A problem arising in taste testing, medical, and parapsychology experiments can be modeled as follows. A deck of $n$ cards contains $c_i$ cards labeled $i, 1 \leq i \leq r$. A subject guesses at the cards sequentially. After each guess the subject is told the card just guessed (or at least if the guess was correct or not). We determine the optimal and worst case strategies for subjects and the distribution of the number of correct guesses under these strategies. We show how to use skill scoring to evaluate such experiments in a way which (asymptomatically) does not depend on the subject's strategy.

In a multivariate "errors in variables" regression model, the unknown mean vectors $\mathbf{u}_{1i}: p \times 1, \mathbf{u}_{2i}: r \times 1$ of the vector observations $\mathbf{x}_{1i}, \mathbf{x}_{2i}$, rather than the observations themselves, are assumed to follow the linear relation: $\mathbf{u}_{2i} = \alpha + B\mathbf{u}_{1i}, i = 1,2,\cdots, n$. It is further assumed that the random errors $\mathbf{e}_i = \mathbf{x}_i - \mathbf{u}_i, \mathbf{x}'_i = (\mathbf{x}'_{1i}, \mathbf{x}'_{2i}), \mathbf{u}'_i = (\mathbf{u}'_{1i}, \mathbf{u}'_{2i})$, are i.i.d. random vectors with common covariance matrix $\Sigma_e$. Such a model is a generalization of the univariate $(r = 1)$ "errors in variables" regression model which has been of interest to statisticians for over a century. In the present paper, it is shown that when $\Sigma_e = \sigma^2I_{p+r}$, a wide class of least squares approaches to estimation of the intercept vector $\alpha$ and slope matrix $B$ all lead to identical estimators $\hat{\alpha}$ and $\hat{B}$ of these respective parameters, and that $\hat{\alpha}$ and $\hat{B}$ are also the maximum likelihood estimators (MLE's) of $\alpha$ and $B$ under the assumption of normally distributed errors $\mathbf{e}_i$. Formulas for $\hat{\alpha}, \hat{B}$ and also the MLE's $\hat{U}_1$ and $\hat{\sigma}^2$ of the parameters $U_1 = (\mathbf{u}_{11}, \cdots, \mathbf{u}_{1n})$ and $\sigma^2$ are given. Under reasonable assumptions concerning the unknown sequence $\{\mathbf{u}_{1i}, i = 1,2,\cdots\}, \hat{\alpha}, \hat{B}$ and $r^{-1}(r + p)\hat{\sigma}^2$ are shown to be strongly (with probability one) consistent estimators of $\alpha, B$ and $\sigma^2$, respectively, as $n \rightarrow \infty$, regardless of the common distribution of the errors $\mathbf{e}_i$. When this common error distribution has finite fourth moments, $\hat{\alpha}, \hat{B}$ and $r^{-1}(r + p)\hat{\sigma}^2$ are also shown to be asymptotically normally distributed. Finally large-sample approximate $100(1 - \nu){\tt\%}$ confidence regions for $\alpha, B$ and $\sigma^2$ are constructed.

This paper is concerned with basic problems of statistical inference. The thesis is in three parts: (1) that inference is a procedure whereby one passes from a population (or sample) to a new individual; (2) that this connection can be established using de Finetti's idea of exchangeability or Fisher's concept of a subpopulation; (3) in making the connection use must be made of the appropriate probability. These three principles are used in a variety of situations and the topics discussed include analysis of variance and covariance, contingency tables, and calibration. Some comments on randomization are also included.

This paper relaxes the conventional subjective probability setup by allowing the $\sigma$-algebra on which probabilities are defined to be subjective along with the probability measure. First, the role of the probability domain in existing statistical decision theory is examined. Then the existing theory is extended by characterizing the individual's selection of a probability domain as the outcome of a decision process.

Smoothing of the data by averaging is suggested in order to study the maximum. The maximum of the smoothed data is approximated by that of a Gaussian sample and thus is more robust against outliers.

In this paper influence curves for censored data estimators are considered. The influence curve of the Kaplan-Meier estimate of survival time is calculated, and it is shown that the influence curve provides an alternative derivation of the asymptotic variance of this estimate. A chain rule for influence curves is established and is used to calculate the influence curves of censored data estimators that are functions of the Kaplan-Meier estimate. The robustness properties and asymptotic variances of these estimates follow directly. Some examples of this approach to calculating variances are given. In particular, it is shown how the theory developed for $M$-estimation and $L$-estimation can be extended to the censored data case. The necessary differentiability conditions are verified in the appendix.

Strong consistency and asymptotic normality are established for the maximum partial likelihood estimate of the regression parameter in Cox's regression model. Estimates are also derived for the underlying cumulative hazard function and survival distribution. We establish the asymptotic normality of these estimates and calculate the limiting variances.

For the Cox regression model, the partial likelihood functions involve linear combinations of induced order statistics. Some invariance principles pertaining to such linear combinations of induced order statistics are studied and the theory is incorporated in the formulation of some repeated significance tests (for the hypothesis of no regression) based on these partial likelihoods.

Let $X_1, \cdots, X_n$ be i.i.d. $P(X > u) = F(u)$ and $Y_1, \cdots, Y_n$ be i.i.d. $P(Y > u) = G(u)$, where both $F$ and $G$ are unknown continuous distributions. For $i = 1, \cdots, n$ set $\delta_i = 1$ if $X_i \leq Y_i$ and 0 if $X_i > Y_i$ and $Z_i = \min \{X_i, Y_i\}$. One way to estimate $F$ from the observations $(Z_i, \delta_i) i = 1, \cdots, n$ is by means of the product limit (PL) estimator, $F^\ast_n$ (Kaplan-Meier, [8]). In this paper it is shown that $F^\ast_n$ is uniformly almost sure consistent with rate $O(\sqrt{\log n} / \sqrt n)$, that is $P(\sup_{0 \leq u \leq T}|F^\ast_n(u) - F(u)| = O(\sqrt{\log n/n})) = 1.$ Assuming that $F$ is distributed according to a Dirichlet process (Ferguson, [3]) with parameter $\alpha$, Susarla and Van Ryzin ([11]) obtained the Bayes estimator $F^\alpha_n$ of $F$. In the present paper a similar result is established for the Bayes estimator, namely: $P(\sup_{0 \leq u \leq T}| F^\alpha_n(u) - F(u)| = O(\sqrt{(\log n)^{1 + \gamma}} / \sqrt n)) = 1 \quad (\gamma > 0).$

The Bayesian bootstrap is the Bayesian analogue of the bootstrap. Instead of simulating the sampling distribution of a statistic estimating a parameter, the Bayesian bootstrap simulates the posterior distribution of the parameter; operationally and inferentially the methods are quite similar. Because both methods of drawing inferences are based on somewhat peculiar model assumptions and the resulting inferences are generally sensitive to these assumptions, neither method should be applied without some consideration of the reasonableness of these model assumptions. In this sense, neither method is a true bootstrap procedure yielding inferences unaided by external assumptions.

A random sample is divided into the $k$ clusters that minimise the within cluster sum of squares. Conditions are found that ensure the almost sure convergence, as the sample size increases, of the set of means of the $k$ clusters. The result is proved for a more general clustering criterion.

Given a sample of independent random variables $Z_1, Z_2, \cdots, Z_n$ with identical distribution $p$ on a compact metric space $(M, d)$, a measure of central tendency is a sample centroid (of order $r > 0$) defined as a point $\hat{X}_n$ in $M$ satisfying $\frac{1}{n} \sum^n_{i=1} d^r(\hat{X}_n, Z_i) = \inf_{x \in M} \frac{1}{n} \sum^n_{i=1} d^r(x, Z_i).$ A (population) centroid of $Z$ is any point $x^\ast$ in $M$ such that $\int_M d^r(x^\ast, z) dp(z) = \inf_{x \in M} \int_M d^r(x, z) dp(z).$ The quantity $\frac{1}{n} \sum^n_{i=1} d^r(\hat{X}_n, Z_i)$ itself is called the sample variation, whereas $\int_M d^r(x^\ast, z) dp(z)$ is the variation of $Z$. This paper establishes almost sure convergence for the sample centroid and variation to the corresponding population values for all orders $r > 0$. Convergence is also proved for the case when the sample centroid is restricted to be one of the sample values.

Let $f$ be a density possessing some smoothness properties and let $X_1,\cdots, X_n$ be independent observations from $f$. Some desirable properties of orthogonal series density estimates $f_{n,m,\lambda}$ of $f$ of the form $f_{n,m,\lambda}(t) = \sum^n_{\nu = 1} \frac{\hat{f}_\nu}{(1 + \lambda\nu^{2m})} \phi_\nu(t)$ where $\{\phi_\nu\}$ is an orthonormal sequence and $\hat{f}_\nu = (1/n)\sum^n_{j=1} \phi_\nu(X_j)$ is an estimate of $f_\nu = \int \phi_\nu(t)f(t) dt$, are discussed. The parameter $\lambda$ plays the role of a bandwidth or "smoothing" parameter and $m$ controls a "shape" factor. The major novel result of this note is a simple method for estimating $\lambda$ (and $m$) from the data in an objective manner, to minimize integrated mean square error. The results extend to multivariate estimates.

In the classical theory of competing risks (as well as in many reliability models and incomplete data problems) it is assumed that (A1) the risks (i.e., the random variables of interest) are independent and that (A2) death does not result from simultaneous causes. Employing our probabilistic solution to a related problem in probability modelling, we obtain strongly consistent estimators for the unobservable marginal distributions of interest. These estimators are analogous to those of Kaplan and Meier [J. Amer. Statist. Assoc. (1958) 63] and are appropriate when the assumptions of independence and no simultaneous causes of death [(A1) and (A2), above] fail to hold. We show how our methods can be used to unify and simplify the nonparametric approach toward estimation in the competing risks model. As a consequence we obtain an elementary proof of the strong consistency of the Kaplan-Meier estimator. Our results extend and simplify the work of Peterson [J. Amer. Statist. Assoc. (1977) 72] and Desu and Narula [The Theory and Applications of Reliability, I (I. Shimi and C. P. Tsokos, eds.) (1977)].

Association of random variables [Ann. Math. Statist. 38 1466-1474] is generalized by considering increasing functions on certain partially ordered spaces. Generalized association enjoys many of the basic properties possessed by association. An essential connection links the two notions. Generalized association is inherited under certain mixing processes. Applications of this result follow for absolute value ordering, unimodal ordering, and linear ordering of the real numbers.

We are concerned with the admissibility of nonlinear estimators of the form $(aX + b)/(cX + d)$ in the one-parameter exponential family, in estimating $g(\theta)$ with quadratic loss. Our method will be reminiscent of that of Karlin who gave sufficient conditions for admissibility of linear estimators $aX$ in estimating the mean in the one-parameter family. Our results generalize those of Ghosh and Meeden who studied admissibility of $aX + b$ for estimating an arbitrary function $g(\theta)$. Particular cases of estimators of the form, $c/X$ are studied and several examples are given. We show that $(n - 2)/(X + a), a \geq 0$ is admissible in estimating the parameter of an exponential density. We also discuss the case of truncated parameter space.

Let $\{X_n, n \geq 1\}$ be a sequence of independent identically distributed random variables with mean $\mu$ and unknown variance $\sigma^2$. We want to estimate $\mu$ by $\bar{X}_n$ with a loss function of $\sigma^{2 \delta - 2}(\bar{X}_n - \mu)^2 + \lambda n$, where $\delta > 0$ and $\lambda \rightarrow 0 +$. For $n_\lambda = o(\lambda^{-1/2})$ and $n_\lambda(\log \lambda)^{-1} \rightarrow -\infty$ as $\lambda \rightarrow 0+$, set $T = \inf\{n \geq n_\lambda: n^{-1} \sum^n_{i=1} (X_i - \bar{X}_n) + b_n \leq \lambda^{1/\delta} a_n\}$. If $a_n n^{-2/\delta} \rightarrow 1$ and $0 < b_n \rightarrow 0$ as $n \rightarrow \infty$, we prove that $T$ is asymptotically risk efficient, that is, as $\lambda \rightarrow 0+, E \lbrack(2\lambda^{1/2}\sigma^\delta)^{-1}(\sigma^{2\delta-2}(\bar{X}_T - \mu)^2 + \lambda T)\lbrack \rightarrow 1$. When the $X_n$'s are normal, the asymptotic risk efficiency of $T$ was established by Starr. By introducing the delay factor $n_\lambda$, we are able to drop the condition of $X_n$'s being normal.

With the ranking after alignment procedure, a conditional permutational argument is introduced which gives rise to a conditional permutation distribution for the test statistic in randomized block designs. In this paper, the almost sure and uniform convergence of this distribution is derived for a wide class of test statistics as the number of blocks goes to infinity.

Let $U_N$ be a one sample $U$-statistic with kernel $h$ of degree two, such that $Eh(X_1, X_2) = \vartheta$ and $\operatorname{Var} E\lbrack h(X_1, X_2)\mid X_1 \rbrack > 0$. It is shown that for a studentized $U$-statistic $\sup_x|P(N^{1/2}S^{-1}_N(U_N - \vartheta) \leqslant x) - \Phi(x)| = O(N^{-1/2})$ as $N \rightarrow \infty$, where $N^{-1}S^2_N = 4N^{-1}(N - 1)(N - 2)^{-2}\sum^N_{i=1} \lbrack (N - 1)^{-1} \sum_{j\neq i} h(X_i, X_j) - U_N \rbrack^2$ is the jackknife estimator of $\operatorname{Var} U_N$. The condition needed to obtain this order bound is the existence of the 4.5th absolute moment of the kernel $h$. As in Helmers' Ph.D. thesis on linear combinations of order statistics, the analogous result for a studentized sum of i.i.d. random variables arises as a special case.

In the two-way heterogeneity setting, the optimality of a generalized Youden design (GYD) has been proved by Kiefer (1975a). A GYD is a design which is a balanced block design (BBD) when each of {rows} and {columns} is considered as blocks. It is observed in the present paper that when the number of rows is equal to the number of columns, a design is optimal as long as the rows and columns together form a BBD. Such a design is termed a pseudo-Youden design (PYD). A square GYD is also a PYD, but the converse is not true. Thus, the stringent conditions imposed on the definition of a GYD are substantially relaxed. A PYD is easier to construct and has the same efficiency as a GYD if they exist simultaneously. Patchwork and geometric methods are combined to construct a family of PYD's. It is also indicated when the construction of a GYD is impossible. A $6 \times 6$ PYD with 9 varieties is constructed. This design has the property that the number of rows is less than the number of varieties, which is never achieved by a square GYD. There is also an analogous theory for higher-dimensional designs.

We extend Bickel's tests for heteroscedasticity to include wider classes of test statistics and fitting methods. The test statistics include those based on Huber's function, while the fitting techniques include Huber's Proposal 2 for robust regression.

This paper shows that Hotelling's $T^2$-test for testing $\mu = 0$ in the one-sample problem is robust against departures from normality in the following sense. It is still UMPI in a broad class of distributions, and the null distribution under any member of the class is the same as that under normality.

Suppose we have to decide on the basis of appropriately drawn samples which of $k$ treatment populations are "good" compared to either given control values or to a control population from which an additional sample is available. The unknown parameters are assumed to vary randomly according to a prior distribution about which we only have the partial knowledge that it is contained in a given class $\Gamma$ of priors. Though we derive in both cases (under the assumption of monotone likelihood ratios) $\Gamma$-minimax procedures which by definition attain minimal supremal risk over $\Gamma$, the emphases are different: while we try to demonstrate in the "known controls case" how well known results from the theory of testing hypotheses can be utilized to solve the problem, our main purpose in the "unknown control case" is to give a new proof for a theorem which was stated but only partially proved by Randles and Hollander.

This paper establishes the asymptotic optimality (in the sense of Robbins) of the empirical Bayes distribution function created from the Bayes rule relative to the Dirichlet process prior with unknown parameter $\alpha(.)$. It will follow that the same result applies to the estimation of the mean of a distribution function.

The results of Wald on the consistency of the maximum likelihood estimate are extended. Applications are made to mixture distributions and to clustering when the number of clusters is not known.

A sampling strategy consists of a sampling design and an estimator. Of special importance are strategies combining simple random sampling with the sample mean as an estimator. We consider this class of strategies and the smaller class of strategies with simple random sampling of (almost) fixed sample size. The convexity of the underlying loss function does not imply the completeness of the smaller class. However the strict convexity of the loss function together with the convexity of its second derivative, is sufficient for this completeness.