An unknown constant matrix M is observed with additive random error. The basic problem considered is to devise an estimator of M that trades off bias against variance so as to achieve relatively low quadratic risk. This paper develops an adaptive total least squares estimator and an adaptive total shrinkage estimator of M that minimize estimated risk over certain large classes of linear estimators. It is shown that the asymptotic risk of the adaptive total least squares…
This paper constructs improved estimators of the means in the Gaussian saturated one-way layout with an ordinal factor. The least squares estimator for the mean vector in this saturated model is usually inadmissible. The hybrid shrinkage estimators of this paper exploit the possibility of slow variation in the dependence of the means on the ordered factor levels but do not assume it and respond well to faster variation if present. To motivate the development, candidate…
REACT estimators for the mean of a linear model involve three steps: transforming the model to a canonical form that provides an economical representation of the unknown mean vector, estimating the risks of a class of candidate linear shrinkage estimators, and adaptively selecting the candidate estimator that minimizes estimated risk. Applied to one- or higher-way layouts, the REACT method generates automatic scatterplot smoothers that compete well on standard datasets with…
REACT estimators for the mean of a linear model involve three steps: transforming the model to a canonical form that provides an economical representation of the unknown mean vector, estimating the risks of a class of candidate linear shrinkage estimators, and adaptively selecting the candidate estimator that minimizes estimated risk. Applied to one- or higher-way layouts, the REACT method generates automatic scatterplot smoothers that compete well on standard datasets with…
An unknown signal plus white noise is observed at $n$ discrete time points. Within a large convex class of linear estimators of $\xi$, we choose the estimator $\hat{\xi}$ that minimizes estimated quadratic risk. By construction, $\hat{\xi}$ is nonlinear. This estimation is done after orthogonal transformation of the data to a reasonable coordinate system. The procedure adaptively tapers the coefficients of the transformed data. If the class of candidate estimators satisfies…
Samples of directional or axial measurements arise in geophysical, biological and econometric contexts. We represent the rotational difference between two mean directions (or two mean axes) as a direction (or axis). We then construct nonparametric simultaneous confidence sets for all pair-wise rotational differences among the mean directions or mean axes of $s$ samples. By specialization, this methodology yields nonparametric simultaneous tests for pairwise equality of…
Rudolf Beran · Andrey Feuerverger · Peter Hall·刊期 1996-12-01 · 卷 24（6）·The Annals of Statistics已编入正刊
随机系数回归中截距与斜率分布的非参数估计AI 解读
An experiment records stimulus and response for a random sample of cases. The relationship between response and stimulus is thought to be linear, the values of the slope and intercept varying by case. From such data, we construct a consistent, asymptotically normal, nonparametric estimator for the joint density of the slope and intercept. Our methodology incorporates the radial projection-slice theorem for the Radon transform, a technique for locally linear nonparametric…
Consider the problem of constructing a prediction region $D_n$ for a potentially observable variable $X$ on the basis of a learning sample of size $n$. Usually, the requirement that $D_n$ contain $X$ with probability $\alpha$, conditionally on the learning sample, does not uniquely determine $D_n$. This paper develops a general probability-centering concept for prediction regions that extends to vector-valued or function-valued $X$ the classical notion of an equal-tailed…
SUMMARY In several important statistical problems, prediction intervals and confidence intervals can be constructed with coverage levels which are known precisely but cannot be rendered equal to predetermined levels such as 0.95. One solution to this difficulty is to interpolate between such intervals. We show that simple linear interpolation reduces the order of coverage error, but that higher orders of interpolation produce no further improvement. The error is reduced by a…
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.
Suppose that the variable X to be predicted and the learning sample Yn that was observed have a joint distribution, which depends on an unknown parameter θ. The parameter θ can be finite- or infinite-dimensional. A prediction region Dn for X is a random set, depending on Yn , that contains X with prescribed probability α. This article studies methods for controlling simultaneously the conditional coverage proability of Dn , given Yn , and the overall (unconditional) coverage…
Simultaneous confidence sets for a collection of parametric functions may be constructed in several different ways. These ways include: (a) the exact pivotal method that underlies Tukey's (1953) and Scheffé's (1953) simultaneous confidence intervals for linear parametric functions in the normal linear model; (b) the method of asymptotic pivots, which is an approximate extension of the pivotal method; (c) the method of bootstrapped roots developed in Beran (1988). These three…
Suppose that T(θ) = {Tu (θ)} is a family of parametric functions indexed by the variable u. For each u, an approximate confidence set Cn,u for Tu (θ) may be obtained by referring a function of Tu (θ) and the sample to an estimated quantile dn,u of that function's sampling distribution. This approach is sometimes called the pivotal method for constructing a confidence set, even when it is not based on a true pivot. A simultaneous confidence set Cn for T(θ) is then obtained by…
Approximate tests for a composite null hypothesis about a parameter θ may be obtained by referring a test statistic to an estimated critical value. Either asymptotic theory or bootstrap methods can be used to estimate the desired quantile. The simple asymptotic test ϕA refers the test statistic to a quantile of its asymptotic null distribution after unknown parameters have been estimated. The bootstrap approach used here is based on the concept of prepivoting. Prepivoting is…
Approximate confidence sets for a parameter θ may be obtained by referring a function of θ and of the sample to an estimated quantile of that function's sampling distribution. We call this function the root of the confidence set. Either asymptotic theory or bootstrap methods can be used to estimate the desired quantile. When the root is not a pivot, in the sense of classical statistics, the actual level of the approximate confidence set may differ substantially from the…
Approximate confidence sets for a parameter θ may be obtained by referring a function of θ and of the sample to an estimated quantile of that function's sampling distribution. We call this function the root of the confidence set. Either asymptotic theory or bootstrap methods can be used to estimate the desired quantile. When the root is not a pivot, in the sense of classical statistics, the actual level of the approximate confidence set may differ substantially from the…
Tests for a null hypothesis whose specification involves an unknown nuisance parameter may be obtained by inverting a bootstrap confidence region for the parameter being tested or by constructing a simulated null distribution for the test statistic. The power of either test against certain alternatives involving the same unknown nuisance parameter can itself be estimated by simulation.
Bootstrap tests and confidence regions for functions of the population covariance matrix have the desired asymptotic levels, provided model restrictions, such as multiple eigenvalues in the covariance matrix, are taken into account in designing the bootstrap algorithm.
Let $\hat{T}_n$ be an estimate of the form $\hat{T}_n = T(\hat{F}_n)$, where $\hat{F}_n$ is the sample $\operatorname{cdf}$ of $n \operatorname{iid}$ observations and $T$ is a locally quadratic functional defined on $\operatorname{cdf's}$. Then, the normalized jackknife estimates for bias, skewness, and variance of $\hat{T}_n$ approximate closely their bootstrap counterparts. Each of these estimates is consistent. Moreover, the jackknife and bootstrap estimates of variance…
This paper concerns robust estimation of the parameter $\theta$ which indexes a parametric model for independent non-identically distributed data. For reasonable choices of contamination neighborhood and of what is to be estimated when the parametric model does not hold, we characterize asymptotically minimax robust estimates of $\theta$. When applied to the normal regression model, the theory yields recipes for the influence curves of optimal robust regression and scale…
Let $X_1, X_2, \cdots, X_n$ be i.i.d random variables with d.f. $F$. Suppose the $\{\hat{T}_n = \hat{T}_n(X_1, X_2, \cdots, X_n); n \geq 1\}$ are real-valued statistics and the $\{T_n(F); n \geq 1\}$ are centering functionals such that the asymptotic distribution of $n^{1/2}\{\hat{T}_n - T_n(F)\}$ is normal with mean zero. Let $H_n(x, F)$ be the exact d.f. of $n^{1/2}\{\hat{T}_n - T_n(F)\}$. The problem is to estimate $H_n(x, F)$ or functionals of $H_n(x, F)$. Under…
Robustness and efficiency of a parameter estimate $T$ can be assessed by comparing the fitted parametric distribution $P_T$ with the actual distribution, which is assumed to lie near the parametric family $\{P_\theta:\theta\in\Theta\}$. Asymptotic lower bounds are established for the minimax risk over distributions near the parametric model, taking as loss function a monotone increasing function of the Hellinger distance between the actual distribution of the sample and the…
A rotationally invariant exponential model, which includes the Fisher-von Mises and Bingham distributions as special cases, is proposed for directional data in $R^p(p \geqslant 2)$. A new regression estimator for the model parameters is developed as a competitor to the maximum likelihood estimator. Both the new estimator and the MLE are asymptotically efficient at the postulated model and are robust under small departures from that model. Computationally, the regression…
An unknown signal plus white noise is observed at n discrete time points. Within a large convex class of linear estimators of ξ, we choose the estimator ξ^ that minimizes estimated quadratic risk. By construction, ξ^ is nonlinear. This estimation is done after orthogonal transformation of the data to a reasonable coordinate system. The procedure adaptively tapers the coefficients of the transformed data. If the class of candidate estimators satisfies…
Samples of directional or axial measurements arise in geophysical, biological and econometric contexts. We represent the rotational difference between two mean directions (or two mean axes) as a direction (or axis). We then construct nonparametric simultaneous confidence sets for all pair-wise rotational differences among the mean directions or mean axes of s samples. By specialization, this methodology yields nonparametric simultaneous tests for pairwise equality of…
Consider the problem of constructing a prediction region Dn for a potentially observable variable X on the basis of a learning sample of size n. Usually, the requirement that Dn contain X with probability α, conditionally on the learning sample, does not uniquely determine Dn. This paper develops a general probability-centering concept for prediction regions that extends to vector-valued or function-valued X the classical notion of an equal-tailed…
Let T^n be an estimate of the form T^n=T(F^n), where F^n is the sample cdf of niid observations and T is a locally quadratic functional defined on cdf′s. Then, the normalized jackknife estimates for bias, skewness, and variance of T^n approximate closely their bootstrap counterparts. Each of these estimates is consistent. Moreover, the jackknife and bootstrap estimates of variance…
This paper concerns robust estimation of the parameter θ which indexes a parametric model for independent non-identically distributed data. For reasonable choices of contamination neighborhood and of what is to be estimated when the parametric model does not hold, we characterize asymptotically minimax robust estimates of θ. When applied to the normal regression model, the theory yields recipes for the influence curves of optimal robust regression and scale…
Let X1,X2,⋯,Xn be i.i.d random variables with d.f. F. Suppose the {T^n=T^n(X1,X2,⋯,Xn);n≥1} are real-valued statistics and the {Tn(F);n≥1} are centering functionals such that the asymptotic distribution of n1/2{T^n−Tn(F)} is normal with mean zero. Let Hn(x,F) be the exact d.f. of n1/2{T^n−Tn(F)}. The problem is to estimate Hn(x,F) or functionals of Hn(x,F). Under…
Robustness and efficiency of a parameter estimate T can be assessed by comparing the fitted parametric distribution PT with the actual distribution, which is assumed to lie near the parametric family {Pθ:θ∈Θ}. Asymptotic lower bounds are established for the minimax risk over distributions near the parametric model, taking as loss function a monotone increasing function of the Hellinger distance between the actual distribution of the sample and the…
A rotationally invariant exponential model, which includes the Fisher-von Mises and Bingham distributions as special cases, is proposed for directional data in Rp(p⩾2). A new regression estimator for the model parameters is developed as a competitor to the maximum likelihood estimator. Both the new estimator and the MLE are asymptotically efficient at the postulated model and are robust under small departures from that model. Computationally, the regression…