The dip test measures multimodality in a sample by the maximum difference, over all sample points, between the empirical distribution function, and the unimodal distribution function that minimizes that maximum difference. The uniform distribution is the asymptotically least favorable unimodal distribution, and the distribution of the test statistic is determined asymptotically and empirically when sampling from the uniform.

Wetherill's estimator $\bar{w}$ is asymptotically equivalent to the mean of peaks and valleys in the sequence of responses in an up-and-down experiment. The asymptotic distribution of $\bar{w}$ is derived and the asymptotic variance expression is simplified. Some values of asymptotic means and variances are calculated for logistic response. They are compared with analogous values for the estimator suggested by Dixon and Mood. These estimators are also compared by some computer simulation. For the conditions investigated, Dixon and Mood's estimator is to be preferred to $\bar{w}$.

Bootstrap tests and confidence regions for functions of the population covariance matrix have the desired asymptotic levels, provided model restrictions, such as multiple eigenvalues in the covariance matrix, are taken into account in designing the bootstrap algorithm.

A general procedure for multistage modification of pivotal statistics is developed to improve the normal approximation. Bootstrapping a first stage modified statistic is shown to be equivalent, in terms of asymptotic order, to the normal approximation of a second stage modification. Explicit formulae are given for some basic cases involving independent random samples and samples drawn without replacement. The Hodges-Lehmann deficiency is calculated to compare the regular $t$-statistic with its one-step correction.

A conjecture of Prentice is established which states that for censored linear rank test, exact scores based on conditional expectations can be replaced by approximate scores obtained by evaluating the score function at an estimate of the survival function. We show that under minimal conditions, asymptotically equivalent tests are obtained when either the Kaplan-Meier, Altshuler, or moment estimator of the survival function is used. Asymptotic normality is also established for a general random censorship model under the null hypothesis, and for contiguous alternatives. This is used to calculate efficacies, and when the censoring times are i.i.d., an expression for the asymptotic relative efficiency is given which is a natural generalization of the one for classical uncensored linear rank tests.

The paper considers the Kaplan-Meier estimator $F^{\mathrm{KM}}_n$ from a combinatoric viewpoint. Under the assumption that the estimated distribution $F$ and the censoring distribution $G$ are continuous, the combinatoric results are used to show that $\int |\theta(z)| dF^{\mathrm{KM}}_n(z)$ has expectation not larger than $\int |\theta(z)| dF(z)$ for any sample size $n$. This result is then coupled with Chebychev's inequality to demonstrate the weak convergence of the former integral to the latter if the latter is finite, if $F$ and $G$ are strictly less than 1 on $\mathscr{R}$ and if $\theta$ is continuous.

A key identity for the product-limit estimator due to Aalen and Johansen (1978) and Gill (1980) is shown to be a consequence of the exponential formula of Doleans-Dade (1970). The basic counting processes in the censored data problem are shown to converge jointly to Poisson processes under "heavy-censoring": $G_n \rightarrow_d \delta_0$, but $n(1 - G_n) \rightarrow \alpha$ where $G_n$ is the censoring distribution. The Poisson limit theorem for counting processes implies Poisson type limit theorems under heavy censoring for the cumulative hazard function estimator and product limit estimator. The latter, in combination with the key identity of Aalen-Johansen and Gill and martingale properties of the limit processes, yields a new approximate variance formula for the product limit estimator which is compared numerically with recent finite sample calculations for the case of proportional hazard censoring due to Chen, Hollander, and Langberg (1982).

Let $\mathscr{P}$ be a finite population with $N \geq 1$ elements; for each $e \in \mathscr{P}$, let $X_e$ and $Y_e$ be independent, positive random variables with unknown distribution functions $F$ and $G$; and suppose that the pairs $(X_e, Y_e)$ are i.i.d. We consider the problem of estimating $F, G$, and $N$ when the data consist of those pairs $(X_e, Y_e)$ for which $e \in \mathscr{P}$ and $Y_e \leq X_e$. The nonparametric maximum likelihood estimators (MLEs) of $F$ and $G$ are described; and their asymptotic properties as $N \rightarrow \infty$ are derived. It is shown that the MLEs are consistent against pairs $(F, G)$ for which $F$ and $G$ are continuous, $G^{-1}(0) \leq F^{-1}(0)$, and $G^{-1}(1) \leq F^{-1}(1). \sqrt N \times$ estimation error for $F$ converges in distribution to a Gaussian process if $\int^\infty_0 (1/G) dF < \infty$, but may fail to converge if this integral is infinite.

The following problem is treated: Given $s$ not-necessarily-random samples from an unknown distribution $F$, and assuming that we know the sampling rule of each sample, is it possible to combine the samples in order to estimate $F$, and if so what is the natural way of doing it? More formally, this translates to the problem of determining whether there exists a nonparametric maximum likelihood estimate (NPMLE) of $F$ on the basis of $s$ samples from weighted versions of $F$, with known weight functions, and if it exists, how to construct it? We give a simple necessary and sufficient condition, which can be checked graphically, for the existence and uniqueness of the NPMLE and, under this condition, we describe a simple method for constructing it. The method is numerically efficient and mathematically interesting because it reduces the problem to one of solving $s - 1$ nonlinear equations with $s - 1$ unknowns, the unique solution of which is easily obtained by the iterative, Gauss-Seidel type, scheme described in the paper. Extensions for the case where the weight functions are not completely specified and for censored samples, applications, numerical examples, and statistical properties of the NPMLE, are discussed. In particular, we prove under this condition that the NPMLE is a sufficient statistic for $F$. The technique has many potential applications, because it is not limited to the case where the sampled items are univariate. A FORTRAN program for the described algorithm is available from the author.

We show that it is helpful to generalize Vardi's problem, estimating an unknown distribution from selection-biased samples, by allowing the sample sizes to be random. The problem then assumes an elegant symmetry, leading to a transparent demonstration of the convergence of Vardi's algorithm.

Suppose one is able to observe sequentially a series of independent observations $X_1, X_2, \cdots$ such that $X_1, X_2, \cdots, X_{\nu-1}$ are iid distributed according to a known distribution $F_0$ and $X_\nu, X_{\nu+1}, \cdots$ are iid distributed according to a known distribution $F_1$. Assume that $\nu$ is unknown and the problem is to raise an alarm as soon as possible after the distribution changes from $F_0$ to $F_1$. Formally, the problem is to find a stopping rule $N$ which in some sense minimizes $E(N - \nu\mid N\geq \nu)$ subject to a restriction $E(N\mid\nu = \infty) \geq B$. A stopping rule that is a limit of Bayes rules is first derived. Then an almost minimax rule is presented; i.e. a stopping rule $N^\ast$ is described which satisfies $E(N^\ast\mid\nu = \infty) = B$ for which \begin{equation*}\begin{split}\sup_{1\leq\nu < \infty}E(N^\ast - \nu\mid N^\ast \geq \nu) \\ - \inf_{\{\text{stopping rules} N|E(N| \nu=\infty)\geq B\}} \sup_{1\leq\nu < \infty}E(N - \nu \mid N \geq \nu) = o(1)\end{split}\end{equation*} where $o(1) \rightarrow 0$ as $B \rightarrow \infty$.

A sequential fixed-width confidence interval for the mean of a $U$-statistic, having coverage probability approximately equal to preassigned $\alpha$, is presented. The main result, Theorem 2, shows that the sequential procedure is asymptotically efficient in the sense of Chow and Robbins (1965) and assumes only finiteness of the second moment of the kernel, the weakest possible condition. The paper follows naturally from Sproule (1974) and Sproule (1969), the primary reference.

Suppose that $f$ is a function from $\mathbb{R}^k$ to $\mathbb{R}^k$ and for some $\theta, f(\theta) = 0$. Initially $f$ is unknown, but for any $x$ in $\mathbb{R}^k$ we can observe a random vector $Y(x)$ with expectation $f(x)$. The unknown $\theta$ can be estimated recursively by Blum's (1954) multivariate version of the Robbins-Monro procedure. Blum's procedure requires the rather restrictive assumption that infimum of the inner product $(x - \theta)^tf(x)$ over any compact set not containing $\theta$ be positive. Thus at each $x, f(x)$ gives information about the direction towards $\theta$. Blum's recursion is $X_{n+1} = X_n - a_n Y_n$ where the conditional expectation of $Y_n$ given $X_1, \cdots, X_n$ is $f(X_n)$ and $a_n > 0$. Unlike Blum's method, the procedure introduced in this paper does not necessarily attempt to move in a direction that decreases $\|X_n - \theta\|$, at least not during the initial stage of the procedure. Rather, except for random fluctuations it moves in a direction which decreases $\|f\|^2$, and it may follow a circuitous route to $\theta$. Consequently, it does not require that $(x - \theta)^tf(x)$ have a constant signum. This new procedure is somewhat similar to the multivariate Kiefer-Wolfowitz procedure applied to $\|f\|^2$, but unlike the latter it converges to $\theta$ at rate $n^{-1/2}$. Deterministic root finding methods are briefly discussed. The method of this paper is a stochastic analog of the Newton-Raphson and Gauss-Newton techniques.

A new estimator for smoothing towards log-linear models for Poisson means is introduced. The estimator is a generalization of one of Peng (1975). The theoretical basis for the choice of estimator is developed from an approximation to the mean square error of any estimator of Poisson means. The new estimator and its competitors are evaluated in simulations. The method has widespread application in contingency table analysis.

Consider the problem of estimating an unknown distribution function $F$ from the class of all distribution functions given a random sample of size $n$ from $F$. It is proved that the empirical distribution function is admissible for the loss functions $L(F, \hat{F}) = \int (\hat{F}(t) - F(t))^2(F(t))^\alpha(1 - F(t))^b dW(t)$ for any $a < 1$ and $b < 1$ and finite measure $W$. Related results for simultaneous estimation of distribution functions and for finite population sampling are also given.

For a continuous and diagonally symmetric multivariate distribution, incorporating the idea of preliminary test estimators, a variant form of the James-Stein type estimation rule is used to formulate some shrinkage estimators of location based on rank statistics and $U$-statistics. In an asymptotic setup, the relative risks for these shrinkage estimators are shown to be smaller than their classical counterparts.

Admissible linear estimators $Mx + \gamma$ must be pointwise limits of Bayes estimators. Using properties of Bayes estimators preserved by taking limits, the structure of $M$ and $\gamma$ can be determined. Among $M, \gamma$ with this structure, a necessary and sufficient condition for admissibility is obtained. This condition is applied to the case of linear (mixture) models. It is shown that only the most trivial such models admit linear estimators of full rank which are admissible or are even limits of Bayes estimators.

In Wald's statistical decision theory, the criterion of domination (or uniform betterness) is defined with respect to a specific loss. In practice, however, the exact form of a loss function is difficult to specify. Hence, it is important to study the domination criterion simultaneously under a class of loss functions. In this paper we focus on estimation problems. We mainly investigate the possibility of domination simultaneously under the class of loss functions $L(|\theta - \delta|)$, where $L$ is an arbitrary nondecreasing function. As usual, $\theta$ and $\delta$ (both in $p$-dimensional Euclidean space $R^p$) are, respectively, the unknown parameter of nature and the statistician's estimate. Domination simultaneously under this class of losses is called universal domination under Euclidean error. Several theoretical questions are resolved in this paper. In particular the criterion of universal domination is shown to be equivalent to the criterion of stochastic domination that compares the estimators by the stochastic ordering of their Euclidean distances from the estimators to the true parameter. Concrete results about universal domination relating to the usual estimator are also established. In particular when $X - \theta$ has a $p$-variate $t$ distribution, and $p = 1, 2$, there exists no estimator for $\theta$ that universally dominates $X$; however, for $p \geq 3$, estimators (of the type of James-Stein positive part estimators) that universally dominate $X$ are specified. When $X$ has a $p$-variate normal distribution with mean $\theta$ and identity covariance matrix, we show that for any dimension $p$, no James-Stein positive part estimators universally dominate $X$. However, under slightly smaller classes of losses, some James-Stein positive part estimators are shown to simultaneously dominate $X$. These hitherto unstudied losses are bounded and fairly practical.

A class of goodness-of-fit tests based on sums of squares of $L$-statistics is proposed for testing a composite parametric location and/or scale null hypothesis versus a general parametric alternative. It is shown that such tests can be constructed optimally to have the same asymptotic power against sequences of local alternatives as the generalized likelihood ratio statistics [G.L.R.S.] and, in fact, under suitable regularity conditions to be asymptotically equivalent to the G.L.R.S. One advantage of the proposed test statistic over the G.L.R.S. is that only an estimate of the scale parameter is needed in the computation of the statistic. No other parameter estimates are required. Also, an example of the practical implementation of the proposed hypothesis testing procedure is given.

The problem of estimating shape and scale parameters for a distribution with regularly varying tails is related to that of nonparametrically estimating a density at a fixed point, in that optimal construction of the estimators depends substantially upon unknown features of the distribution. We show how to overcome this problem by using adaptive methods. Our main results hold very generally, for a large class of adaptive estimators. Later we consider specific versions of adaptive estimators, and describe their performance both in theory and by means of simulation studies. We also examine a technique proposed by Hill (1975) for solving similar problems.

Generalized linear models are used for regression analysis in a number of cases, including categorical responses, where the classical assumptions are violated. The statistical analysis of such models is based on the asymptotic properties of the maximum likelihood estimator. We present mild general conditions which, respectively, assure weak or strong consistency or asymptotic normality. Most of the previous work has been concerned with natural link functions. In this case our normality condition, though obtained by a different approach, is closely related to a condition of Haberman (1977a). Examples show how the general conditions reduce to weak requirements for special exponential families. Further, for regressors with a compact range, sufficient conditions are given which do not involve the unknown parameter, and are therefore easy to check in practice. Responses with a bounded range, e.g. categorical responses, and stochastic regressors also are treated.

Consider a trigonometric regression of order $m$ or a polynomial regression of degree $m$. Explicit $D_s$-optimal designs are given for some subsets of the coefficients. Lauter type optimal designs are given for various models involving the order or the degree. The designs are calculated using canonical moments.

Let $G$ be a locally compact group acting continuously on the left of a locally compact space $\mathscr{X}$. It is assumed that $G = HK$ where $H$ and $K$ are closed subgroups. It is shown that if $K$ acts properly on $\mathscr{X}$ and $H$ acts properly on $\mathscr{X}/K$, then $G$ acts properly on $\mathscr{X}$. Under a mild additional condition the converse is also true. Several examples are given to show how these results can help decide the properness of composite actions. Proper action can be used to justify the representation of the density ratio of a maximal invariant as a ratio of integrals over the group.

Several authors have proposed empirical Bayes tests (EBT) for the continuous one-parameter exponential family for the case that the prior distribution is completely unspecified. They investigated the convergence rate of the (unconditional) Bayes risk, and gave upper bounds for this convergence rate. In this paper it is proposed to study the convergence of the conditional Bayes risk. A method is presented which makes it possible to derive the exact convergence rate of the conditional risk and its limit distribution. Several results are given. Also the question is considered whether monotonizing an empirical Bayes test influences its asymptotic properties.

Marshall and Proschan (1965) showed that the MLE for a life distribution with increasing failure rate is strongly consistent. In this note we show that the MLE for a life distribution with increasing failure rate average is not consistent; in fact, maximum likelihood estimation in the IFRA case yields estimators of the average failure rate and of the distribution function which, in general, converge a.s. to values other than the true values. In the decreasing failure rate average case, the MLE fails to exist.

A version of the two-armed bandit with two states of nature and two repeatable experiments is studied. With an infinite horizon and with or without discounting, an optimal procedure is to perform one experiment whenever the posterior probability of one of the states of nature exceeds a constant $\xi^\ast$, and perform the other experiment whenever the posterior is less than $\xi^\ast$ with indifference when the posterior equals $\xi^\ast. \xi^\ast$ is expressed in terms involving expectations of ladder variables and can be calculated using Spitzer series.

Convergence of densities implies convergence of their distribution functions via Scheffe's theorem. This paper is concerned with the converse: what are sufficient conditions to obtain convergence of densities from convergence of distribution functions? A general lemma is given and local limit results are obtained for translation and scale statistics.

Multidimensional asymptotic normality of the kernel quantile estimator is established under fairly general conditions on the underlying distribution function and on the kernel. Sharpening these assumptions, one can utilize the proof to achieve also a bound for the rate of convergence which entails the comparison of the kernel estimator with the empirical quantile on the basis of their covering probabilities.