For random vectors taking values in $\mathbb{R}^d$ we introduce a notion of multivariate quantiles defined in terms of a class of sets and study an associated process which we call the generalized quantile process. This process specializes to the well known univariate quantile process. We obtain functional central limit theorems for our generalized quantile process and show that both Gaussian and non-Gaussian limiting processes can arise. A number of interesting example are…
Sliced inverse regression [Li (1989), (1991) and Duan and Li (1991)] is a nonparametric method for achieving dimension reduction in regression problems. It is widely applicable, extremely easy to implement on a computer and requires no nonparametric smoothing devices such as kernel regression. If $Y$ is the response and $X \in \mathbf{R}^p$ is the predictor, in order to implement sliced inverse regression, one requires an estimate of $\Lambda = E\{\operatorname{cov}(X\mid Y)\} = \operatorname{cov}(X) - \operatorname{cov}\{E(X\mid Y)\}$…
Consider an incomplete data problem with the following specifications. There are three independent samples $(X_1, \ldots, X_m), (Z_1, \ldots, Z_n)$ and $(U_1, \ldots, U_n)$. The first two samples are drawn from a common lifetime distribution function $G$, while the third sample is drawn from the uniform distribution over the interval $(0,1)$. In this paper we derive the large sample properties of $\hat{G}_{m,n}$, the nonparametric maximum likelihood estimate of $G$ based on…
This article studies the large sample behavior of the censored data least squares estimator derived from the synthetic data method proposed by Leurgans and Zheng. The asymptotic distributions are derived by representing the estimator as a martingale plus a higher-order remainder term. Recently developed counting process techniques are used. The results are then compared to the censored regression estimator of Koul, Susarla and Van Ryzin.
A method of deriving asymptotics for linear processes is introduced which uses an explicit algebraic decomposition of the linear filter. The technique is closely related to Gordin's method but has some advantages over it, especially in terms of its range of application. The method offers a simple unified approach to strong laws, central limit theory and invariance principles for linear processes. Sample means and sample covariances are covered. The results accommodate both…
Simple renormalization arguments can often be used to calculate optimal rates of convergence for estimating linear functionals from indirect measurements contaminated with white noise. This allows one to quickly identify optimal rates for certain problems of density estimation, nonparametric regression, signal recovery and tomography. Optimal kernels may also be derived from renormalization; we give examples for deconvolution and tomography.
A semiparametric method for estimating densities of normal mean mixtures is presented. This consistent data-driven method of estimation is based on probability spacings. The estimation technique involves iteratively fixing the standard deviation of the normal kernel that serves as a smoothing parameter, and then maximizing a function of the probability spacings over all mixing distributions. Based on the distribution of uniform spacings, a distribution free goodness-of-fit…
We introduce a simple procedure for obtaining robust estimates of the concentration parameter of the von Mises-Fisher distribution on the $q$-dimensional unit sphere. The procedure is based on the median deviation from a location parameter. Its influence function is derived and standardized bias robustness is proved. The asymptotic efficiency is calculated and an example is given.
In the age-dependent minimal repair model of Block, Borges and Savits (BBS), a system failing at age $t$ undergoes one of two types of repair. With probability $p(t)$, a perfect repair is performed and the system is returned to the "good-as-new" state, while with probability $1 - p(t)$, a minimal repair is performed and the system is repaired, but is only as good as a working system of age $t$. Whitaker and Samaniego propose an estimator for the system life distribution $F$ w…
Peter J. Bickel · Vijayan N. Nair · Paul C. C. Wang·刊期 1992-06-01 · 卷 20（2）·The Annals of Statistics已编入正刊
有限总体偏倚抽样下的非参数推断AI 解读
We consider a biased sampling model that has been found useful in incorporating size biases inherent in many types of discovery data. The model postulates that the data are obtained from a finite population by selecting successively without replacement and with probability proportional to some measure of size. Unlike the ppswor scheme in survey sampling, it is assumed here that the size measure is a function of the unknown population values. In this article, we consider…
Suppose that the mean $\tau$ of a vector of Poisson variates is known to lie in a bounded domain $T$ in $\lbrack 0,\infty)^p$. How much does this a priori information increase precision of estimation of $\tau$? Using error measure $\sum_i(\hat\tau_i - \tau_i)^2/\tau_i$ and minimax risk $\rho(T)$, we give analytical and numerical results for small intervals when $p = 1$. Usually, however, approximations are needed. If $T$ is "rectangularly convex" at 0, there exist linear…
Suppose $X_1,X_2,\ldots,X_n$ are i.i.d. with unknown density $f$. There is a well-known expression for the asymptotic mean integrated squared error (MISE) in estimating $f$ by a kernel estimate $\hat{f}_n$, under certain conditions on $f$, the kernel and the bandwidth. Suppose that one would like to choose a sample size so that the MISE is smaller than some preassigned positive number $w$. Based on the asymptotic expression for the MISE, one can identify an appropriate…
A Poisson approximation of a truncated, empirical point process enables us to reduce conditional statistical problems to unconditional ones. Let $(\mathbf{X,Y})$ be a $(d + m)$-dimensional random vector and denote by $F(\cdot\mid\mathbf{x})$ the conditional d.f. of $\mathbf{Y}$ given $\mathbf{X} = \mathbf{x}$. Applying our approach, one may study the fairly general problem of evaluating a functional parameter $T(F(\cdot\mid\mathbf{x}_1),\ldots,F(\cdot\mid\mathbf{x}_p))$ based…
It is argued that mean integrated squared error is not a useful measure of the performance of a variable bandwidth density estimator based on Abramson's square root law. The reason is that when the unknown density $f$ has even moderately light tails, properties of those tails drive the formula for optimal bandwidth, to the virtual exclusion of other properties of $f$. We suggest that weighted integrated squared error be employed as the performance criterion, using a weight…
Estimators for location and size of a discontinuity or change-point in an otherwise smooth regression model are proposed. The assumptions needed are much weaker than those made in parametric models. The proposed estimators apply as well to the detection of discontinuities in derivatives and therefore to the detection of change-points of slope and of higher order curvature. The proposed estimators are based on a comparison of left and right one-sided kernel smoothers. Weak…
Several authors have developed bootstrap methods for constructing confidence intervals in nonparametric regression. On each occasion a nonpivotal approach has been employed. Nonpivotal methods are still the overwhelmingly popular choice when statisticians use the bootstrap to compute confidence intervals, but they are not necessarily the most appropriate. In this paper we point out some of the theoretical advantages of pivoting. They include a reduction in the error of the…
Bootstrap distributional limit theorems for $U$ and $V$ statistics are proved. They hold a.s., under weak moment conditions and without restrictions on the bootstrap sample size (as long as it tends to $\infty$), regardless of the degree of degeneracy of $U$ and $V$. A testing procedure based on these results is outlined.
The paper develops a one-step triangular array asymptotic expansion for continuous martingales which are asymptotically normal. Mixing conditions are not required, but the quadratic variations of the martingales must satisfy a law of large numbers and a central limit type condition. From this result we derive expansions for the distributions of estimators in asymptotically ergodic differential equation models, and also for the bootstrapping estimators of these distributions.
We consider a class of $U$-statistics type estimates for multivariate location. The estimates extend some $R$-estimates to multivariate data. In particular, the class of estimates includes the multivariate median considered by Gini and Galvani (1929) and Haldane (1948) and a multivariate extension of the well-known Hodges-Lehmann (1963) estimate. We explore large sample behavior of these estimates by deriving a Bahadur type representation for them. In the process of…
An exact and easily computable expression for the mean integrated squared error (MISE) for the kernel estimator of a general normal mixture density, is given for Gaussian kernels of arbitrary order. This provides a powerful new way of understanding density estimation which complements the usual tools of simulation and asymptotic analysis. The family of normal mixture densities is very flexible and the formulae derived allow simple exact analysis for a wide variety of density…
The bootstrap is a poor estimator of bias in problems of curve estimation, and so bias must be corrected by other means when the bootstrap is used to construct confidence intervals for a probability density. Bias may either be estimated explicitly, or allowed for by undersmoothing the curve estimator. Which of these two approaches is to be preferred? In the present paper we address this question from the viewpoint of coverage accuracy, assuming a given number of derivatives…
In this note we study comparison of experiments via the positive dependence of normal variables with a common univariate marginal distribution. We show that positive dependence has an adverse effect on the information concerning the common mean $\theta$, and give a partial ordering of the information via a majorization ordering of the correlation matrices. In the special case when the random variables are equally correlated, the main theorem yields a result for the…
This paper is concerned with the size of confidence intervals for parameters of stochastic processes based on limit laws with two competing normalizations, one producing asymptotic normality and the other asymptotic mixed normality. It is shown that, in a certain sense, the interval based on asymptotic normality is preferable on average. Applications to estimation of parameters in nonergodic stochastic processes and to estimation of steady-state parameters in a simulation…
A strong adaptive criteria is defined for density estimation problems. In a particular case it is shown that there is no strongly adaptive sequence of estimators. In contrast Woodroofe has shown that a weakly adaptive result holds.
For random vectors taking values in Rd we introduce a notion of multivariate quantiles defined in terms of a class of sets and study an associated process which we call the generalized quantile process. This process specializes to the well known univariate quantile process. We obtain functional central limit theorems for our generalized quantile process and show that both Gaussian and non-Gaussian limiting processes can arise. A number of interesting example are…
Sliced inverse regression [Li (1989), (1991) and Duan and Li (1991)] is a nonparametric method for achieving dimension reduction in regression problems. It is widely applicable, extremely easy to implement on a computer and requires no nonparametric smoothing devices such as kernel regression. If Y is the response and X∈Rp is the predictor, in order to implement sliced inverse regression, one requires an estimate of Λ=E{cov(X∣Y)}=cov(X)−cov{E(X∣Y)}…
Consider an incomplete data problem with the following specifications. There are three independent samples (X1,…,Xm),(Z1,…,Zn) and (U1,…,Un). The first two samples are drawn from a common lifetime distribution function G, while the third sample is drawn from the uniform distribution over the interval (0,1). In this paper we derive the large sample properties of G^m,n, the nonparametric maximum likelihood estimate of G based on…
We introduce a simple procedure for obtaining robust estimates of the concentration parameter of the von Mises-Fisher distribution on the q-dimensional unit sphere. The procedure is based on the median deviation from a location parameter. Its influence function is derived and standardized bias robustness is proved. The asymptotic efficiency is calculated and an example is given.
In the age-dependent minimal repair model of Block, Borges and Savits (BBS), a system failing at age t undergoes one of two types of repair. With probability p(t), a perfect repair is performed and the system is returned to the "good-as-new" state, while with probability 1−p(t), a minimal repair is performed and the system is repaired, but is only as good as a working system of age t. Whitaker and Samaniego propose an estimator for the system life distribution F w…
Suppose that the mean τ of a vector of Poisson variates is known to lie in a bounded domain T in [0,∞)p. How much does this a priori information increase precision of estimation of τ? Using error measure ∑i(τ^i−τi)2/τi and minimax risk ρ(T), we give analytical and numerical results for small intervals when p=1. Usually, however, approximations are needed. If T is "rectangularly convex" at 0, there exist linear…
Suppose X1,X2,…,Xn are i.i.d. with unknown density f. There is a well-known expression for the asymptotic mean integrated squared error (MISE) in estimating f by a kernel estimate f^n, under certain conditions on f, the kernel and the bandwidth. Suppose that one would like to choose a sample size so that the MISE is smaller than some preassigned positive number w. Based on the asymptotic expression for the MISE, one can identify an appropriate…
A Poisson approximation of a truncated, empirical point process enables us to reduce conditional statistical problems to unconditional ones. Let (X,Y) be a (d+m)-dimensional random vector and denote by F(⋅∣x) the conditional d.f. of Y given X=x. Applying our approach, one may study the fairly general problem of evaluating a functional parameter T(F(⋅∣x1),…,F(⋅∣xp)) based…
It is argued that mean integrated squared error is not a useful measure of the performance of a variable bandwidth density estimator based on Abramson's square root law. The reason is that when the unknown density f has even moderately light tails, properties of those tails drive the formula for optimal bandwidth, to the virtual exclusion of other properties of f. We suggest that weighted integrated squared error be employed as the performance criterion, using a weight…
On the Bootstrap of U and V Statistics
Bootstrap distributional limit theorems for U and V statistics are proved. They hold a.s., under weak moment conditions and without restrictions on the bootstrap sample size (as long as it tends to ∞), regardless of the degree of degeneracy of U and V. A testing procedure based on these results is outlined.
Multivariate Location Estimation Using Extension of R-Estimates Through U-Statistics Type Approach
We consider a class of U-statistics type estimates for multivariate location. The estimates extend some R-estimates to multivariate data. In particular, the class of estimates includes the multivariate median considered by Gini and Galvani (1929) and Haldane (1948) and a multivariate extension of the well-known Hodges-Lehmann (1963) estimate. We explore large sample behavior of these estimates by deriving a Bahadur type representation for them. In the process of…
In this note we study comparison of experiments via the positive dependence of normal variables with a common univariate marginal distribution. We show that positive dependence has an adverse effect on the information concerning the common mean θ, and give a partial ordering of the information via a majorization ordering of the correlation matrices. In the special case when the random variables are equally correlated, the main theorem yields a result for the…