Sheffer polynomials are used for proving algorithms and closed forms for a broad class of nonlinear one-sided order and rank distributions. For two-sided tests a more general concept results in a representation theorem for piecewise Sheffer polynomial functions.

Techniques are presented for stepwise selection of observations in order of discrepancy from a linear model, and for assigning a $P$ value to each selected observation. A general class of criteria for discrepancy is defined, and leading members of the class are discussed. Two ways to define the sequence of $P$ values for a general selection criterion are discussed, and their relative advantages are compared. The techniques are illustrated on several data sets using only the computationally simpler of the two approaches to $P$ values.

We demonstrate the admissible Bayes character of two residual error criteria for subset selection of independent variables in normal multivariate regression models. In particular, suppose a linear model includes the independent variable list $X$, and suppose all additional independent variable subsets of size $s$ (fixed) from list $Z$ are under consideration for inclusion in the model. Let $\mathbf{\Sigma}$ be the regression error covariance matrix and $\hat{\mathbf{\Sigma}}(\hat{\mathbf{\Sigma}}_\mathscr{J})$ the usual unbiased estimator of $\Sigma$ which assumes a model fitting the variables in list $X (X \cup \mathscr{J}$, where $\mathscr{J} \subset Z$ of size $s$). Then two best subsets of $Z$ of size $s$ may be characterized as minimizing $\operatorname{tr} \hat{\mathbf{\Sigma}}^{-1} \hat{\mathbf{\Sigma}}_\mathscr{J}$ and $|\hat{\mathbf{\Sigma}}_\mathscr{J}|/|\hat{\mathbf{\Sigma}}|$ over all subsets $\mathscr{J}$ of size $s$ in $Z$. We show that the significance tests for including-excluding these two best subsets are admissible proper Bayes rules for fixed effect variables in list $Z$. If $Z$ is allowed to encompass random and mixed effects, then the latter test is admissible proper Bayes in a class of location and scale invariant tests. Special cases of the general selection problem include multiple outlier detection and slippage tests where the best subset criteria above lead to Studentized residual outlier and slippage detection criteria. These tests are derived using models which explain outliers and slippage as locational biases and/or inflated variances.

The purpose of this paper is to develop diagnostic tests for open-loop transfer function models with autoregressive-moving average stochastic disturbances using the efficient scores procedure. The tests proposed are asymptotically equivalent to those based upon the likelihood ratio principle and have the advantage that they do not involve a heavy computational burden. Consideration is given both to the standard case of nonsingular information matrices and to the situation obtaining when there are identifiability problems and application of conventional large sample test procedures is not feasible. Relationships between score tests and portmanteau tests in time series analysis are also investigated. Some simulation evidence on the finite sample behaviour of the tests is presented and it is seen that the tests of this paper perform well.

Each of $n$ arms generates an infinite sequence of Bernoulli random variables. The parameters of the sequences are themselves random variables, and are independent with a common distribution satisfying a mild regularity condition. At each stage we must choose an arm to observe (or pull) based on past observations, and our aim is to maximize the expected discounted sum of the observations. In this paper it is shown that as the discount factor approaches one the optimal policy tends to the rule of least failures, defined as follows: pull the arm which has incurred the least number of failures, or if this does not define an arm uniquely select from amongst the set of arms which have incurred the least number of failures an arm with the largest number of successes.

When classifying an observation into one of $k$ multivariate normal distributions based on samples of correctly classified observations, two estimates of the probability of correct classification, called the apparent and plug-in correct classification rates, are considered. Asymptotic expansions are found for the means and variances of these estimates. It is shown that these expansions can be used to help reduce the bias of the estimates. In the course of finding the expansions, an asymptotic expansion for the conditional joint density of two observations given the sample mean and pooled covariance matrix is found.

The asymptotic normality of both linear and nonlinear statistics and the consistency of the variance estimators obtained using the linearization, jackknife and balanced repeated replication (BRR) methods in stratified samples are established. The results are obtained as $L \rightarrow \infty$ within the context of a sequence of finite populations $\{\Pi_L\}$ with $L$ strata in $\Pi_L$ and are valid for any stratified multistage design in which the primary sampling units (psu's) are selected with replacement and in which independent subsamples are taken within those psu's selected more than once. In addition, some exact analytical results on the bias and stability of these alternative variance estimators in the case of ratio estimation are obtained for small $L$ under a general linear regression model.

A new non-parametric stochastic approximation procedure for estimating the roots of a non-decreasing regression function is described. The observations are taken on a discrete lattice and the estimation is based on the roots of the sample isotonic regression function fitted to the observed values. Asymptotic properties of the estimator are proven. When specialized to the bio-assay case it gives asymptotic results similar to those obtained by Derman for his up-and-down method but under weaker assumptions than Derman required.

In practice, comparisons with the "best" are often the ones of primary interest. In this paper, parametric and nonparametric simultaneous upper confidence intervals for all distances from the "best" are derived under the location model. Their improvement upon the results of Bechhofer (1954), Gupta (1956, 1965), Fabian (1962), and Desu (1970) in the parametric case is discussed. In the nonparametric case, no comparable confidence statements were available previously.

Total positivity properties of multivariate densities are useful in deducing positive dependence of random vector components and related probability inequalities. In this paper we determine necessary and sufficient conditions for total positivity of absolute value multinormal variables. The results are applied to obtain positive dependence and associated inequalities for the multinormal and related distributions, e.g., the multivariate $t$ and Wishart distributions. Inequalities of this type yield bounds for multivariate confidence set probabilities.

An attempt is made to model the appearance times of metastases as a nonstationary Poisson process. Three algorithms are developed for this task. The first follows the kernel approach used in probability density estimation by Parzen and Rosenblatt; the second extends the work of Grenander on mortality measurements to a more general censoring scheme appropriate for the present application; the third employs a discrete maximum penalized likelihood approach. We obtain estimates using both stratification and the proportional hazards model. Contrary to customary belief, it seems that the intensity functions associated with the tumor systems under investigation are nonincreasing.

Ferguson and Phadia have recently discussed the nonparametric Bayesian estimation of a distribution function from a right-censored random sample using process priors that are neutral to the right. The more general problem of estimating the baseline distribution function with right censored data from the proportional hazards model has been studied by Kalbfleisch who uses the more restrictive class of gamma process priors. This note shows that, with a simple modification, the analysis of Ferguson and Phadia can be extended to deal with the proportional hazards situation with constant covariates. Extensions to time dependent covariates and other regression models are also considered.

We consider estimators for a multivariate probability density at a point. Efficient choices require knowledge of the density and its second derivatives although these are not known. We use consistent, but not necessarily efficient, estimators for these and use them to replace the unknown values in the choices for an efficient estimator. Our second stage estimators and the unattainable efficient choices are asymptotically equivalent. This follows because we show that an entire class of estimators converges weakly to a limiting stochastic process. We find asymptotically efficient estimators of kernel type.

Hajek (1974) has shown that in the two-sample problem the best exact slope for a test of randomness against any particular member of a large class of alternative hypotheses is attained by a linear rank test. Here a new class of two-sample rank tests is constructed, and it is shown that for each linear test there exists a test within the new class which is always at least as efficient in terms of exact Bahadur efficiency irrespective of which alternative hypothesis, is tested. Conditions are provided under which the new test is strictly more efficient than the linear rank test. Some comments are made about the practical applicability of the new class of tests.

The problem is considered of testing symmetry of a bivariate distribution $\mathscr{L}(X, Y)$ against "asymmetry towards high $X$-values," subject to the restriction of invariance under the transformations $(x_i, y_i) \mapsto (g(x_i), g(y_i)) (1 \leq i \leq n)$ for increasing bijections $g$. This invariance restriction prohibits the common reduction to the differences $x_i - y_i$. The intuitive concept of "asymmetry towards high $X$-values" is approached in several ways, and a mathematical formulation for this concept is proposed. Most powerful and locally most powerful invariant similar tests against certain subalternatives are characterized by means of a Hoeffding formula. Asymptotic normality and consistency results are obtained for appropriate linear rank tests.

A robust version of Neyman's optimal $C(\alpha)$ test is proposed for contamination neighborhoods. The proposed robust test is shown to be asymptotically locally maximin among all asymptotic level $\alpha$ tests. Asymptotic efficiency of the test procedure at the ideal model is investigated. An outlier resistant version of Student's $t$-test is proposed.

It is proved in the paper that every possible set of conditions which implies consistency of the maximum likelihood method also implies consistency of Bayes estimates for a large class of prior distributions. The method of the proof is based on a necessary and sufficient condition for the consistency of maximum likelihood estimates which is due to Perlman (1972). Perlman's result is improved as far as certain questions of measurability are concerned.

The limit distribution of the least squares estimator $\hat{\alpha}$ of the parameter $\alpha$ of the first order stochastic difference equation, in the boundary case $\alpha = 1$, is calculated.

Consider the problem of estimating the mean of a finite population on the basis of a simple random sample. It was proved by Aggarwal (1954) that the sample mean minimizes the maximum expected squared error divided by the population variance $\tau^2$. Aggarwal also stated, but did not successfully prove, that the sample mean minimizes the maximum expected squared error over the populations satisfying $\tau^2 \leq M$ for any fixed positive $M$. It is the purpose of this paper to give a proof of this second result, and to indicate some generalizations.

A re-expression of the usual representation of the multinomial distribution as the conditional distribution of independent Poisson random variables given fixed sum provides a convenient new way to compute multinomial cumulative distribution functions.

An example is presented of an inadmissible best invariant point estimator of a location parameter. The extra moment condition of Brown and Fox (1974b) is violated. The example involves two independent, identically distributed observations, an improvement over Perng's (1970) example.

Identifiability of finite mixtures of von Mises distributions is proved.