A common reaction among applied statisticians is that the Bayesian statistician's energies in an applied problem must be directed at the a priori elicitation of one model specification from which an optimal design and all inferences follow automatically by applying Bayes's theorem to calculate conditional distributions of unknowns given knowns. I feel, however, that the applied Bayesian statistician's tool-kit should be more extensive and include tools that may be usefully labeled frequency calculations. Three types of Bayesianly justifiable and relevant frequency calculations are presented using examples to convey their use for the applied statistician.

Consider estimation of a Poisson mean $\lambda$ based on a single observation $x$, using estimator $d(x)$ and loss function $(d(x) - \lambda)^2/\lambda$. The goal is to decide (in)admissibility of $d(x)$. To every generalized Bayes estimator there corresponds a unique reversible birth and death process $\{X_t\}$ on $\mathbb{Z}_+$. Under side conditions $d(x)$ is admissible if and only if it is generalized Bayes and $\{X_t\}$ is recurrent. Explicit equivalent conditions exist in terms of difference equations and minimization problems. The theory is a discrete, univariate counterpart to Brown's (1971) diffusion characterization of admissibility in estimation of a multivariate normal mean. A companion paper discusses simultaneous estimation of several Poisson means.

Simultaneous confidence bounds for multilinear regression functions over subregions $X$ of Euclidean space are defined to be $\mu$-optimal in a class of bounds $C$, if they minimize average width with respect to $\mu$ over $X$, among all bounds in $C$ with equal coverage probability. We show that for certain simultaneous confidence bounds we can find a measure $\mu$ relative to which the bounds are $\mu$-optimal in $C$, where $C$ is a large class of bounds. Such results are obtained for bounds over finite sets, and for bounds for simple linear regression functions over finite intervals.

This paper is concerned with the problem of choosing a bandwidth parameter for nonparametric regression. We analyze a tapered Fourier series estimate and discuss the relationship of this estimate to a kernel estimate. We first consider a method based on an unbiased estimate of mean square error, and show that the bandwidth thus chosen is asymptotically optimal. Other methods are examined as well and are shown to be asymptotically equivalent. A small simulation shows, however, that for small or moderate sample size, the methods perform quite differently.

We consider the Parzen-Rosenblatt kernel density estimate on $\mathbb{R}^d$ with data-dependent smoothing factor. Sufficient conditions on the asymptotic behavior of the smoothing factor are given under which the estimate is pointwise consistent almost everywhere for all densities $f$ to be estimated. When the smoothing factor is a function only of the sample size $n$, it is shown that these conditions are also necessary, a generalization of results by Deheuvels. The consistency of various automatic kernel density estimates is a simple consequence of these theorems.

We consider the kernel estimate on the real line, $f_n(x) = (nh)^{-1} \sum^n_{i=1} K((X_i - x)/h),$ where $K$ is a bounded even density with compact support, and $X_1, \cdots, X_n$ are independent random variables with common density $f$. We treat the problem of placing a lower bound on the $L$1 error $J_n = E(\int |f_n - f|)$ which holds for all $f$. In particular, we show that there exist $A(K) \geq (9/125)^{1/5}$ depending only upon $K$, and $B^\ast(f) \geq 1$ depending only upon $f$ such that (i) for all $f:\inf_{h > 0}n^{2/5}J_n \geq CA(K)B^\ast(f) + o(1) \geq 0.6076703\cdots + o(1)$ where $C = 1.028493\cdots$ is a universal constant; (ii) for all $f$ with compact support and two bounded continuous absolutely integrable derivatives, $\inf_{h > 0}n^{2/5}J_n \leq C^\ast A(K)B^\ast(f) + o(1)$ where $C^\ast = 1.3768102\cdots$ is another universal constant. For this class of densities, we also obtain the exact asymptotic behavior of $J_n$.

A class of maximum penalized likelihood estimators (MPLE) of the density function $f$ is constructed, through the use of a rather general roughness-penalty functional. This class contains all the density estimates in the literature that arise as solutions to MPLE problems with penalties on $f^{1/2}$. In addition, the flexibility of the penalty functional permits the construction of new spline estimates with improved performance at the peaks and valleys of the density curves. The consistency of the estimators in probability and a.s., in the $L_p(\mathbb{R}) -$ norms, $p = 1, 2, \infty$, in the Hellinger metric and Sobolev norms is established in a unified manner. A class of penalty functionals is identified which leads to estimators which approach the optimal rates of convergence predicted in Farrell (1972). Based on the above estimates, a class of MPLE regression estimators is introduced which has the appealing property of reducing to the classical nonparametric regression estimates when a smoothing parameter goes to zero. Finally, a theoretically justifiable and numerically efficient method for a data based choice of the smoothing parameter is proposed for further study. A number of numerical examples and graphs are presented.

Kernel estimates of an unknown multivariate density are investigated, with mild restrictions being placed on the kernel. A window selection rule is considered, which can be interpreted in terms of cross-validation. Under the mild assumption that the unknown density and its one-dimensional marginals are bounded, the rule is shown to be asymptotically optimal. This strengthens recent results of Peter Hall.

Consider the general linear model $Y = x\beta + R$ with $Y$ and $R n$-dimensional, $\beta p$-dimensional, and $X$ an $n \times p$ matrix with rows $x'_i$. Let $\psi$ be given and let $\hat\beta$ be an $M$-estimator of $\beta$ satisfying $0 = \sum x_i\psi(Y_i - x'_i\hat\beta)$. Previous authors have considered consistency and asymptotic normality of $\hat\beta$ when $p$ is permitted to grow, but they have required at least $p^2/n \rightarrow 0$. Here the following result is presented: in typical regression cases, under reasonable conditions if $p(\log p)/n \rightarrow 0$ then $\|\hat{\beta} - \beta\|^2 = \mathscr{O}_p(p/n)$. A subsequent paper will show that $\hat{\beta}$ has a normal approximation in $R^p$ if $(p \log p)^{3/2}/n \rightarrow 0$ and that $\max_i|x'_i(\hat{\beta} - \beta)| \rightarrow_p 0$ (which would not follow from norm consistency if $p^2/n \rightarrow \infty$). In ANOVA cases, $\hat{\beta}$ is not norm consistent, but it is shown here that $\max|x'_i(\hat{\beta} - \beta)| \rightarrow_p 0$ if $p \log p/n \rightarrow 0$. A normality result for arbitrary linear combinations $a'(\hat{\beta} - \beta)$ is also presented in this case.

We consider the Bayes estimator $\delta_0$ for a Gaussian signal process observed in the presence of additive Gaussian noise under contamination of the signal or noise by QN-laws, introduced by Gualtierotti (1979). Upper bounds on the increase in the mean square error of $\delta_0$ over the minimum possible mean square error under contaminated noise or contaminated signal are given. It is shown that the performance of $\delta_0$ is relatively close to optimal for small amounts of contamination.

Theory for finding designs in estimating a linear functional of a regression function is developed for classes of regression functions which are infinite dimensional. These classes can be viewed as representing possible departures from an "ideal" simple model and thus describe a model robust setting. The estimates are restricted to be linear and the design (and estimate) sought is minimax for mean square error. The structure of the design is obtained in a variety of cases; some asymptotic theory is given when the functionals are integrals. As to be expected, optimal designs depend critically on the particular functional to be estimated. The associated estimate is generally not a least squares estimate but we note some examples where a least squares estimate, in conjunction with a good design, is adequate.

We study robust estimation in the general normal regression model with random carriers permitting small departures from the model. The framework is that of Bickel (1981). We obtain solutions of Huber (1982), Krasker-Hampel (1980) and Krasker-Welsch (1982) as special cases as well as some new procedures. Our calculations indicate that the optimality properties of these estimates are more limited than suggested by Krasker and Welsch.

A general class of statistical functionals is introduced and the asymptotic normality of the corresponding estimators is established. Included are certain measures of spread, proposed by Bickel and Lehmann (1979), for which the asymptotic distribution theory has been an open problem. The proofs involve some new results, of independent interest, for empirical and quantile processes.

Suppose $F$ is a mixture of a known parametric family of distributions with an unknown nonparametric mixing distribution. Consider the problem of estimating the value $T(dF)$ of a smooth functional $T$ at the mixed distribution $F$. One nonparametric estimator of $T(dF)$ is $T(dF_n)$ where $F_n$ denotes the empirical distribution. Under a differentiability condition on $T$ given in this note, $T(dF_n)$ is shown to be fully efficient asymptotically in the sense that the limiting distribution of any other estimator of $T(dF)$ that is regular in the sense of Hajek and Beran can be expressed as a convolution of the limiting normal distribution of $T(dF_n)$ with another distribution. The differentiability condition is verified for the case that the parametric family being mixed is a one-parameter exponential family and the support of the mixing distribution is an infinite set with a finite accumulation point. Regularity is verified for the maximum likelihood estimator of the probability of zero in a mixed Poisson distribution under certain conditions on the mixing distribution. Moreover, the maximum likelihood estimator and the sample proportion of zeroes are both shown to be fully efficient in this example.

This paper considers the asymptotic behavior of the maximum likelihood estimators (mle's) of the probabilities of a mixed Poisson distribution with a nonparametric mixing distribution. The vector of estimated probabilities is shown to converge in probability to the vector of mixed probabilities at rate $n^{1/2-\varepsilon}$ for any $\varepsilon > 0$ under a generalized $\chi^2$ distance function. It is then shown that any finite set of the mle's has the same joint limiting distribution as does the corresponding set of sample proportions when the support of the mixing distribution $G_0$ is an infinite set with a known upper bound and $G_0$ satisfies a certain condition at zero.

Different estimates of the mean squared error of prediction for linear regression models are derived by the bootstrap and cross-validation approaches. A comparison is made under normal error distributions, especially by the biases and the mean square errors. The results indicate that the bias corrected bootstrap estimator is best unbiased and should be the first choice, while its simulated variant has approximately the same behaviour. On the other hand, if only a comparison between uncorrected estimators is made (with implications for nonlinear regression models in mind), then other variants of bootstrap estimates are preferable for a large or a small dimension of the model parameter. For a small dimension, the cross-validation estimate and sometimes grouped variants of it seem also to be acceptable if the model error is known to be small.

In Hannan (1980), some limiting properties of the order selection criteria, AIC, BIC, and $\phi(p, q)$ for modeling stationary time series were derived. In this paper, we generalize these properties to the case in which the underlying process follows a nonstationary autoregressive model. We show that BIC and $\phi(p, 0)$ are weakly consistent. For the AIC, we prove that the asymptotic distribution given by Shibata (1976) for the stationary autoregressive models continues to hold.

Consider the problem of estimating, in a Bayesian framework and in the presence of additive Gaussian noise, a signal which is a step function. Best linear estimates and Bayes estimates are derived, evaluated and compared. A characterization of the Bayes estimates is presented. This characterization has an intuitive interpretation and also provides a way to compute the Bayes estimates with a number of operations of the order of $T^3$ where $T$ is the fixed time span. An approximation to the Bayes estimates is proposed which reduces the total number of operations to the order of $T$. The results are applied to the case where the Bayesian model fails to be satisfied using an empirical Bayes approach.

We consider linear estimators for the mean $\theta$ of a Gaussian distribution $N(\theta, C)$ on a Hilbert space, when the covariance operator $C$ is known. It was argued in a previous work that the natural class of linear estimators is the class of measurable linear transformations. Using the simplest quadratic loss we prove that the linear estimator $L$ is admissible if and only if the operator $C^{-1/2}LC^{1/2}$ is Hilbert-Schmidt, self-adjoint, its eigenvalues are all between 0 and 1 and two are equal to 1 at the most. As an application of the general theory, we investigate some linear estimators for the drift function of a Brownian motion.

An estimate of the upper tail of a distribution function which is based on the upper $m$ order statistics from a sample of size $n(m \rightarrow \infty, m/n \rightarrow 0$ as $n \rightarrow \infty)$ is shown to be consistent for a wide class of distribution functions. The empirical mean residual life of the $\log$ transformed data and the sample $1 - m/n$ quantile play a key role in the estimate. The joint asymptotic behavior of the empirical mean residual life and sample $1 - m/n$ quantile is determined and rates of convergence of the estimate to the tail are derived.

An analysis of variance (ANOVA) is defined to be a partition of the total sum of squares into independent terms which, when suitably scaled, are chi-squared variables. A partition of less than the total sum of squares, but with these properties, will often suffice and is referred to as a partial ANOVA. Conditions for an ANOVA, and for partial ANOVAs selected to contain only specific parameters, are given. Implications for estimation of variance components from an ANOVA are also discussed. These results are largely an extension of work by Graybill and Hultquist (1961). With unbalanced data, conditions for an ANOVA and the number of terms in it both can depend on which effects in the model are fixed and which are random. This is not taken into account by those procedures for partitioning a sum of squares which distinguish between random and fixed effects only in the calculation of expected mean squares. Several examples are given.

A simple method of directly obtaining generators for the intrablock subgroup and for the interblock subgroup of a confounding plan is presented. The method is applicable to completely general factorial experiments and can be used either to construct the design which confounds a specified set of treatment effects or to determine the confounding pattern of a specified design. This procedure avoids the usual trial and error approach to confounding and involves nothing more complicated than addition and multiplication modulo a prime number.

In this paper we consider exponential families of distributions and obtain under certain conditions a uniform large deviation result about the tail probability $P_\partial(\phi_\partial(\bar{X}_n) > \varepsilon), \varepsilon > 0$, where $\partial$ is the natural parameter and $\phi_\partial(\bar{X}_n)$ is the $\log$ likelihood ratio statistic for testing the null hypothesis $\{\partial\}$. The technique involves approximating certain convex compact sets in $R^k$ by polytopes, then estimating the probability contents of associated closed halfspaces, and counting the number of these half-spaces. Some examples are given, among them the multivariate normal distribution with unknown mean vector and covariance matrix.

A theorem of Bahadur on the asymptotic optimality of the likelihood ratio statistic has been extended to sequential analysis by Berk and Brown (1978) in the context of testing one-sided hypotheses about the mean of a normal distribution with known variance. In this work, Bahadur's theorem is extended to sequential analysis for general hypotheses about the parameters of an exponential family of distributions. Specifically, it is shown that, under certain conditions, modifications of the likelihood ratio statistic analogous to those exhibited by Berk and Brown (1978) in the above normal context are optimal for any family of stopping times approaching $\infty$. These results indicate that Bahadur efficiency has a limited impact in sequential analysis.

In 1974 Karlin introduced the concept of random replacement schemes and conjectured that the componentwise monotonicity of the replacement probabilities (condition A) is equivalent to a corresponding ordering of expectations of all functions $\phi$ from a certain class $\mathscr{C}_K$ (condition B). In this paper it is shown that A implies B for sample sizes $n \leq 5$ and--provided the sample space is sufficiently large--also for $n \geq 6$. By a counterexample it is shown that $\mathscr{C}_K$ is not suitable for A being implied by B, i.e. one direction of Karlin's conjecture is disproved.

As an estimator of the population mean, the sample mean based only on the distinct units possesses a remarkable invariance property. Under three forms of simple random sampling, viz. simple random sampling without replacement (SRSWOR), simple random sampling with replacement (SRSWR), and fixed cost simple random sampling (SRSFC), it is admissible and unbiased; and asymptotically normally distributed if and only if the Erdos-Renyi-Hajek condition is satisfied. An important implication of this invariance is that for estimating the population mean, these forms of simple random sampling are asymptotically equally cost-efficient. However, from a practical point of view SRSFC does seem to provide greater flexibility in large surveys.

The problem of estimating an endpoint of a distribution is revisited, using the bootstrap and random subsample methods. Contrary to an example in Bickel and Freedman (1981) suggesting that these methods do not work here, it is shown that one can in fact construct asymptotically valid confidence intervals in some situations.

Suppose that you specify your prior probability that an unknown quantity $\theta$ lies in each member of a disjoint partition of the values of $\theta$. What does this imply about your prior mean and variance for $\theta$, and your posterior mean and variance, given sample information? We provide a partial answer by modifying a suggestion of Manski for incorporating the cost of specification of prior probabilities into the analysis of decision problems. This modification leads to a simple explicit solution in the problem of estimating the mean of a distribution, with quadratic loss, in the class of linear functions of the sample, and this solution is related to the problem of turning probabilities into expectations.

An extension of Ferguson's (Fourth Berkeley Symposium on Probability and Mathematical Statistics, 1961, Volume 1) univariate normal results and Schwager and Margolin's (1982) multivariate normal results for detection of outliers is made to the multivariate elliptically symmetric case with mean slippage. The main result can be viewed as a robustness property of the use of Mardia's multivariate kurtosis statistic as a locally optimum test statistic to detect outliers against nonnormal multivariate distributions.

Under a quadratic subspace condition, Pukelsheim (1981a) has proved that for estimating linear combinations of variance components, either standard methods provide a nonnegative quadratic unbiased estimate or such an estimate does not exist. This result is proved, replacing the quadratic subspace condition by a weaker condition. This answers in the negative a question raised by Pukelsheim (1981b).

An estimate $\sum^n_{i=1} Y_iK((x - X_i)/h)/\sum^n_{j=1} K((x - X_j)/h)$, calculated from a sequence $(X_1, Y_1), \cdots, (X_n, Y_n)$ of independent pairs of random variables distributed as a pair $(X, Y)$, converges to the regression $E\{Y\mid X = x\}$ as $n$ tends to infinity in probability for almost all $(\mu) x \in R^d$, provided that $E|Y| < \infty, h \rightarrow 0$ and $nh^d \rightarrow \infty$ as $n \rightarrow \infty$. The result is true for all distributions $\mu$ of $X$. If, moreover, $|Y| \leq \gamma < \infty$ and $nh^d/\log n \rightarrow \infty$ as $n \rightarrow \infty$, a complete convergence holds. The class of applicable kernels includes those having unbounded support.

An asymptotic expansion of the nonnull distribution of the Wilks statistic for testing the linear hypothesis in multivariate analysis of variance is obtained up to the order $N^{-2}$ where $N$ is the sample size, for the first time in terms of noncentral beta distributions. The asymptotic distributions are better than the ones available in Anderson (1958) in the null case and in Sugiura and Fujikoshi (1969) and Posten and Bergman (1964) in the nonnull case. In fact, for certain parameters the asymptotic expansion reduces to the first term and we get the exact distribution.

Consistency in the undominated location model is investigated from a Bayesian point of view and a proof on the consistency of the Bayes procedures with respect to the invariant prior is provided. The consistency of Bayes procedures with respect to other prior measures is established as a corollary.