ to hold for all $\phi \in C$ and all increasing functions $x$, with $x(0) = 0$. This inequality is used to develop comparisons (i) between combinations of order statistics, and (ii) between combinations of the expected values of the order statistics, arising from distributions $F$ and $G$ in the case that $G^{-1} F \in C$. If $F(0) = G(0) = \frac{1}{2}$ and the inequality on the gradient of $F^{-1} F, (G^{-1} F)' (x) \leqq (G^{1-} F)'(-x)$ for $x > 0$, is satisfied, then $G^{-1} F \in C$ implies $F <_s G$. The inequalities presented preserve the ordering. A weaker ordering of distributions, called $r$-ordering, is defined: $F <_r G$ if and only if $F(0) = G(0) = \frac{1}{2}$ and $G^{-1} F(x)/x$ is increasing (decreasing) for $x$ positive (negative) on the support of $F$. For symmetric $r$-ordered distributions, the ratio of the expected values of the order statistics preserve the ordering.

Let $\{\mathbf{y}_1, \cdots, \mathbf{y}_N\}$ be $N$ independent repetitions of an experiment for which $\mathscr{E}\mathbf{y} = \mathbf{X\beta}$ and $\mathscr{E}(\mathbf{y} - \mathbf{X\beta})(\mathbf{y} - \mathbf{X\beta} ' = \sigma^2\mathbf{I}$, where $\mathbf{X}$ is nonstochastic of full rank and $\mathbf{\beta'} = \lbrack \mathbf{\beta_1'}, \cdot, \mathbf{\beta}_r'\rbrack$ and $\sigma^2$ are unknown parameters. We investigate some large-sample properties of $N^{\frac{1}{2}} (\hat{\mathbf{\beta}}_N - \mathbf{\beta})$, of $V_N = N(\hat{\mathbf{\beta}}_N - \mathbf{\beta}_0)'\mathbf{T}^{-1} (\hat{\mathbf{\beta}}_N - \mathbf{\beta}_0)/\sigma^2$, and of $V_{Nj} = N(\hat{\mathbf{\beta}}_{Nj} - \mathbf{\beta}_{j0})'\mathbf{T}^{-1}_{jj} (\hat{\mathbf{\beta}}_{Nj} - \mathbf{\beta}_{j0})/\sigma^2;\quad 1 \leqq j \leqq r$ where $\hat{\mathbf{\beta}}_N = N^{-1} \mathbf{TX}'(\mathbf{y}_1 + \cdots + \mathbf{y}_N)$ and $\mathbf{T} = \lbrack \mathbf{T}_{ij}\rbrack = (\mathbf{X'X})^{-1}$. Our conclusions thus apply to problems of inference regarding $\mathbf{\beta}$ and $\{ \mathbf{\beta}_1,\cdots, \mathbf{\beta}_r\}$. Given certain higher-order moments of $\mathbf{y}$, we provide bounds of the distributions of $N^{\frac{1}{2}}(\hat{mathbf{\beta}}_N - \mathbf{\beta})$, of $V_n$, of $\{ V_{N1}, \cdots, V_{Nr}\}$, and of the variance is the sample variance. ratios $U_N = \sigma^2V_N/\hat{\sigma}_N^2$ and $\{U_{N1}, \cdots, U_{Nr}\}$, where $U_{Nj} = \sigma^2\mathbf{V}_{Nj}/\hat{\sigma}_N^2$ and $\hat{\sigma}_N^2$ is the sample variance.

A Kunt Vik design of order $n$ can be defined as an $n \times n$ array of elements, chosen from a set of $n$ elements (treatments) such that with respect to rows and columns the array is a Latin square and in addition each treatment appears once in each of the $n$ left and right diagonals. These designs are useful for eliminating sources of variation in four directions. This paper is concerned with the existence and nonexistence of these designs. Specifically, (i) it is shown that no such design exists for $n$ even, (ii) these designs exist for all odd orders except possibly for $n \equiv 0 (\mod 3)$, (iii) the Kronecker product of two Knut Vik designs in a Kunt Vik design and (iv) the concept of semi Knut Vik design is also defined and it is shown that while these designs do not exist for even orders, they exist for all odd orders.

In 1960, Charles Stein conjectured that the best invariant estimate of a single co-ordinate of a $p$-dimensional location parameter would be admissible if $p \leqq 3$ but inadmissible if $p \geqq 4$. This appear present a class of examples which supports Stein's conjecture.

Minimax interval estimators of location parameters analogous to the Pitman point estimators are found by employing invariance in decision theory. The half normal distribution is used as an illustrative example.

Let $\eta = f(\mu)$ and $\theta = Ef(Y)$, where $f$ is a monotone function and $Y \sim N(\mu, \sigma^2)$. In this note we use the method of convolution transforms to show that the UMVU estimators of $\eta$ and $\theta$ based on a pair of independent sufficient statistics $T \sim N(\mu, \alpha \sigma^2)$ and $S^2 \sim \sigma^2 \chi^2_{(\nu)}$ are related to each other in a simple manner: the replacement $\alpha$ by $\alpha - 1$ in the expression of the UMVU estimator of $\eta$ gives the corresponding expression of the UMVU estimator of $\theta$. In addition, we show that a similar relationship also exists among the estimators of the variances.

The generalization of the standard model for MANOVA which allows for the possibility of different "design" matrices for the variates is known as the multiple design multi variate linear model. For example, in multivariate regression analysis we might have polynomial models of different degree in the "independent" variates. In this paper, new tests are given for the general linear hypothesis under the multiple design multivariate model and in one case the corresponding critical region is "inverted" to obtain simultaneous confidence intervals on certain functions of the location parameters.

In this paper it is shown that the distribution of the product of independently distributed beta random variables can be represented as a mixture of beta distributions. This result is then applied to derive the distributions of Wilks' statistic in the central and the noncentral linear cases. In the noncentral linear case the results of computations agree with those of Gupta (1971).

The problem of determining upper limits to the range of means of several independent normal variables with possibly different variances is considered. Simultaneous confidence regions based on others' work yield confidence limits which are rather conservative. We then approach the problem from a fiducial viewpoint and derive the equation which yields the exact fiducial upper limit.

Pfanzagl (1972) has shown that under suitable regularity conditions a family of probability measures which is generated by a transformation group and which for some sample size greater than one admits a sufficient statistic which is continuous, real-valued, and equivariant, is equivalent to the location parameter family of normal distributions or to a scale parameter family of Gamma distributions. This was proved under the assumption that the transformation group is Abelian. In this not commutativity of the group is replaced by local compactness.

In a previous paper the admissibility was proved of a generalized version of the confidence sets, commonly used in practice, which are based on the sample mean and the sample standard deviation. A stronger definition of admissibility is obtained, if instead of the length of the interval for each individual observable sample $s$, only the expected length for all samples together for each $x \in R_N$, is taken into consideration for defining the permissible alternatives to the given set of confidence intervals. This stronger definition corresponds exactly to the definition of strong admissibility formulated by the author (1969) for confidence procedures for the parameter $\theta$ in a uni- or multivariate population. Using the stronger definition it is shown that confidence sets centered at the sample mean but having a fixed length are strongly admissible. The question of the strong admissibility of the usual confidence intervals with length proportional to the sample deviation remains open.

Let $p_0$ be a given sampling design, and $p$ be any sampling design with support contained in that of $p_0$. It is shown that if an estimator, $e$, of some finite population quantity is admissible with respect to $p_0$ it is also admissible with respect to $p$. Similarly if the pair $(e, p_0)$ is uniformly admissible in the sense of Joshi (1966), then so is the pair $(e, p)$.

Let $C'$ be a class of sampling designs of fixed expected sample size $n$ and fixed inclusion probabilities $\pi_i$ and $C$ be the subclass of $C'$ consisting of designs of fixed size $n$ and inclusion probabilities $\pi_i$. Then it is established that the pair $(e^\ast, p^\ast)$ where $p^\ast \in C$ and $e^\ast(x, \mathbf{x}) = \sigma_{i \in s} b_i x_i, b_1 > 1$, and $\sigma^N_1 (b_i)^{-1} = E(n(s)) = n$, is strictly uniformly admissible among pairs $(e_1, p_1)$ where $p_1 \in C'$ and $e_1$ is any measurable estimate.

Suppose a finite population is sampled with replacement until the sample contains a fixed number $n$ of distinct units. Let $v$ denote the total number of draws. It is known that $\bar{y}_n$, the mean for the $n$ distict units, and $\bar{y}_v$, the total sample mean, are both unbiased estimators of the population means and that $V(\bar{y}_n) \leqq V (\bar{y}_v)$. In this paper the relative difference $\delta = \lbrack V(\bar{y}_v) - V)\bar{y}_n)\rbrack/V(\bar{y}_n)$ is approximate by a quantity $\delta_1$ which is easy to compute. Upper and lower bounds for $\delta - \delta_1$ are given and it is shown that $\delta < (\lambda + \varepsilon_n) f$ for $n \geqq 3$ and $f \leqq \frac{3}{4}$, where $f = n/N, N$ is the population size, $\lambda = \lbrack (1 - f)^{-\frac{1}{2}} - 1 \rbrack/f,$ and $\varepsilon_n = (1 - f)^{-1}/(n - 1)$.

Dempster (1966) defines two sampling models which he labels structures of the first and second types. In this paper we consider the application of the type one model to the problem of finding upper and lower posterior probabilities when sampling from discrete univariate distributions whose suppower consists of the set of noinnegative integers. Specific results are obtained for the Poisson and geometric distributions.

The problem of making inferences about real functions of a probability distribution of unknown form is examined in a Bayesian nonparameteric framework. With respect to a general quadratic loss function, Bayes estimates within the class of linear combinations of a given set of functions on the sample space are obtained for general functions on the distribution space. The result is then used to derive Bayes polynomial estimates of the moments of the distribution.

The problem of statistical comparison of two distributions, continuous as well as discrete, is considered. Very slight and reasonable modifications of traditional parameteric models, e.g. `normal distributions with equal variances', are shown to result in permutation tests, only. Fisher's permutations test is shown to have optimum properties which mean a good merit for its practical use. Further, an accurate method of determining the $p$-value of Fisher's test is proposed.

For several linear regression lines $Y_{ij} = \alpha_i + \beta_i(x_{ij} - x{_i.}) + Z_{ij}, i = 1,\cdots, k; j = 1, \cdot, n_i$, a statistic for testing $\alpha_i = \alpha, \beta_i = \beta$ is constructed based on the simultaneous ranking of all the observations. The asymptotic properties of the criterion are also studied. The results are, however, not directly applicable to the general design model $Y_{ij} = \alpha_i + \beta_i x_{ij} + Z_{ij}$, unless it is assumed that the group means $x_i$. are all equal.

Any triangular array of row independent random vectors with continuous df's has a standard reduction to random vectors with values in the unit cube. The reduced empirical process belonging to the transformed random vectors is always relatively compact. Weak convergence to a (necessarily Gaussian) process holds iff the corresponding covariance kernel converges pointwise.

Let $\mathbf{\Sigma}$ denote the convariance matrix of a vector $\mathbf{x} = (x_1, \cdots, x_T)'$ of $T$ successive observations from a stationary process $\{ x_t\}$ with continuous positive spectral density $f(\lambda)$. Let $\mathbf{\Gamma}$ be the $T \times T$ matrix with elements $\gamma(s, t) = (2\pi)^{-2} \int^\pi_{-\pi} e^{i\lambda (s-t)} f^{-1}(\lambda) d\lambda$. The properties of $\mathbf{\Gamma}$ considered as an approximate inverse of $\mathbf{\Sigma}$ are studied. When $\{ x_t\}$ is a$(n)$ moving average (autoregressive) process of order $q$, rows (columns) $q + 1, \cdots, T - q$ of $\mathbf{\Sigma\Gamma} - \mathbf{I}$ are zero vectors. In this case $\mathbf{\Sigma\Gamma} - \mathbf{I}$ has $2q$ positive characteristic roots which approach paired positive limiting values as $T \rightarrow \infty$ if the roots of $\sum^q_{j=0} \beta_j z^{q-j} = 0$ are less than 1 in absolute value, where $\beta_1, \cdots, \beta_q$ are the coefficients of the process. Statistical properties of $\mathbf{x'Tx} - \mathbf{x'\Sigma}^{-1} \mathbf{x}$ and $\mathbf{x'\Gamma x}/ \mathbf{x'\Sigma}^{-1} \mathbf{x}$ are also discussed.

We consider families of stochastic processes indexed by a finite number of alternative parameter values. For general classes of stochastic processes it is shown that maximum likelihood estimates convergence almost surely to the correct parameter value. This established by use of a submartingale property of the sequence of maximized likelihood ratios together with a technique first employed by Wald [24] in the case of independently identically distributed random variables.

Standard finite state and action continuous time Markov decision processes with discounting are studied using a new optimality criterion called moment optimality. A policy is moment optimal if it lexicographically maximizes the sequence of signed moments of total discounted return with a positive (negative) sign if the moment is odd (even). It is shown constructively that a stationary policy is moment optimal among the class of piecewise constant policies by examining the negative of the Laplace transform of the total return random variable and its Taylor series expansion.

Shapely's stochastic game is considered in a more general setting, with the accumulated payoff being regarded as a function on the space of infinite trajectories, and the set of states of the system taken as a compact metric space. It has been shown that any game with a lower semicontinuous payoff has value and one of the players has an optimal strategy. As a consequence, in Shapley's game both players have optimal strategies.