Rotationally invariant tests based on test statistics of the von Mises type are proposed under the hypothesis of spherical symmetry of a multivariate distribution. The tests are distribution-free when the hypothesis of spherical symmetry is true. The asymptotic distribution of the test statistics are derived under the null hypothesis and under any fixed alternative. A simple criterion for consistency is given. The results are illustrated by numerous examples of test…
The first anomaly in the L.R.T. for testing restricted hypotheses was observed by Warrack and Robertson. They found the L.R.T. for testing an order restriction in a normal model to be dominated by a different test. In this paper we deal with a more general situation in which the L. R. T. for testing a face of an acute cone is dominated by a different test that does not take into account some of the information in the model.
Richard Dykstra · Subhash Kochar · Tim Robertson·刊期 1991-06-01 · 卷 19（2）·The Annals of Statistics已编入正刊
多个总体一致随机序的统计推断AI 解读
Stochastic ordering between probability distributions is a widely studied concept. It arises in numerous settings and has useful applications. Since it is often easy to make value judgments when such orderings exist, it is desirable to recognize their occurrence and to model distributional structure under such orderings. Unfortunately, the necessary theory for statistical inference procedures has not been developed for many problems involving stochastic ordering and this…
The problem of maximum likelihood estimation of Lowner ordered covariance matrices is considered. It is shown that a dual formulation of this problem is tractable and important in its own right. The interplay between the primal and dual problems suggests a general algorithm for computing the solutions to these problems. This algorithm has application to some estimation problems in balanced multivariate variance components models. The speed of convergence is also discussed…
In a general setting in which prior distributions that may take on the value $\infty$ are admitted, an inference based on a posterior for a prior, $\mu$, that is "minimally compatible" with the inference is shown to have a strong property of expectation consistency, that implies a corresponding property of coherence: A nonnegative expected payoff function for a gambler's strategy is necessary 0 almost everywhere $(\mu)$. In the converse direction, under appropriate…
It has been observed that to a smoothing spline operator there corresponds an equivalent kernel operator; these two operators have been compared in a variety of norms [Cox (1984), Silverman (1984)]. In this paper, we refine the existing bounds for the particular case of the spline estimator considered in Rice and Rosenblatt (1983) and its corresponding equivalent kernel estimator. We obtain detailed asymptotic expressions for the bias and covariance functions of the two…
Biased sampling regression models were introduced by Jewell, generalizing the truncated regression model studied by Bhattacharya, Chernoff and Yang. If the independent variable takes on only a finite number of values (as does the stratum variable), we show: 1. That if the slope of the underlying regression model is assumed known, then the nonparametric maximum likelihood estimates of the distribution of the independent and dependent variables (a) can be calculated from…
Simultaneous error bars are constructed for nonparametric kernel estimates of regression functions. The method is based on the bootstrap, where resampling is done from a suitably estimated residual distribution. The error bars are seen to give asymptotically correct coverage probabilities uniformly over any number of gridpoints. Applications to an economic problem are given and comparison to both pointwise and Bonferroni-type bars is presented through a simulation study.
Let $(X, Y)$ be a random vector such that $X$ is $d$-dimensional, $Y$ is real valued and $Y = \theta(X) + \varepsilon$, where $X$ and $\varepsilon$ are independent and the $\alpha$th quantile of $\varepsilon$ is $0$ ($\alpha$ is fixed such that $0 < \alpha < 1$). Assume that $\theta$ is a smooth function with order of smoothness $p > 0$, and set $r = (p - m)/(2p + d)$, where $m$ is a nonnegative integer smaller than $p$. Let $T(\theta)$ denote a derivative of $\theta$ of…
We propose a new nonparametric regression estimate. In contrast to the traditional approach of considering regression functions whose $m$th derivatives lie in a ball in the $L_\infty$ or $L_2$ norm, we consider the class of functions whose $(m - 1)$st derivative consists of at most $k$ monotone pieces. For many applications this class seems more natural than the classical ones. The least squares estimator of this class is studied. It is shown that the speed of convergence is…
The problem of estimating a smooth monotone regression function $m$ will be studied. We will consider the estimator $m_{SI}$ consisting of a smoothing step (application of a kernel estimator based on a kernel $K$) and of a isotonisation step (application of the pool adjacent violator algorithm). The estimator $m_{SI}$ will be compared with the estimator $m_{IS}$ where these two steps are interchanged. A higher order stochastic expansion of these estimators will be given…
We consider here spline estimates of the optimal transformations of variables for multiple correlation and regression as dealt with in a recent paper by Breiman and Friedman. We show that we can construct estimates of the optimal transformations which have the same optimal rate of convergence as in the usual nonparametric estimation of a univariate function.
We establish upper and lower bounds on the asymptotic minimax risk in estimating (1) a density at a point when the density is known to be decreasing with a Lipschitz condition; (2) a density at a point when the density satisfies a local second-order smoothness (Sacks-Ylvisaker) condition; and (3) the $k$th derivative of the density at a point, when the density satisfies a local $L_p$ constraint on the $m$th derivative. In (1), (2) and (3) the upper and lower bounds differ…
An approximate maximum likelihood estimate is known to be consistent under some compactness and integrability conditions. In this paper we study its convergence rate and its asymptotic efficiency in estimating smooth functionals of the parameter. We provide conditions under which the rate of convergence can be established. This rate is essentially governed by the size of the space of score functions as measured by an entropy index. We also show that, for a large class of…
This paper discusses the minimum Hellinger distance estimation (MHDE) of the parameter that gives the "best fit" of a parametric family to a density when the data are randomly censored. In studying the MHDE, the tail behavior of the product-limit (P-L) process is investigated, and the weak convergence of the process on the real line is established. An upper bound on the mean square increment of the normalized P-L process is also obtained. With these results, the asymptotic…
We study the asymptotics of the ML estimators for the Curie-Weiss model parametrized by the inverse temperature $\beta$ and the external field $h$. We show that if both $\beta$ and $h$ are unknown, the ML estimator of $(\beta, h)$ does not exist. For $\beta$ known, the ML estimator $\hat{h}_n$ of $h$ exhibits, at a first order phase transition point, superefficiency in the sense that its asymptotic variance is half of that of nearby points. At the critical point $(\beta = 1)$…
A class of rank estimators is introduced for regression analysis in the presence of both left-truncation and right-censoring on the response variable. By making use of martingale theory and a tightness lemma for stochastic integrals of multiparameter empirical processes, the asymptotic normality of the estimators is established under certain assumptions. Adaptive choice of the score functions to give asymptotically efficient rank estimators is also discussed.
Consider a general regression model of the form $y = g(\alpha + \mathbf{x}'\beta, \varepsilon)$, with an arbitrary and unknown link function $g$. We study a link-free method, the slicing regression, for estimating the direction of $\beta$. The method is easy to implement and does not require iterative computation. First, we estimate the inverse regression function $E(\mathbf{x}\mid y)$ using a step function. We then estimate $\Gamma = \operatorname{Cov}\lbrack E(\mathbf{x}\mid y)\rbrack$…
This note studies the asymptotic behavior of the delete-$d$ jackknife in the irregular case of a sample $p$-quantile, calculated from a sample of $n$ iid r.v.'s. Two results are obtained: (a) an almost sure rate of convergence of the delete-$d$ jackknife histogram to the normal distribution; (b) almost sure convergence of the delete-$d$ jackknife variance estimate to the asymptotic variance of the sample $p$-quantile.
An asymptotic expansion of length 2 is established for the coverage probabilities of confidence intervals for the underlying $q$-quantile which are derived by bootstrapping the sample $q$-quantile. The corresponding level error turns out to be of order $O(n^{-1/2})$ which is unexpectedly low. A confidence interval of even more practical use is derived by using backward critical points. The resulting confidence interval is of the same length as the one derived by ordinary…
The asymptotic accuracy of the estimated one-term Edgeworth expansion and the bootstrap approximation for a Studentized $U$-statistic is investigated. It is shown that both the Edgeworth expansion estimate and the bootstrap approximation are asymptotically closer to the exact distribution of a Studentized $U$-statistic than the normal approximation. The conditions needed to obtain these results are weak moment assumptions on the kernel $h$ of the $U$-statistic and a…
Estimation of the finite population distribution function and related statistics, such as the median and interquartile range, is considered. Large-sample properties of estimators constructed from stratified cluster samples, and properties of large-sample confidence intervals, are established. The results are obtained within the context of a sequence of finite populations generated from a superpopulation.
Conditions under which the bivariate Kaplan-Meier estimate of Dabrowska is not a proper survival function are given. All points assigned negative mass are identified under the assumption that the observations follow an absolutely continuous distribution. The number of points assigned negative mass increases as $n^2$ and the total amount of negative mass does not disappear as $n \rightarrow \infty$, where $n$ is the sample size. A simulation study is reported which shows that…
A minor modification of the product-limit estimator is proposed for estimating a distribution function (not necessarily continuous) when the data are subject to either truncation or censoring, or to both, by independent but not necessarily identically distributed truncation-censoring variables. Making use of martingale integral representations and empirical process theory, uniform strong consistency of the estimator is established and weak convergence results are proved for…
In a general setting in which prior distributions that may take on the value ∞ are admitted, an inference based on a posterior for a prior, μ, that is "minimally compatible" with the inference is shown to have a strong property of expectation consistency, that implies a corresponding property of coherence: A nonnegative expected payoff function for a gambler's strategy is necessary 0 almost everywhere (μ). In the converse direction, under appropriate…
Let (X,Y) be a random vector such that X is d-dimensional, Y is real valued and Y=θ(X)+ε, where X and ε are independent and the αth quantile of ε is 0 (α is fixed such that 0<α<1). Assume that θ is a smooth function with order of smoothness p>0, and set r=(p−m)/(2p+d), where m is a nonnegative integer smaller than p. Let T(θ) denote a derivative of θ of…
We propose a new nonparametric regression estimate. In contrast to the traditional approach of considering regression functions whose mth derivatives lie in a ball in the L∞ or L2 norm, we consider the class of functions whose (m−1)st derivative consists of at most k monotone pieces. For many applications this class seems more natural than the classical ones. The least squares estimator of this class is studied. It is shown that the speed of convergence is…
The problem of estimating a smooth monotone regression function m will be studied. We will consider the estimator mSI consisting of a smoothing step (application of a kernel estimator based on a kernel K) and of a isotonisation step (application of the pool adjacent violator algorithm). The estimator mSI will be compared with the estimator mIS where these two steps are interchanged. A higher order stochastic expansion of these estimators will be given…
We establish upper and lower bounds on the asymptotic minimax risk in estimating (1) a density at a point when the density is known to be decreasing with a Lipschitz condition; (2) a density at a point when the density satisfies a local second-order smoothness (Sacks-Ylvisaker) condition; and (3) the kth derivative of the density at a point, when the density satisfies a local Lp constraint on the mth derivative. In (1), (2) and (3) the upper and lower bounds differ…
We study the asymptotics of the ML estimators for the Curie-Weiss model parametrized by the inverse temperature β and the external field h. We show that if both β and h are unknown, the ML estimator of (β,h) does not exist. For β known, the ML estimator h^n of h exhibits, at a first order phase transition point, superefficiency in the sense that its asymptotic variance is half of that of nearby points. At the critical point (β=1)…
Consider a general regression model of the form y=g(α+x′β,ε), with an arbitrary and unknown link function g. We study a link-free method, the slicing regression, for estimating the direction of β. The method is easy to implement and does not require iterative computation. First, we estimate the inverse regression function E(x∣y) using a step function. We then estimate Γ=Cov[E(x∣y)]…
This note studies the asymptotic behavior of the delete-d jackknife in the irregular case of a sample p-quantile, calculated from a sample of n iid r.v.'s. Two results are obtained: (a) an almost sure rate of convergence of the delete-d jackknife histogram to the normal distribution; (b) almost sure convergence of the delete-d jackknife variance estimate to the asymptotic variance of the sample p-quantile.
An asymptotic expansion of length 2 is established for the coverage probabilities of confidence intervals for the underlying q-quantile which are derived by bootstrapping the sample q-quantile. The corresponding level error turns out to be of order O(n−1/2) which is unexpectedly low. A confidence interval of even more practical use is derived by using backward critical points. The resulting confidence interval is of the same length as the one derived by ordinary…
On the Edgeworth Expansion and the Bootstrap Approximation for a Studentized U-Statistic
The asymptotic accuracy of the estimated one-term Edgeworth expansion and the bootstrap approximation for a Studentized U-statistic is investigated. It is shown that both the Edgeworth expansion estimate and the bootstrap approximation are asymptotically closer to the exact distribution of a Studentized U-statistic than the normal approximation. The conditions needed to obtain these results are weak moment assumptions on the kernel h of the U-statistic and a…
Conditions under which the bivariate Kaplan-Meier estimate of Dabrowska is not a proper survival function are given. All points assigned negative mass are identified under the assumption that the observations follow an absolutely continuous distribution. The number of points assigned negative mass increases as n2 and the total amount of negative mass does not disappear as n→∞, where n is the sample size. A simulation study is reported which shows that…