Asymptotic expansions are established for the power of distribution-free tests in the two-sample problem. These expansions are then used to obtain deficiencies in the sense of Hodges and Lehmann for distribution-free tests with respect to their parametric competitors and for the estimators of shift associated with these tests.

An asymptotic expansion is obtained for the distribution function of the standardized mean of a sample of $s$ observations taken randomly without replacement from a finite population of $n$ numbers. The expansion is given to order $1/n$ and agrees with the formal Edgeworth expansion. The proof of the result is obtained using an approximation to the characteristic function of the standardized sum.

This paper considers the general linear regression model $Y_i = \sum_j \beta_j x_{ij} + \varepsilon_i$, and studies the problem of testing hypotheses about some of the $\beta$'s while regarding others as nuisance parameters. The test criteria discussed, which are based on ranks of residuals, are shown to be asymptotically distribution-free.

In this paper Pitman's method of constructing and comparing tests based on statistics which are asymptotically normal under the null hypothesis and the local alternatives is extended to sequential tests of statistical hypotheses. The asymptotic normality assumption in Pitman's theory is replaced in its sequential analogue by the weak convergence of normalized processes formed from these statistics under the null hypothesis and the local alternatives. Uniform invariance principles are developed for a large class of statistics, and as an immediate corollary of these results, the desired weak convergence assumption is shown to hold. Furthermore uniform large deviation theorems are obtained for the test statistics and these results guarantee that the sequential tests under consideration have finite expected sample sizes under the null hypothesis and the local alternatives. As an illustration of the general method, the two-sample location problem is studied in detail, and the asymptotic relative efficiencies of the sequential Wilcoxon test, the sequential van der Waerden test and the sequential normal scores test relative to the two-sample sequential $t$-test are easily obtained since one of our key results (Theorem 1) implies that the asymptotic relative efficiencies of these sequential tests coincide with the corresponding Pitman efficiencies of their nonsequential analogues.

The rate of convergence of the point estimators for the parameter based on the sample mean of i.i.d. random vectors is obtained. The result is used to prove the Bahadur's efficiency of the regular best asymptotically normal estimator when the underlying distribution is in the exponential family. An example is given to show that if the distribution is not in the exponential family, then the regular best asymptotically normal estimator is not necessarily efficient in Bahadur's sense.

Consider a time-continuous nonhomogeneous Markovian stochastic process $V$ having state space $A^0$. Let $A \subset A^0$ and let $P_{Aij}(\tau, t)$ be the $i \rightarrow j$ transition probability of the Markovian stochastic process $V_A$ arising in the hypothetical situation where states $A^0 - A$ have been eliminated from the state space of $V$. Based upon the concept of Kaplan and Meier's product-limit estimator, a nonparametric estimator $\hat{P}_{Aij}(\tau, t)$ is formulated which is proved to be uniformly strongly consistent and asymptotically unbiased. These results generalize those by Aalen for the special case in which $A^0$ has one transient state.

Consider a time-continuous nonhomogeneous Markovian process $V$ having state space $A^0$. For $A \subset A^0$ and $i, j \in A, P_{Aij}(\tau, t)$ is the $i \rightarrow j$ transition probability of the Markovian process $V_A$ which arises in the hypothetical situation where states $A^0 - A$ have been eliminated from the state space of $V$. Let $\hat{P}_{Aij}(\tau, t)$ be the generalized product-limit estimator of $P_{Aij}(\tau,t)$. It is shown that the vector consisting of components in $\{N^\frac{1}{2}(\hat{P}_{Aij}(\tau, t) - P_{Aij}(\tau, t)): i, j \in A; i \neq j\}$ converges weakly to a vector of dependent Gaussian processes. The structure of this limiting vector process is studied. Finally these results are applied to the estimation of certain biometric functions.

In order to obtain quantitative results about the influence of outliers on tests, their maximum size and minimum power over certain neighborhoods are evaluated asymptotically. The neighborhoods are defined in terms of $\varepsilon$-contamination and total variation, the tests considered are based on statistics $n^{-\frac{1}{2}} \sum^n_{i=1} IC(x_i), IC$ a rather arbitrary function. Furthermore, the unique $IC^\ast$ is determined that leads to a maximin test with respect to this subclass of tests. A comparison with the likelihood ratio of least favorable pairs shows that the test based on $n^{-\frac{1}{2}}\sum^n_{i=1} IC^\ast(x_i)$ is in fact maximin among all tests at a given level. Tests based on $(M)$-statistics are also considered.

A decision problem is characterized by a loss function $V$ and opinion $H$. The pair $(V, H)$ is said to be strongly stable iff for every sequence $F_n \rightarrow_\omega H, G_n \rightarrow_\omega H$ and $L_n\rightarrow V, W_n\rightarrow V$ uniformly, $\lim_{\varepsilon \downarrow 0} \lim \sup_{n\rightarrow \infty} \lbrack \int L_n(\theta, D_n(\varepsilon)) dF_n(\theta) - \inf_D \int L_n(\theta, D) dF_n(\theta)\rbrack = 0$ for every sequence $D_n(\varepsilon)$ satisfying $\int W_n(\theta, D_n(\varepsilon)) dG_n(\theta) \leqq \inf_D \int W_n(\theta, D) dG_n(\theta) + \varepsilon.$ We show that squared error loss is unstable with any opinion if the parameter space is the real line and that any bounded loss function $V(\theta, D)$ that is continuous in $\theta$ uniformly in $D$ is stable with any opinion $H$. Finally we examine the estimation or prediction case $V(\theta, D) = h(\theta - D)$, where $h$ is continuous, nondecreasing in $(0, \infty)$ and nonincreasing in $(-\infty, 0)$ and has bounded growth. While these conditions are not enough to assure strong stability, various conditions are given that are sufficient. We believe that stability offers the beginning of a Bayesian theory of robustness.

In generalized ridge estimation, the components of the ordinary least squares (OLS) regression coefficient vector which lie along the principal axes of the given regressor data are rescaled using known ridge factors. Generalizing a result of Swindel and Chapman, it is shown that, if each ridge factor is nonstochastic, nonnegative, and less than one, then there is at most one unknown direction in regression coefficient space along which ridge coefficients have larger mean squared error than do OLS coefficients. Then, by decomposing the mean squared error of a ridge estimator into components parallel to and orthogonal to the unknown true regression coefficient vector, new insight is gained about definitions for optimal factors. Estimators of certain unknown quantities are displayed which are maximum likelihood or unbiased under normal theory or which have correct range.

An approximate linear model is proposed to allow for deviations from an underlying ideal linear model as follows: If, in standard notation, $Y = A\beta + \varepsilon$ is the ideal model then $Y = A\beta + r + \varepsilon$ where $|r_i| \leqq M_i$ for $M$ a given vector is an approximate linear model. The problem solved here is that of finding a linear estimate of a single linear function of $\beta$ which minimaxes mean square error in the approximate model. The estimate obtained may be the standard one from the ideal model, but in general it is not. The estimate is calculated as a solution to a set of nonlinear equations (generalizing the usual normal equations) and an algorithm is given for obtaining the solution.

A spline density function estimate is considered. Limit theorems are obtained for a quadratic norm of the normalized deviation of the estimate from its expected value. Results obtained can be used in a test of goodness of fit. Some of the commonly used boundary conditions for the spline function are considered. It appears that in some situations certain boundary conditions are undesirable.

A method of estimating the parameters of a linear regression model when the covariance matrix is an unknown diagonal matrix is investigated. It is assumed that the observations fall into $k$ groups with constant error variance for a group. The estimation is carried out in two steps, the first step being an ordinary least squares regression. The least squares residuals are used to estimate the covariance matrix and the second step is the calculation of the generalized least squares estimator using the estimated covariance matrix. The large sample properties of the estimator are derived for increasing $k$, assuming the numbers in the groups form a fixed sequence.

We study the class of scale mixtures of normal distributions with mean zero. Given that the $\operatorname{cdf} F(x)$ of such a mixture is fixed at two points, say $F(x_1) = \alpha_1, F(x_2) = \alpha_2$, we answer the question of how widely $F(x_3)$ can vary at some third point $x_3$. A brief final section mentions extensions of our theorem.

The following result of Lukacs is known: let $X_1, X_2$ be independent, positive random variables, having the nondegenerate distributions $P_1$ and $P_2$. Suppose that $X_1/X_2$ and $X_1 + X_2$ are independent. Then $P_1$ and $P_2$ are gamma distributions with the same scale parameter. Lukacs' original deduction requires details from complex analysis. Here a simpler proof is given. Instead of $P_1$ and $P_2$ two other probability measures $\mu_1$ and $\mu_2$ are shown to be determined by the independence properties of $X_1$ and $X_2$. It is possible to express $P_i$ and $\mu_i$ by each other, and $\mu_i$ is chosen such that all moments of $\mu_i$ are finite $(i = 1,2)$. Thus the proof reduces to a straight-forward calculation of moments.

Suppose a finite population is stratified such that the variate values, appropriately scaled, are exchangeable within strata and independent from stratum to stratum. Consider designs with fixed sample sizes within strata, and unbiased estimators of the population total. Minimum expected variance is achieved with a stratified $\pi ps$ design and the usual (Horvitz-Thompson) unbiased estimator.