Exponential response models are a generalization of logit models for quantal responses and of regression models for normal data. In an exponential response model, $\{F(\theta): \theta \in \Theta\}$ is an exponential family of distributions with natural parameter $\theta$ and natural parameter space $\Theta \subset V$, where $V$ is a finite-dimensional vector space. A finite number of independent observations $S_i, i \in I$, are given, where for $i \in I, S_i$ has distribution $F(\theta_i)$. It is assumed that $\mathbf{\theta} = \{\theta_i: \mathbf{i} \in \mathbf{I}\}$ is contained in a linear subspace. Properties of maximum likelihood estimates $\hat\mathbf{\theta}$ of $\mathbf{\theta}$ are explored. Maximum likelihood equations and necessary and sufficient conditions for existence of $\hat\mathbf{\theta}$ are provided. Asymptotic properties of $\hat\mathbf{\theta}$ are considered for cases in which the number of elements in $I$ becomes large. Results are illustrated by use of the Rasch model for educational testing.

The autoregressive moving average model is a stationary stochastic process $\{y_t\}$ satisfying $\sum^p_{k=0} \beta_ky_{t-k} = \sum^q_{g=0} \alpha_g\nu_{t-g}$, where the (unobservable) process $\{v_t\}$ consists of independently identically distributed random variables. The coefficients in this equation and the variance of $v_t$ are to be estimated from an observed sequence $y_1, \cdots, y_T$. To apply the method of maximum likelihood under normality the model is modified (i) by setting $y_0 = y_{-1} = \cdots = y_{1-p} = 0$ and $\nu_0 = v_{-1} = \cdots = v_{1-q} = 0$ and alternatively (ii) by setting $y_0 \equiv y_T, \cdots, y_{1-p} \equiv y_{T+1-p}$ and $v_0 \equiv v_T, \cdots, v_{1-q} \equiv v_{T+1-q}$; the former lead to procedures in the time domain and the latter to procedures in the frequency domain. Matrix methods are used for a unified development of the Newton-Raphson and scoring iterative procedures; most of the procedures have been obtained previously by different methods. Estimation of the covariances of the moving average part is also treated.

This paper studies the properties of open-ended power-one tests of $H_0: \theta \leqq \theta_0$ versus $H_1: \theta > \theta_0$ or of $H: \theta = \theta_0$ versus $K: \theta \neq \theta_0$ based on sample sums stopped at moving boundaries. The behavior of the expected sample size is analyzed and certain asymptotic results as $\theta \rightarrow \theta_0$ are obtained in the case of a location parameter and also in the case of an exponential family of distributions.

A conditional confidence property is examined in the context of invariant statistical models, for set estimators of equivariant functions of the parameter. Set estimators deduced from likelihood considerations are then identical to Bayes credible sets induced from a right invariant prior. It is shown that amenability of the group ensures that these intervals satisfy a betting interpretation of confidence sets due essentially to Buehler and Tukey. As a corollary, a level $\alpha$ Bayes set estimator is of level at-most-$\alpha$ as a Neyman-Pearson confidence interval if the group is amenable.

A confidence level is sometimes treated as a probability and accordingly given substantial "confidence." And with certain statistical models a confidence level can also be an objective posterior probability (Fraser and MacKay, 1975). An instance involving such a probability with a binary error was discussed at the Symposium on Foundations of Statistical Inference, University of Waterloo, 1970 (Godambe and Sprott, 1971, page 49). An example by Buehler (ibid., page 337) of a betting strategy and a generalization by Rubin (ibid., page 340) were subsequently presented to support a claim against the objective posterior probability and against too much "confidence." The ordinary Student confidence interval in appropriate contexts is also an objective posterior probability interval. An example by Buehler and Feddersen (1963) has been cited frequently as evidence against the validity of the objective posterior probability and against too much "confidence." This note censures the common procedure of assessment in terms of betting strategies and introduces a modified balanced procedure for such betting assessments. When assessed by this balanced procedure the Buehler and Buehler-Feddersen strategies are faced with large losses and thus do not support claims against the objective posteriors and confidence levels.

Let $X$ be a real valued random variable with a family of possible distributions belonging to a one parameter exponential family with the natural parameter $\theta \in(\underline{\theta}, + \infty)$. Let $g$ be a prior probability density for $\theta$ with unbounded support. Under some additional assumptions it is shown that for large values of $x$ the posterior distribution of $\theta$ given $X = x$ is approximately normally distributed about its mode. If $\delta_g$ denotes the Bayes estimator for squared error loss of some function $\gamma(\theta)$ against $g$ then the rate at which $\delta_g(x)$ approaches infinity as $x$ approaches infinity is found. The rate is shown to depend on the behavior of the prior density $g(\theta)$ for large values of $\theta$.

The least favorable pair (LFP) that Huber (1965), (1968) wrote down when he considered minimax test problems between neighborhoods of single probability measures $P_0, P_1$ defined in terms of $\varepsilon$-contamination and total variation is a canonical but only one possible choice of an LFP. We treat these neighborhoods by means of special capacities. The minimax test statistic is obtained by explicitly solving a minimization program, all LFP's are characterized by their $(P_0 + P_1)$-densities, another LFP is given explicitly. The technique is similar to that used by Huber and Strassen (1973), but is simpler and more constructive in this special situation.

Let $(\mathbf{M}, \mathbf{T}, \mathbf{S})$ be random matrices such that $\mathbf{M}$ and $\mathbf{S}$ are Hermitian positive definite almost everywhere. Let $\mathbf{M}_{(t)} = \lbrack m_{ij}; 1 \leqq i, j \leqq t\rbrack, \mathbf{S}_{(t)} = \lbrack s_{ij}; 1 \leqq i, j \leqq t\rbrack$ and $\mathbf{T}_{(r,s)} = \lbrack t_{ij}; 1 \leqq i \leqq r, 1 \leqq j \leqq s\rbrack$, and define $Q(r, s) = P\lbrack G((\mathbf{M}_{(r)})^{-\frac{1}{2}}\mathbf{T}_{(r,s)}(\mathbf{S}_{(s)}) ^{-\frac{1}{2}}) \leqq c\rbrack$ for some $G$ belonging to the class $\mathscr{G}$ of monotone unitarily invariant functions. The main result is that, for any $c$ and $G \in \mathscr{G}, Q(r, s)$ is a decreasing function of $r$ and $s$. Applications yield simultaneous confidence bounds for a variety of multivariate and multiparameter problems.

Consider BIB designs with parameters $v, b, r, k$ and $\lambda$. Define the support of a BIB design to be the set of its distinct blocks and let the cardinality of the support be $b^\ast$. If $b^\ast < b$ then the design is said to be a BIB design with repeated blocks. Some potential applications of such designs to experimental design and controlled sampling are given. Some necessary and sufficient conditions for the existence of these designs and some algorithms for their constructions are provided. Bounds on $b^\ast$ have been obtained. A necessary and sufficient condition under which a set of blocks can be the support of a BIB design are found. A table of BIB designs with $22 \leqq b^\ast \leqq 56$ for $v = 8$ and $k = 3$ is included.

Renewal theory is developed for processes of the form $Z_n = S_n + \xi_n$, where $S_n$ is the $n$th partial sum of a sequence $X_1, X_2, \cdots$ of independent identically distributed random variables with finite positive mean $\mu$ and $\xi_n$ is independent of $X_{n+1}, X_{n+2}, \cdots$ and has sample paths which are slowly changing in an appropriate sense. Applications to sequential analysis are given.

For fixed $\theta$, let $X_1, X_2, \cdots$ be a sequence of independent identically distributed random variables having density $f_\theta(x)$. Using a sequential Bayes decision theoretic approach we consider the problem of estimating any strictly monotone function $g(\theta)$ when the error incurred by a wrong estimate is measured by squared error loss and the sampling cost is $c$ units per observation. A heuristic stopping rule is suggested. It is shown that the excess risk which results when using it is bounded above by terms of order $c$.

The problem of selecting a subset of $k$ populations $\pi_1, \cdots, \pi_k,$ which contains the "best" population, is considered. The unknown values $\theta_1, \cdots, \theta_k$ are the characteristics associated with $\pi_1, \cdots, \pi_k$ and the unknown population associated with $\theta_{\lbrack k\rbrack} = \max_i\theta_i$ is called the "best." It is assumed that, given $\mathbf{\theta} = (\theta_1, \cdots, \theta_k)$ the $\operatorname{pdf}$ of the independent random variables $X_1, \cdots, X_k$ belong to a monotone likelihood ratio family, the prior distribution of $\mathbf{\theta}$ is exchangeable, and the loss function is a linear combination of two components, namely the subset size $|s|$ and the distance between the "best" and the "best" in the selected subset $s,$ i.e., $\mathbf{L}(\mathbf{\theta}, s) = c|s| + \lbrack\theta_{\lbrack k\rbrack} - \max_{j : \pi_j \in s} \theta_j\rbrack.$ It is shown that the Bayes rule depends on at most $(k - 1)$ computable expressions. Some lower and upper bounds on the differences of Bayes risks are given to help reduce the amount of computation for the Bayes rule. If $X_i$ has a normal distribution with mean $\theta_i$ and known variance $\sigma^2,$ then it is shown that (i) for $k = 2,$ the Bayes rule with vague prior knowledge and the classical rule are the same if the probability of correct selection, $P^\ast,$ is chosen as a suitable function of $c,$ and (ii) if $c/\sigma \geqq 1/\pi^\frac{1}{2},$ then the Bayes rule selects only one population and if $.2821 \leqq c/\sigma < 1/\pi^\frac{1}{2},$ then it selects at most two populations. The tables for implementing the Bayes rule for normal populations are also given.

Several stopping times which arise from problems of sequential estimation may be written in the form $t_c = \inf\{n \geqq m: S_n < cn^\alpha L(n)\}$ where $S_n, n \geqq 1,$ are the partial sums of i.i.d. positive random variables, $\alpha > 1, L(n)$ is a convergent sequence, and $c$ is a positive parameter which is often allowed to approach zero. In this paper we find the asymptotic distribution of the excess $R_c = ct_c^\alpha - S_{t_c}$ as $c \rightarrow 0$ and use it to obtain sharp estimates for $E\{t_c\}.$ We then apply our results to obtain second order approximations to the expected sample size and risk of some sequential procedures for estimation.

Let $(X_i, Y_i) (i = 1, 2, \cdots, n)$ be $n$ independent $\mathrm{rv}$'s from some bivariate distribution. If $X_{r:n}$ denotes the $r$th ordered $X$-variate, then the $Y$-variate $Y_{\lbrack r:n\rbrack}$ paired with $X_{r:n}$ is termed the concomitant of the $r$th order statistics. The exact and asymptotic distribution theory of $Y_{\lbrack r:n\rbrack}$ and of its rank are studied. The results obtained are applied to a prediction problem in a Round Robin tournament.

$X_1, \cdots, X_n$ are i.i.d. Uniform (0, 1) rv's with empirical df $\Gamma_n$ and order statistics $0 < U_1 < \cdots < U_n < 1.$ Define random variables $U_\ast, i_\ast$ (for $n \geqq 2$) by $\max_{1\leqq i \leqq n - 1} \frac{U_{i + 1}}{i} = \frac{U_{i_\ast} + 1}{i_\ast}, U_\ast = U_{i_\ast + 1};$ $i_\ast + 1$ is the (random) index of the order statistic at which the maximum is achieved and $U_\ast$ is the value of that order statistic. The distributions of $(U_\ast, i_\ast)$ and of $U_\ast$ and $i_\ast$ are found for all $n \geqq 2,$ extending and complementing earlier results due to Birnbaum and Pyke, Chang, and Dempster. The limiting distributions are found and related to the corresponding sums of exponential rv's by a Poisson type invariance result for the empirical df $\Gamma_n$ and its inverse $\Gamma_n^{-1}$.

Let $X_1, X_2, \cdots, X_n$ be independent identically distributed random variables defined on the unit interval. The generalized $j$th orthogonal component is defined as $V_{nj} = n^{-\frac{1}{2}} \sum^n_{i = 1} d_j(X_i),$ where $\{1, d_1, d_2, \cdots\}$ is an orthonormal basis for $\mathscr{L}_2(\lbrack 0, 1\rbrack).$ These statistics are a generalization of the orthogonal components of the Cramer-von Mises statistic [2]. Linear combinations of the $V_{nj}$ are applied to the problem of testing the null hypothesis of a uniform distribution against the alternative density $p_n(x) = 1 + h(x)/n^\frac{1}{2} + k_n(x)/n$ where $h(x)$ is square integrable and $k_n(x)$ is dominated by a square integrable function. When $a_j = \int h(x) d_j(x) dx,$ tests based on $\sum^m_{j = 1} a_j V_{nj}$ are shown to be asymptotically most powerful as $\min (m, n) \rightarrow \infty.$ The asymptotic power and efficiency of these tests are computed. A procedure is developed for choosing among possible density functions when a goodness-of-fit test rejects the null hypothesis.

Let $\mathbf{X} = (X_1, X_2)'$ be a bivariate random vector distributed according to an absolutely continuous distribution function $F(\mathbf{x})$ which has first partial derivatives. Let $\bar{F}(\mathbf{x}) = P(X_1 > x_1, X_2 > x_2).$ The vector-valued bivariate failure rate is defined as $\mathbf{r}(\mathbf{x}) = (r_1(\mathbf{x}), r_2(\mathbf{x}))',$ where $r_i(\mathbf{x}) = -\partial \ln \bar{F}(\mathbf{x})/\partial x_i (i = 1, 2)$. In this paper, we propose a smooth nonparametric estimate $\hat\mathbf{r}(\mathbf{x})$ of $\mathbf{r}(\mathbf{x})$ using Cacoullos' (Ann. Inst. Statist. Math. 18 (1966), 181-190) multivariate density estimate. Regularity conditions are obtained under which $\hat\mathbf{r}(\mathbf{x})$ is shown to be pointwise strongly consistent. A set of sufficient conditions is given for the strong uniform consistency of $\hat\mathbf{r}(\mathbf{x})$ over a subset $S$ of $R^2$ where $\bar{F}(\mathbf{x})$ is bounded below by $\varepsilon > 0.$ The joint asymptotic normality of the estimate evaluated at $q$ distinct continuity points of the failure rate is established. The methods and results presented in this paper can be generalized to any finite dimensional case in a straightforward manner.

An extension of a central limit theorem for a mean of a random number of observations is given. A natural application occurs in the area of fixed-width confidence intervals. We provide an example which shows that the standard procedure does not preserve the intended coverage probability uniformly over nontrivial sets of distribution functions. The major weak convergence result is used to provide conditions for and a simple proof of such uniformity. The results are also shown to hold for $M$-estimates of location.

For $x_i \sim N(\mu, \sigma_i^2) (i = 1, 2,\cdots, n)$ and the $x_i$'s independent, this paper gives necessary and sufficient conditions under which the weighted average of the $x_i$'s, with weights proportional to inverses of the sample variances, has uniformly smaller variance than any of the $x_i$'s.

Two previous results of the author, regarding the admissibility of the sample mean as estimator of the population mean in the class of all estimators for arbitrary sampling designs with the squared error as loss function, and for a wide class of loss functions, are shown to remain valid in the more general case when the population means of many variates are simultaneously under estimation.