A flexible class of prior distributions is proposed, for the covariance matrix of a multivariate normal distribution, yielding much more general hierarchical and empirical Bayes smoothing and inference, when compared with a conjugate analysis involving an inverted Wishart distribution. A likelihood approximation is obtained for the matrix logarithm of the covariance matrix, via Bellman's iterative solution to a Volterra integral equation. Exact and approximate Bayesian, empirical and hierarchical Bayesian estimation and finite sample inference techniques are developed. Some risk and asymptotic frequency properties are investigated. A subset of the Project Talent American High School data is analyzed. Applications and extensions to multivariate analysis, including a generalized linear model for covariance matrices, are indicated.

This paper addresses the problem of quantifying expert opinion about a normal linear regression model when there is uncertainty as to which independent variables should be included in the model. Opinion is modeled as a mixture of natural conjugate prior distributions with each distribution in the mixture corresponding to a different subset of the independent variables. It is shown that for certain values of the independent variables, the predictive distribution of the dependent variable simplifies from a mixture of $t$-distributions to a single $t$-distribution. Using this result, a method of eliciting the conjugate distributions of the mixture is developed. The method is illustrated in an example.

As a first step toward developing statistical models based on upper and lower probabilities, we study upper probabilities and upper expectations on the unit interval that are symmetric, by which we mean invariant with respect to equimeasurability. These upper probabilities are generalizations of uniform probability measures. We give some characterizations of these upper probabilities. Specifically, we show that symmetry of the upper expectation functional is equivalent to the underlying set of densities being closed under majorization. We also show that a function is the upper distribution for a symmetric upper probability if and only if its lower graph is star-shaped with respect to the origin and to the point (1,1). We derive inner and outer approximations to symmetric classes of probabilities based on the upper probability. The class of symmetric upper expectations that are completely determined by their values on the indicator functions is characterized. We provide a geometric characterization of a hierarchy of upper probabilities including Fine's generalized upper probabilities and 2-alternating Choquet capacities. In particular, we establish a 1-1 correspondence between symmetric, 2-alternating capacities and nonincreasing density functions. We prove that undominated generalized upper probabilities do not exist in the symmetric case. Examples from robust statistics are considered. An example is given that shows that symmetry of upper probabilities does not imply symmetry of upper expectations. A corollary is that symmetry of the Choquet integral does not imply symmetry of the upper expectation functional.

This paper presents a quasi-Bayesian model of subjective uncertainty in which beliefs which are represented by lower and upper probabilities qualified by numerical confidence weights. The representation is derived from a system of axioms of binary preferences which differs from standard axiom systems insofar as completeness is not assumed and transitivity is weakened. Confidence-weighted probabilities may be elicited through the acceptance of bets with limited stakes, a generalization of the operational method of de Finetti. The model is applicable to the reconciliation of inconsistent probability judgments and to the sensitivity analysis of Bayesian decision models.

In this paper, we outline a general approach to estimating the parametric component of a semiparametric model. For the case of a scalar parametric component, the method is based on the idea of first estimating a one-dimensional subproblem of the original problem that is least favorable in the sense of Stein. The likelihood function for the scalar parameter along this estimated subproblem may be viewed as a generalization of the profile likelihood for the problem. The scalar parameter is then estimated by maximizing this "generalized profile likelihood." This method of estimation is applied to a particular class of semiparametric models, where it is shown that the resulting estimator is asymptotically efficient.

We describe multivariate generalizations of the median, trimmed mean and $W$ estimates. The estimates are based on a geometric construction related to "projection pursuit." They are both affine equivariant (coordinate-free) and have high breakdown point. The generalization of the median has a breakdown point of at least $1/(d + 1)$ in dimension $d$ and the breakdown point can be as high as $1/3$ under symmetry. In contrast, various estimators based on rejecting apparent outliers and taking the mean of the remaining observations have breakdown points not larger than $1/(d + 1)$ in dimension $d$.

Rousseeuw's minimum volume estimator for multivariate location and dispersion parameters has the highest possible breakdown point for an affine equivariant estimator. In this paper we establish that it satisfies a local Holder condition of order $1/2$ and converges weakly at the rate of $n^{-1/3}$ to a non-Gaussian distribution.

We provide a solution to the smoothing parameter selection problem involved in the construction of adaptive estimates for the symmetric location model and the general linear model. Linear $B$-splines are used to give a simple form of the estimate of the score function of the underlying density. New empirical methods are proposed to locate the knots optimally and to select the number of knots. We also give asymptotic bounds for the empirical selection method and show that an estimate with an empirically selected smoothing parameter is adaptive. Our estimates are easy to compute and possess useful computational features. Simulation studies reveal that our estimates perform well in comparison with some well-known estimates.

Orthogonal regression $M$-estimates are considered from a bias robust point of view. Their maximum bias over epsilon-contamination neighborhoods is characterized, and maximum bias curves are computed. The most bias robust orthogonal regression $M$-estimate is derived and shown to be a "mode type" estimate; for instance, in the two-dimensional case this estimate can be computed by locating a strip of fixed width covering the maximum number of data points. It will be shown that, although orthogonal regression $M$-estimates with bounded loss function have unbounded influence function, the derivative of their maximum bias curve at zero is finite. Finally, an implicit formula for an upper bound for the breakdown point of all orthogonal regression $M$-estimates is found. The upper bound, which depends on the signal-to-noise ratio, is sharp and attained by the most bias robust estimate.

We study the asymptotic behavior of Mardia's measure of (sample) multivariate skewness. In the special case of an elliptically symmetric distribution, the limit law is a weighted sum of two independent $\chi^2$-variates. A normal limit distribution arises if the population distribution has positive skewness. These results explain some curiosities in the power performance of a commonly proposed test for multivariate normality based on multivariate skewness.

By providing a probabilistic model for nested case-control sampling in epidemiologic cohort studies, consistency and asymptotic normality of the maximum partial likelihood estimator of regression parameters in a Cox proportional hazards model can be derived using process and martingale theory as in Andersen and Gill. A general expression for the asymptotic variance is given and used to calculate asymptotic relative efficiencies relative to the full cohort variance in some important special cases.

We consider the effects of misspecifying the covariate vector on tests of association between a particular covariate and the response variable in logistic regression models. Misspecification can include random measurement error, discretizing a continuous explanatory variable and mismodeling the functional form of an explanatory variable. The effects of misspecification are examined by deriving the asymptotic distribution of the test statistic under a sequence of models. Applications of this result are discussed.

Let $(\mathbf{X}_i, Y_i)$ be independent, identically distributed observations that satisfy a logistic regression model; that is, for each $i, \log \lbrack P(Y_i = 1 | \mathbf{X}_i)/P(Y_i = 0 |\mathbf{X}_i)\rbrack = \mathbf{X}^T_i \beta_0$, where $Y_i \in \{0, 1\}, \mathbf{X}_i \in \mathbf{R}^p$ and $\beta_0 \in \mathbf{B}^p$ is the unknown parameter vector of the model. The marginal distribution of the covariate vectors $\mathbf{X}_i$ is assumed to be unknown. Sequential procedures for constructing fixed size and fixed proportional accuracy confidence regions for $\beta_0$ are proposed and shown to be asymptotically efficient as the size of the region becomes small.

Random coefficient regression models are important in representing linear models with heteroscedastic errors and in unifying the study of classical fixed effects and random effects linear models. For prediction intervals and for bootstrapping in random coefficient regressions, it is necessary to estimate the distributions of the random coefficients consistently. We show that this is often possible and provide practical representative estimators of these distributions.

In 1989 Kunsch introduced a modified bootstrap and jackknife for a statistic which is used to estimate a parameter of the $m$-dimensional joint distribution of stationary and $\alpha$-mixing observations. The modification amounts to resampling whole blocks of consecutive observations, or deleting whole blocks one at a time. Liu and Singh independently proposed (in 1988) the same technique for observations that are $m$-dependent. However, many time-series statistics, notably estimators of the spectral density function, involve parameters of the whole (infinite-dimensional) joint distribution and, hence, do not fit in this framework. In this report we generalize the "moving blocks" resampling scheme of Kunsch and Liu and Singh; a still modified version of the nonparametric bootstrap and jackknife is seen to be valid for general linear statistics that are asymptotically normal and consistent for a parameter of the whole joint distribution. We then apply this result to the problem of estimation of the spectral density.

In this paper we introduce an appealing nonparametric method for estimating the mean regression function. The proposed method combines the ideas of local linear smoothers and variable bandwidth. Hence, it also inherits the advantages of both approaches. We give expressions for the conditional MSE and MISE of the estimator. Minimization of the MISE leads to an explicit formula for an optimal choice of the variable bandwidth. Moreover, the merits of considering a variable bandwidth are discussed. In addition, we show that the estimator does not have boundary effects, and hence does not require modifications at the boundary. The performance of a corresponding plug-in estimator is investigated. Simulations illustrate the proposed estimation method.

We consider the Akaike-Parzen-Rosenblatt density estimate $f_{nh}$ based upon any superkernel $L$ (i.e., an absolutely integrable function with $\int L = 1$, whose characteristic function is 1 on $\lbrack -1, 1\rbrack)$, and compare it with a kernel estimate $g_{nh}$ based upon an arbitrary kernel $K$. We show that for a given subclass of analytic densities, $\inf_L \sup_K \lim \sup_{n\rightarrow \infty} \frac{\inf_h \mathbb{E} \int |f_{nh} - f|}{\inf_h \mathbb{E} \int |g_{nh} - f |} = 1,$ where $h > 0$ is the smoothing factor. Thus, asymptotically, the class of superkernels is as good as any other class of kernels when certain analytic densities are estimated. We also obtain exact asymptotic expressions for the expected $L_1$ error of the kernel estimate when superkernels are used.

For the data based choice of the bandwidth of a kernel density estimator, several methods have recently been proposed which have a very fast asymptotic rate of convergence to the optimal bandwidth. In particular the relative rate of convergence is the square root of the sample size, which is known to be the best possible. The point of this paper is to show how semiparametric arguments can be employed to calculate the best possible constant coefficient, that is, an analog of the usual Fisher information, in this convergence. This establishes an important benchmark as to how well bandwidth selection methods can ever hope to perform. It is seen that some existing methods attain the bound, others do not.

Two new statistics for testing goodness-of-fit are derived from the viewpoint of nonparametric density estimation. These statistics are closely related to the Neyman smooth and Cramer-von Mises statistics but are shown to have superior properties both through asymptotic and small sample analyses. Comparison of the proposed tests with the Cramer-von Mises statistic requires the development of a novel technique for comparing tests that are capable of detecting local alternatives converging to the null at different rates.

It is well known that anomalies are sometimes observed when using the likelihood ratio test (LRT) for testing restricted hypotheses in a normal model. This paper considers a general framework for these anomalies to occur. We provide a condition, that relates the null and the alternative hypotheses, under which the dominance of the LRT is obtained. Conditions are also given which guarantee the equivalence between the LRT and a simpler test. The situations of known and unknown variances are considered and examples are given to illustrate the results.

Statistical concepts of order permeate the theory and practice of statistics. The present paper is concerned with a large class of directional orderings of univariate distributions. (What do we mean by saying that a random variable $Y$ is larger than another random variable $X$?) Attention is restricted to preorders that are invariant under monotone transformations; this includes orderings such as monotone likelihood ratio, hazard ordering, and stochastic ordering. Simple characterizations of these orderings are obtained in terms of a maximal invariant. It is shown how such invariant preorderings can be used to generate concepts of $Y_2$ being further to the right of $X_2$ than $Y_1$ is of $X_1$.

The $k$-armed bandit problem on the Bernoulli dependent arms is discussed. Order relations on the prior distributions of the Bernoulli parameters using moments of the posterior are used to prove a monotonicity property of the value function. When $k = 2$, a stay-with-a-winner rule is derived for negatively correlated arms and for a certain class of positively correlated arms. These results are extensions of those given in Berry and Fristedt for independent Bernoulli arms. They also generalize the results of Benzing, Hinderer and Kolonko and Kolonko and Benzing.

In this paper we find several interesting properties of $2^{n-k}$ fractional factorial designs. An upper bound is given for the length of the longest word in the defining contrasts subgroup. We obtain minimum aberration $2^{n-k}$ designs for $k = 5$ and any $n$. Furthermore, we give a method to test the equivalence of fractional factorial designs and prove that minimum aberration $2^{n - k}$ designs for $k \leq 4$ are unique.

General techniques for the construction of asymmetrical orthogonal arrays of strength 2 are presented. These are then applied to special cases to obtain new families of such arrays. Among these are saturated main-effect plans based on $s^m$ runs with factors at $s^{\nu_i}$ levels, $i = 0, 1, \ldots, r,$ where $m \geq v_r, v_0 = 1, v_{i-1}$ divides $v_i, i = 1,2,\ldots, r$, and $s$ is a prime power.

The importance of optimal filtration problem for Markov chain with two states observed in Gaussian white noise (GWN) for a lot of concrete technical problems is well known. The equation for a posterior probability $\pi(t)$ of one of the states was obtained many years ago. The aim of this paper is to study a simple filtration method. It is shown that this simplified filtration is asymptotically efficient in some sense if the diffusion constant of the GWN goes to 0. Some advantages of this procedure are discussed.

The problem of combining high efficiency with high breakdown properties for regression estimators has piqued the interest of statisticians for some time. One proposal specifically suggested by Rousseeuw and Leroy is to use the least median of squares estimator, omit observations whose residuals are larger than some constant cut-off value and apply least squares to the remaining observations. Although this proposal does retain high breakdown point, it actually converges no faster than the initial estimator. In fact, the reweighted least squares estimator is asymptotically a constant times the initial estimator if the initial estimator converges at a rate strictly slower than $n^{-1/2}$.

The Robbins-Monro process $X_{n+1} = X_n - c_n Y_n$ is a standard stochastic approximation method for estimating the root $\theta$ of an unknown regression function. There is a vast literature on the convergence properties of $X_n$ to $\theta$. In practice, one is also interested in the conditional distribution of the system under the sequential control when the control is set at $\theta$ or near $\theta$. This problem appears to have received no attention in the literature. We introduce an estimator using methods of nonparametric conditional quantile estimation and derive its asymptotic properties.

Density ratio neighborhoods are classes of probabilities that are used in robust Bayesian inference. These classes are invariant under Bayesian updating and marginalization. This makes them computationally convenient in robust Bayesian inference. We show that this is the unique class of probabilities that has these invariance properties. Aside from its theoretical value, this result has computational implications as well.

Let $X_1, X_2, \ldots, X_k$ be $k$ independent gamma random variables with different scale parameters but with a common known shape parameter. Suppose the population corresponding to the largest $X_{(1)}$ [or the smallest $X_{(k)}$] observation is selected. The problem of estimating the scale parameter $\theta_{(1)}$ [or $\theta_{(k)}$] of the selected population is considered. We derive, using the method of differential inequalities, explicit estimators that dominate the natural or the existing estimators. The improved estimators of $\theta_{(1)}$ are similar to that of DasGupta estimators for the usual simultaneous estimation problem. An implication of this result for the simultaneous estimation of the selected subset is also considered.

For the invariant decision problem of estimating a continuous distribution function with the Kolmogorov-Smirnov loss, it is proved that the best invariant estimator is minimax.