Estimation of the means of independent normal random variables is considered, using sum of squared errors as loss. An unbiased estimate of risk is obtained for an arbitrary estimate, and certain special classes of estimates are then discussed. The results are applied to smoothing by use of moving averages and to trimmed analogs of the James-Stein estimate. A suggestion is made for calculating approximate confidence sets for the mean vector centered at an arbitrary estimate.

In a previous paper, it was shown that parameter-effects nonlinearities of a nonlinear regression model-experimental design-parameterization combination can be quantified by means of a parameter-effects curvature array $A$ based on second derivatives of the model function. In this paper, the individual terms of $A$ are interpreted and local compensation methods are suggested. A method of computing the parameter-effects array corresponding to a transformed set of parameters is given and we discuss how this result could be used to determine reparameterizations which have zero local parameter-effects nonlinearity.

A concept of model-robustness is defined in terms of the performance of the design in the presence of model violations. The robustness problem is discussed for several randomization procedures commonly used in experimental design situations. Among them, the balanced completely randomized design, the randomized complete block design and the randomized Latin square design are shown to be model-robust in their own settings. To compare different randomization procedures, we define a concept of efficiency which depends on the particular "pattern" of model violations. This concept, when applied to different designs, gives results which are consistent with the intuitive grounds on which the designs are suggested.

Tests for independence of rows and columns in an $r \times s$ contingency table are developed from canonical correlation analysis and from models of linear-by-linear interaction. The resulting test statistics are asymptotically equivalent under the null hypothesis. They are consistent and asymptotically unbiased. Approximate critical values are available from existing tables. The proposed tests are most appropriate when the matrix of joint probabilities is well approximated by a matrix of rank 2. Against some alternatives which may arise in such tables, the proposed statistics have greater asymptotic power than conventional chi-square tests of independence.

In the non-lattice case it is shown that the bootstrap approximation of the distribution of the standardized sample mean is asymptotically more accurate than approximation by the limiting normal distribution. The exact convergence rate of the bootstrap approximation of the distributions of sample quantiles is obtained. A few other convergence rates regarding the bootstrap method are also studied.

Efron's "bootstrap" method of distribution approximation is shown to be asymptotically valid in a large number of situations, including $t$-statistics, the empirical and quantile processes, and von Mises functionals. Some counter-examples are also given, to show that the approximation does not always succeed.

The regression and correlation models are considered. It is shown that the bootstrap approximation to the distribution of the least squares estimates is valid, and some error bounds are given.

We describe the asymptotic theory of triple sampling as it pertains to the estimation of a mean. We obtain limit theorems for the case of the normal distribution. Our results show that triple sampling combines the simplicity of Stein's double sampling technique with the efficiency of the fully sequential Anscombe-Chow-Robbins procedure.

Assume observations are from a subclass of a one parameter exponential family whose dominating measure is nonatomic. Consider a one-sided sequential testing problem where null and alternative parameter sets have one common boundary point. Let the risk function be a linear combination of probability of error and expected sample size. Our main result is that a sequential test is inadmissible if its continuation region has unbounded width in terms of the natural sufficient statistic. We apply this result to prove that weight function tests, with weight functions that contain the common boundary point in their support, are inadmissible. Furthermore any obstructive test is inadmissible, where obstructive means that the stopping time for the test does not have a finite moment generating function for some parameter point. Specific tests of the above type are cited.

Likelihood ratio tests concerning the parameters of two multinomial populations are discussed. A stochastic ordering restriction is considered as a one sided alternative to equality. The one and two sample tests for equality versus stochastic ordering and stochastic ordering versus all alternatives are derived and their large sample distributions are obtained. The large sample distributions are mixtures of chi-squared distributions. The tests developed provide discrete analogues for the one sided Mann-Whitney-Wilcoxon and Kolmogorov-Smirnov tests.

Minimal complete classes of invariant tests are presented for modifications of the problem of testing the independence of $Y$ and $X$, where $(Y, X) \equiv (Y, X_1, \cdots, X_p)$ is a multivariate normal random vector. One modification involves having extra independent observations on $Y$. Others involve extra variates $Z \equiv (Z_1, \cdots, Z_q)$ such that $(Y,X,Z)$ is multivariate normal. Among other results, locally most powerful invariant tests and asymptotically most powerful invariant tests are found; it is shown that for some problems the likelihood ratio test is admissible among invariant tests only for levels less than a specified one; and it is shown that for the problem of testing the independence of $Y$ and $X$ when it is known that $Y$ and $Z$ are independent, the test based on the sample multiple correlation coefficient of $Y$ and $(X, Z)$ is inadmissible.

This paper gives necessary and sufficient conditions for the null distribution of a test statistic to remain the same in the class of left $\mathscr{O}(n)$-invariant distributions and in the class of elliptically symmetric distributions. Secondly, it is shown that in certain special cases, the usual MANOVA tests are still uniformly most powerful invariant in a class of left $\mathscr{O}(n)$-invariant distributions including elliptically symmetric distributions.

This paper proposes a new estimator of the parameter vector in a linear regression model when the observations are randomly censored on the right and when the error distribution is unknown. This estimator is explicitly defined and easily computable. The paper contains sufficient conditions under which this estimator is mean square consistent and asymptotically normal. A numerical example is given.

This paper contains a complete class theorem (Theorem 3.2) which applies to most statistical estimation problems having a finite sample space. This theorem also applies to many other statistical problems with finite sample spaces. The description of this complete class involves a stepwise algorithm. At each step of the process it is necessary to construct the Bayes procedures in a suitably modified version of the original problem. The complete class is a minimal complete class if the loss function is strictly convex. Some examples are given to illustrate the application of this complete class theorem. Among these is a new result concerning the estimation of the parameters of a multinomial distribution under a normalized quadratic loss function. (See Example 4.5).

If $X$ is a $N(\theta, 1)$ random variable, let $\rho (m)$ be the minimax risk for estimation with quadratic loss subject to $|\theta| \leq m$. Then $\rho (m) = 1 - \pi^2/m^2 + o(m^{-2})$. We exhibit estimates which are asymptotically minimax to this order as well as approximations to the least favorable prior distributions. The approximate least favorable distributions (correct to order $m^{-2}$) have density $m^{-1} \cos^2 \big(\frac{\pi}{2m} s\big), |s| \leq m$ rather than the naively expected uniform density on $\lbrack -m, m \rbrack$. We also show how our results extend to estimation of a vector mean and give some explicit solutions.

Let $(X, Y), (X_1, Y_1), \cdots, (X_n, Y_n)$ be independent identically distributed random vectors from $R^d \times R$, and let $E(|Y|^p) < \infty$ for some $p \geq 1$. We wish to estimate the regression function $m(x) = E(Y \mid X = x)$ by $m_n(x)$, a function of $x$ and $(X_1, Y_1), \cdots, (X_n, Y_n)$. For large classes of kernel estimates and nearest neighbor estimates, sufficient conditions are given for $E\{|m_n(x) - m(x)|^p\} \rightarrow 0$ as $n \rightarrow \infty$, almost all $x$. No additional conditions are imposed on the distribution of $(X, Y)$. As a by-product, just assuming the boundedness of $Y$, the almost sure convergence to 0 of $E\{|m_n(X) - m(X)\| X_1, Y_1, \cdots, X_n, Y_n\}$ is established for the same estimates. Finally, the weak and strong Bayes risk consistency of the corresponding nonparametric discrimination rules is proved for all possible distributions of the data.

Let $(X, Y), (X_1, Y_1), \cdots, (X_n, Y_n)$ be independent identically distributed random vectors from $R^d \times \{0, 1\}$, and let $\hat{Y}$ be the $k$-nearest neighbor estimate of $Y$ from $X$ and the $(X_i, Y_i)$'s. We show that for all distributions of $(X, Y)$, the limit of $L_n = P(\hat{Y} \neq Y)$ exists and satisfies $\lim_{n\rightarrow\infty} L_n \leq (1 + a_k)R^\ast, a_k = \frac{\alpha \sqrt k}{k - 3.25}\big(1 + \frac{\beta}{\sqrt{k-3}}\big), k \text{odd}, k \geq 5,$ where $R^\ast$ is the Bayes probability of error and $\alpha = 0.3399 \cdots$ and $\beta = 0.9749 \cdots$ are universal constants. This bound is shown to be best possible in a certain sense.

Two approximate ancillaries, the Efron-Hinkley ancillary and the (signed) likelihood ratio, are compared in a specific example by means of their level curves, their marginal densities and the conditional and unconditional likelihood functions. It is shown that in the present model the latter of the two ancillaries is the better. It is almost exactly distribution constant and has more stable conditional likelihood functions.

A theorem is proved which gives a necessary and sufficient condition for improving, up to $o(n^{-2})$, the mean squared error of the maximum likelihood estimate $\hat{\theta}$ by using an estimate of the form $\hat{\theta} + d(\hat{\theta})/n$. An application is made to a bioassay problem of Berkson.

We search a population by selecting one member at a time with replacement and observing the species of each selected member. We are interested in predicting the conditional probability of discovering a new species in the next selection after $n$ observations. The existence and asymptotic behavior of the uniformly minimum variance unbiased estimator of the unconditional probability is investigated under the condition that the species are uniformly distributed. We also compare the performance of the estimator as a predictor of the conditional probability with that of a linear unbiased predictor.