The sovereignty of MLE is questioned. Minimum $\chi^2_\lambda$ yields the same estimating equations as MLE. For many cases, as illustrated in presented examples, and further algorithmic exploration in progress may show that for all cases, minimum $\chi^2_\lambda$ estimates are available. In this sense minimum $\chi^2$ is the basic principle of estimation. The criterion of asymptotic sufficiency which has been called "second order efficiency" is rejected as a criterion of goodness of estimate as against some loss function such as the mean squared error. The relation between MLE and sufficiency is not assured, as illustrated in an example in which MLE yields $\infty$ as estimate with samples that have different values of the sufficient statistic. Other examples are cited in which minimal sufficient statistics exist but where the MLE is not sufficient. The view is advanced that statistics is a science, not mathematics or philosophy (inference) and as such requires that any claimed attributes of the MLE must be testable by a Monte Carlo experiment.

The $n^{-2}$ order mean squared errors of the maximum likelihood and the minimum chi-square estimator of the logit regression model are derived and the latter is shown to be superior for many parameter values considered. The maximum likelihood is shown to be better if the bias of each estimator is corrected to the order of $n^{-1}$; however, the difference is shown to be negligibly small in many practical situations.

Let $X_1, X_2, \cdots$ be a sequence of i.i.d. random variables, each having density $f(x, \theta_0)$ where $\{f(x, \theta)\}$ is a family of densities with respect to a dominating measure $\mu$. Suppose $n^{\frac{1}{2}}(\hat{\theta} - \theta)$ and $n^{\frac{1}{2}}(T - \theta)$, where $\hat{\theta}$ is the mle and $T$ is any other efficient estimate, have Edgeworth expansions up to $o(n^{-1})$ uniformly in a compact neighbourhood of $\theta_0$. Then (under certain regularity conditions) one can choose a function $c(\theta)$ such that $\hat{\theta}' = \hat{\theta} + c(\hat{\theta})/n$ satisfies $P_{\theta_0} \{-x_1 \leqslant n^{\frac{1}{2}}(\hat{\theta}' - \theta_0)(I(\theta_0))^{\frac{1}{2}} \leqslant x_2\} \\ \geqslant P_{\theta_0}\{-x_1 \leqslant n^{\frac{1}{2}}(T - \theta_0)(I(\theta_0))^{\frac{1}{2}} \leqslant x_2\} + o(n^{-1}),$ for all $x_1, x_2 \geqslant 0$. This result implies the second order efficiency of the mle with respect to any bounded loss function $L_n(\theta, a) = h(n^{\frac{1}{2}}(a - \theta))$, which is bowl-shaped i.e., whose minimum value is zero at $a - \theta = 0$ and which increases as $|a - \theta|$ increases. This answers a question raised by C. R. Rao (Discussion on Professor Efron's paper).

We use a close connection between the theory of Markov fields and that of log-linear interaction models for contingency tables to define and investigate a new class of models for such tables, graphical models. These models are hierarchical models that can be represented by a simple, undirected graph on as many vertices as the dimension of the corresponding table. Further all these models can be given an interpretation in terms of conditional independence and the interpretation can be read directly off the graph in the form of a Markov property. The class of graphical models contains that of decomposable models and we give a simple criterion for decomposability of a given graphical model. To some extent we discuss estimation problems and give suggestions for further work.

The main theorem of this note is required in a paper of Brown. Briefly, the theorem shows that procedures which can be improved on in a neighborhood of infinity are either inadmissible or are generalized Bayes for a (possibly improper) prior whose rate of growth at infinity is of an appropriate order. This theorem is applied here to show that the risk of the usual estimator of a two dimensional normal mean, $\theta$, cannot be improved on near $\infty$ at order $\|\theta\|^{-2}$.

A general technique is developed for improving upon inadmissible estimators of natural parameters (or integral powers thereof) from continuous exponential families. The technique is to reduce the problem to the study of a differential inequality. Typical differential inequalities are presented and solved. Explicit results are given for the simultaneous estimation of gamma scale parameters (and their inverses) for a variety of natural loss functions. Surprising behavior is observed for many of the estimators improving upon "standard" estimators. For squared error loss (and any continuous exponential family) it is shown explicitly how to establish inadmissibility of an estimator and construct improved estimators.

Two examples are presented. In each, $p$ independent normal random variables having unit variance are observed. It is desired to estimate the unknown means, $\theta_i$, and the loss is of the form $L(\theta, a) = (\Sigma^p_{i=1} \nu(\theta_i))^{-1} \Sigma^p_{i=1}\nu(\theta_i)(\theta_i - a_i)^2$. The usual estimator, $\delta_0(x) = x$, is minimax with constant risk. In the first example $\nu(t) = e^{rt}$. It is shown that when $r \neq 0, \delta_0$ is inadmissible if and only if $p \geqslant 2$ whereas when $r = 0$ it is known to be inadmissible if and only if $p \geqslant 3$. In the second example $\nu(t) = (1 + t^2)^{r/2}$. It is shown that $\delta_0$ is inadmissible if $p > (2 - r)/(1 - r)$ and admissible if $p < (2 - r)/(1 - r)$. (In particular $\delta_0$ is admissible for all $p$ when $r \geqslant 1$ and only for $p = 1$ when $r < 0.$) In the first example the first order qualitative description of the better estimator when $\delta_0$ is inadmissible depends on $r$, while in the second example it does not. An example which is closely related to the first example, and which has more significance in applications, has been described by J. Berger.

Let $\mathbf{S}_{p \times p}$ have a Wishart distribution with scale matrix $\Sigma$ and $k$ degrees of freedom. Estimators of $\Sigma$ are given for each of the loss functions $L_1(\hat{\Sigma}, \Sigma) = \operatorname{tr} (\hat{\Sigma}\Sigma^{-1}) - \log \det (\hat{\Sigma}\Sigma^{-1}) - p$ and $L_2(\hat{\Sigma}, \Sigma) = \operatorname{tr} (\hat{\Sigma}\Sigma^{-1} - I)^2$. The obvious estimators of $\Sigma$ are the scalar multiples of $\mathbf{S}$, i.e., $a\mathbf{S}$ where $0 < a \leqslant 1/k$. (Recall that $(1/k)\mathbf{S}$ is unbiased.) For each problem $(\Sigma, \hat{\Sigma}, L_i), i = 1, 2$, we provide empirical Bayes estimators which dominate $a\mathbf{S}$ by a substantial amount. It is seen that the uniform reduction in the risk function determined by $L_2$ is at least $100(p + 1)/(k + p + 1){\tt\%}$. Dominance results for $L_1$ and $L_2$ were first given by James and Stein.

A general calculus of conditional independence is developed, suitable for application to a wide range of statistical concepts such as sufficiency, parameter-identification, adequacy and ancillarity. A vehicle for this theory is the statistical operation, a structure-preserving map between statistical spaces. Concepts such as completeness and identifiability of mixtures arise naturally and play an important part. Some general theorems are exemplified by applications to ancillarity, including a study of a Bayesian definition of ancillarity in the presence of nuisance parameters.

A parameter expressed as a functional $T(F)$ of a distribution function (df) $F$ may be estimated by the "statistical function" $T(F_n)$ based on the sample df $F_n$. For analysis of the estimation error $T(F_n) - T(F)$, we adapt the differential approach of von Mises (1947) to exploit stochastic properties of the Kolmogorov-Smirnov distance $\sup_x|F_n(x) - F(x)|$. This leads directly to the central limit theorem (CLT) and law of the iterated logarithm (LIL) for $T(F_n) - T(F)$. The adaptation also incorporates innovations designed to broaden the scope of statistical application of the concept of differential. Application to a wide class of robust-type $M$-estimates is carried out.

The theory of general chi-square statistics for testing fit to parametric families of distributions is extended to samples censored at sample quantiles. Data-dependent cells with sample quantiles as cell boundaries are employed. Asymptotic distribution theory is given for statistics in which unknown parameters are estimated by estimators asymptotically equivalent to linear combinations of functions of order statistics. Emphasis is placed on obtaining statistics having a chi-square limiting null distribution. Examples of such statistics for testing the fit of Type II censored samples to the negative exponential, normal, two-parameter uniform and two-parameter Weibull families are given.

Most algorithms in order restricted statistical inference express the estimates in terms of certain summary statistics computed from pooled samples. These algorithms may or may not yield optimal estimates depending on whether or not the Cauchy mean value property holds strictly for the summary statistics. In this paper a minimum lower sets algorithm, which holds generally, is described and used to prove the optimality of estimates described by a max-min formula.

For each $x$ in some factor space $X$ an experiment can be performed whose outcome is $\{Y(x, t): t \in T \rbrack$ where $Y(x, t) = m_x(\theta, t) + \varepsilon(t)$. The zero mean error process $\varepsilon(t)$ has known covariance function $K$ and the maps $m_x$ (of known form) are linear from the parameter space $\Theta$ to the rkhs generated by $K$. Expressions for the variance of the umvlue of $\tau(\theta)$ (where $\tau$ is linear) are given which are analogous to the formulas in the finite dimensional $\Theta$ case. An Elfving's theorem is proved and a number of examples are given.

Let $\tau$ be a prior distribution over the parameter space $\Theta$ for a given parametric model $P_\theta, \theta \in \Theta$. For the sample space $\mathscr{X}$ (over which $P_\theta$'s are probability measures) belonging to a general class of topological spaces, which include the usual Euclidean spaces, it is shown that this parametric Bayes model can be approximated by a nonparametric Bayes model of the form of a mixture of Dirichlet processes prior, so that (i) the nonparametric prior assigns most of its weight to neighborhoods of the parametric model, and (ii) the Bayes rule for the nonparametric model is close to the Bayes rule for the parametric model in the no-sample case. Moreover, any prior parametric or nonparametric, may be approximated arbitrarily closely by a prior which is a mixture of Dirichlet processes. These results have implications in Bayesian inference.

In this note, we give a lemma which shows that the expected squared difference between the Bayes estimator with a Dirichlet process prior and the Kaplan-Meier product limit (PL) estimator for a survival function based on censored data is $O(n^{-2})$. This lemma, together with already proven pointwise consistency properties of the Bayes estimator, is used to establish two properties of the PL estimator; namely, the mean square consistency of the PL estimator with rate $O(n^{-1})$ and strong consistency of the PL estimator with rate $o(n^{-\frac{1}{2}} \log n)$.

A new bound is established for the Euclidean norm of the difference between the least squares estimator and the best linear unbiased estimator of the vector of expectations in the general linear model. The bound is valid regardless of the rank of the dispersion matrix and is expressed in substantially simpler terms than the bounds given earlier by Haberman and by Baksalary and Kala.

Brown, Cohen and Strawderman propose curtailed procedures for the $t$-test and Hotelling's $T^2$. In this paper we present the exact forms of these procedures and examine the expected sample size savings under the null hypothesis. The sample size savings can be bounded by a constant which is independent of the sample size. Tables are given for the expected sample size savings and maximum sample size saving under the null hypothesis for a range of significance levels $(\alpha)$, dimensions $(p)$ and sample sizes $(n)$.

The maximum likelihood estimators (m.l.e.) are obtained for the parameters of a bivariate normal distribution with equal variances when some of the observations are missing on one of the variables. The likelihood equation for estimating $\rho$, the correlation coefficient, may have multiple roots but a result proved here provides a unique root which is the m.l.e. of $\rho$. The problem of estimating the difference $\delta$ of the two means is also considered and it is shown that the m.l.e. of $\delta$ is unbiased.