In a smooth semiparametric estimation problem, the marginal posterior for the parameter of interest is expected to be asymptotically normal and satisfy frequentist criteria of optimality if the model is endowed with a suitable prior. It is shown that, under certain straightforward and interpretable conditions, the assertion of Le Cam’s acclaimed, but strictly parametric, Bernstein–von Mises theorem [Univ. California Publ. Statist. 1 (1953) 277–329] holds in the…
Introduction.- Asymptotic Inference for (Finite-Dimensional) Parametric Models.- Information Bounds for Euclidean Parameters in Infinite-Dimensional Models.- Euclidean Parameters: Further Examples.- Information Bounds for Infinite-Dimensional Parameters.- Infinite-Dimensional Parameters: Further Examples: Construction of Examples.
We consider in this paper two widely studied examples of nonparametric and semiparametric models in which the standard information bounds are totally misleading. In fact, no estimators converge at the $n^{-\alpha}$ rate for any $\alpha > 0$, although the information is strictly positive "promising" that $n^{-1/2}$ is achievable. The examples are the estimation of $\int p^2$ and the slope in the model of Engle et al. A class of models in which the parameter of interest can be…
We first discuss the construction of bootstrap confidence bands for the distribution function F of a population for simple random sampling but do not assume that F is continuous. The known alternative approach is to use the quantiles from the tabled Kolmogorov distribution. This approach is known to be conservative. It has already been shown in Bickel and Freedman (1981) that the bootstrap confidence band has the correct coverage probability asymptotically. We show by…
We consider efficient estimation of the slope in the errors in variables model with normal error when either the ratio of error variances is known and the distribution of the independent is arbitrary and unknown or the distribution of the independent variable is not Gaussian or degenerate. We calculate information bounds and exhibit estimates achieving these bounds using an initial minimum distance estimate and suitable estimates of the efficient score function.
An Edgeworth expansion with remainder $o(N^{-1})$ is established for a $U$-statistic with a kernel $h$ of degree 2. The assumptions involved appear to be very mild; in particular, the common distribution of the summands $h(X_i, X_j)$ is not assumed to be smooth.
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.
This paper is about the asymptotic distribution of linear combinations of stratum means in stratified sampling, with and without replacement. Both the number of strata and their size is arbitrary. Lindeberg conditions are shown to guarantee asymptotic normality and consistency of variance estimators. The same conditions also guarantee the validity of the bootstrap approximation for the distribution of the $t$-statistic. Via a bound on the Mallows distance, situations will be…
We simplify a general heuristic necessary condition of Stein's for adaptive estimation of a Euclidean parameter in the presence of an infinite dimensional shape nuisance parameter and other Euclidean nuisance parameters. We derive sufficient conditions and apply them in the construction of adaptive estimates for the parameters of linear models and multivariate elliptic distributions. We conclude with a review of issues in adaptive estimation.
The optimum design and its efficiency relative to the uniform design for estimating the mean of a stationary First-order autoregressive process plus an independent error are characterized completely. This is related to the work of Jones (1948), Optimum designs and variances of least squares estimates are calculated numerically for the problem of estimating the slope in a simple linear regression when the errors follow the above structure, for a range of values of the sample…
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…
Consider the problem of estimating the mean of a finite population on the basis of a simple random sample. It was proved by Aggarwal (1954) that the sample mean minimizes the maximum expected squared error divided by the population variance $\tau^2$. Aggarwal also stated, but did not successfully prove, that the sample mean minimizes the maximum expected squared error over the populations satisfying $\tau^2 \leq M$ for any fixed positive $M$. It is the purpose of this paper…
A new asymptotic theory for studying the effect of dependence of the observations in experimental design for the linear model is developed. The uniform design is shown to be asymptotically optimal in a strong sense for estimating location and in a weaker sense for estimating the slope of a straight line regression. Numerical results supporting the asymptotics appear in a companion paper.
Asymptotic expansions are established for the power of distribution-free tests in the two-sample problem. These expansions are then used to obtain deficiencies in the sense of Hodges and Lehmann for distribution-free tests with respect to their parametric competitors and for the estimators of shift associated with these tests.
We study the asymptotic power functions of tests for heteroscedasticity and nonlinearity in the linear model which were proposed by Anscombe and introduce and study some competitors robust against gross errors.
Measures of dispersion are defined as functionals satisfying certain equivariance and order conditions. In the main part of the paper attention is restricted to symmetric distributions. Different measures are compared in terms of asymptotic relative efficiency, i.e., the inverse ratio of their standardized variances. The efficiency of a trimmed to the untrimmed standard deviation turns out not to have a positive lower bound even over the family of Tukey models. Positive…
Asymptotic expansions are established for the power of distribution free tests in the one-sample problem. These expansions are then used to obtain deficiencies in the sense of Hodges and Lehmann (1970) for distribution free tests with respect to their parametric competitors and for the estimators of location associated with these tests.
Measures of location (without assumption of symmetry) are defined as functionals satisfying certain equivariance and order conditions. Three classes of such measures are discussed whose estimators are respectively linear functions of order statistics, $R$-estimators and $M$-estimators. It is argued that such measures can be compared in terms of the (asymptotic) efficiencies of their estimators. Of the three classes considered, it is found that trimmed expectations (and…
An overview is given of an approach to the definition of descriptive characteristics of populations and to their estimation. The emphasis is on the robustness and efficiency of the estimators. Detailed summaries will be found in successive papers of the series dealing with the problems of location, scale and kurtosis.
This is a survey of recent work on Edgeworth expansions for $(M)$ estimates, rank tests and some other statistics arising in nonparametric models. A Berry-Esseen theorem for $U$-statistics which seems to be new is also proved.
We consider the general linear model with independent symmetric errors. In this context we propose and examine the large sample behavior of some estimates of the regression parameters. For the location model these statistics are linear combinations of order statistics. In general they depend on a preliminary estimate and the ordered residuals based on it. The asymptotic efficiency of these procedures is independent of the design matrix. Specifically analogues of the median…
We show that, in a weak sense, as the cost of observation tends to zero, the shape of the continuation region of the Bayes solution for the exponential family problem given above is approximated by that of a corresponding problem for the Wiener process with drift. The approach is an extension of that used in Bickel and Yahav, Proc. Sixth Berkeley Symp. Math. Statist. Prob. (1971).
We consider in this paper two widely studied examples of nonparametric and semiparametric models in which the standard information bounds are totally misleading. In fact, no estimators converge at the n−α rate for any α>0, although the information is strictly positive "promising" that n−1/2 is achievable. The examples are the estimation of ∫p2 and the slope in the model of Engle et al. A class of models in which the parameter of interest can be…
The Edgeworth Expansion for U-Statistics of Degree Two
An Edgeworth expansion with remainder o(N−1) is established for a U-statistic with a kernel h of degree 2. The assumptions involved appear to be very mild; in particular, the common distribution of the summands h(Xi,Xj) is not assumed to be smooth.
This paper is about the asymptotic distribution of linear combinations of stratum means in stratified sampling, with and without replacement. Both the number of strata and their size is arbitrary. Lindeberg conditions are shown to guarantee asymptotic normality and consistency of variance estimators. The same conditions also guarantee the validity of the bootstrap approximation for the distribution of the t-statistic. Via a bound on the Mallows distance, situations will be…
If X is a N(θ,1) random variable, let ρ(m) be the minimax risk for estimation with quadratic loss subject to ∣θ∣≤m. Then ρ(m)=1−π2/m2+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−1cos2(2mπs),∣s∣≤m rather…
Consider the problem of estimating the mean of a finite population on the basis of a simple random sample. It was proved by Aggarwal (1954) that the sample mean minimizes the maximum expected squared error divided by the population variance τ2. Aggarwal also stated, but did not successfully prove, that the sample mean minimizes the maximum expected squared error over the populations satisfying τ2≤M for any fixed positive M. It is the purpose of this paper…
Measures of location (without assumption of symmetry) are defined as functionals satisfying certain equivariance and order conditions. Three classes of such measures are discussed whose estimators are respectively linear functions of order statistics, R-estimators and M-estimators. It is argued that such measures can be compared in terms of the (asymptotic) efficiencies of their estimators. Of the three classes considered, it is found that trimmed expectations (and…
This is a survey of recent work on Edgeworth expansions for (M) estimates, rank tests and some other statistics arising in nonparametric models. A Berry-Esseen theorem for U-statistics which seems to be new is also proved.
On the Asymptotic Shape of Bayesian Sequential Tests of θ≦0 Versus θ>0 for Exponential Families