A general minimum distance estimation procedure is presented for nonstationary time series models that have an evolutionary spectral representation. The asymptotic properties of the estimate are derived under the assumption of possible model misspecification. For autoregressive processes with time varying coefficients, the estimate is compared to the least squares estimate. Furthermore, the behavior of estimates is explained when a stationary model is fitted to a…
In a partial linear model, the dependence of a response variate Y on covariates (W, X$ is given by $$Y = W \beta + \eta(X) + \mathscr{E}$$ where $\mathscr{E}$ is independent of $(W, X)$ with densities g and f, respectively. In this paper an asymptotically efficient estimator of $\beta$ is constructed solely under mild smoothness assumptions on the unknown $\eta$, f and g, thereby removing the assumption of finite residual variance on which all least-squares-type estimators…
Universal Gaussian approximations are established for empirical cumulative hazard and product-limit processes under random censorship. They hold uniformly up to some large order statistics in the sample, with the approximation rates depending on the order of these statistics, and require no assumptions on the censoring mechanism. Weak convergence results and laws of the iterated logarithm follow on the whole line if the respective processes are stopped at certain large order…
Linear regression with doubly censored responses is considered. Buckley-James-Ritov-type estimators are proposed. Semiparametric information and projective scores are discussed. An expansion of the estimating equations is obtained under fairly general assumptions. Sufficient conditions are given for the asymptotic consistency and normality of the estimators.
To compare two samples of possibly right-censored failure times, a location-scale model, without assuming the distribution form, is considered for the log-transformed data. This new accelerated failure time model is introduced to accommodate the possible heterogeneity between within-treatment variations of the two groups considered. One distinct feature of this model is that, with the presence of heterogeneity, statistical inferences on both location and scale parameters…
Let $\hat{F}_n$ be the Kaplan-Meier estimator of a distribution function F computed from randomly censored data. It is known that, under certain integrability assumptions on a function $\varphi$, the Kaplan-Meier integral $\int \varphi d \hat{F}_n$, when properly standardized, is asymptotically normal. In this paper it is shown that, with probability 1, the jackknife estimate of variance consistently estimates the (limit) variance.
Maximum likelihood estimators for record-value data with a linear trend are quite sensitive to misspecification of the error distribution. Indeed, incorrect choice of that distribution can lead to inconsistent estimation of the intercept parameter and produce estimators of slope that do not enjoy the asymptotic convergence rate prescribed by the information matrix. These properties and the importance of linearly trended records lead us to suggest a distribution-free approach…
Given data $X_1, \dots, X_n$ and a kernel h with m arguments, Serfling introduced the class of generalized L-statistics (GL-statistics), which is defined by taking linear combinations of the ordered $h(X_{i_1}, \dots, X_{i_m})$ where $(i_1, \dots, i_m)$ ranges over all $n!/(n - m)!$ distinct m-tuples of $(1, \dots, n)$. In this paper we derive a class of incomplete generalized L-statistics (IGL-statistics) by taking linear combinations of the ordered elements from a subset…
A structured linear regression model is one in which there are permanent dependencies among some p row vectors of the $n \times p$ design matrix. To study structured linear regression, we introduce a new class of robust estimators, called D-estimators, which can be regarded as a generalization of the least median of squares and least trimmed squares estimators. They minimize a dispersion function of the ordered absolute residuals up to the rank h. We investigate their…
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…
A general constrained minimum risk inequality is derived. Given two densities $f_{\theta}$ and $f_0$ we find a lower bound for the risk at the point $\theta$ given an upper bound for the risk at the point 0. The inequality sheds new light on superefficient estimators in the normal location problem and also on an adaptive estimation problem arising in nonparametric functional estimation.
We shall study the general regression model $Y = g_0 (X) + \varepsilon$, where X and $varepsilon$ are independent. The available information about $g_0$ can be expressed by $g_0 \epsilon \mathscr{G}$ for some class $\mathscr{G}$. As an estimator of $g_0$ we choose the least squares estimator. We shall give necessary and sufficient conditions for consistency of this estimator in terms of (basically) geometric properties of $\mathscr{G}$. Our main tool will be the theory of…
We define a minimum distance estimate of the smoothing factor for kernel density estimates, based on a methodology first developed by Yatracos. It is shown that if $f_{nh}$ denotes the kernel density estimate on $\mathbb{R}^d$ for an i.i.d. sample of size n drawn from an unknown density f, where h is the smoothing factor, and if $f_n$ is the kernel estimate with the same kernel and with the proposed new data-based smoothing factor, then, under a regularity condition on the…
Let a function f be observed with a noise. We wish to test the null hypothesis that the function is identically zero, against a composite nonparametric alternative: functions from the alternative set are separated away from zero in an integral (e.g., $L_2$) norm and also possess some smoothness properties. The minimax rate of testing for this problem was evaluated in earlier papers by Ingster and by Lepski and Spokoiny under different kinds of smoothness assumptions. It was…
Peter Hall · Ian McKay · Berwin A. Turlach·刊期 1996-12-01 · 卷 24（6）·The Annals of Statistics已编入正刊
含大量不连续点函数的小波方法表现AI 解读
Compared to traditional approaches to curve estimation, such as those based on kernels, wavelet methods are relatively unaffected by discontinuities and similar aberrations. In particular, the mean square convergence rate of a wavelet estimator of a fixed, piecewise-smooth curve is not influenced by discontinuities. Nevertheless, it is clear that as the estimation problem becomes more complex the limitations of wavelet methods must eventually be apparent. By allowing the…
Signal recovery in Gaussian white noise with variance tending to zero has served for some time as a representative model for nonparametric curve estimation, having all the essential traits in a pure form. The equivalence has mostly been stated informally, but an approximation in the sense of Le Cam's deficiency distance $\Delta$ would make it precise. The models are then asymptotically equivalent for all purposes of statistical decision with bounded loss. In nonparametrics,…
The principal result is that, under conditions, to any nonparametric regression problem there corresponds an asymptotically equivalent sequence of white noise with drift problems, and conversely. This asymptotic equivalence is in a global and uniform sense. Any normalized risk function attainable in one problem is asymptotically attainable in the other, with the difference in normalized risks converging to zero uniformly over the entire parameter space. The results are…
On the basis of N i.i.d. random variables with a common unknown distribution P we wish to estimate a functional $\tau_N(P)$. An obvious and very general approach to this problem is to find an estimator $\hat{P}_N$ of P first, and then construct a so-called substitution estimator $\tau_N (\hat{P}_N)$ of $\tau_N(P)$. In this paper we investigate how to choose the estimator $\hat{P}_N$ so that the substitution estimator $\tau_N (\hat{P}_N)$ will be consistent. Although our…
We obtain strong Bahadur representations for a general class of M-estimators that satisfies $\Sigma_i \psi (x_i, \theta) = o(\delta_n)$, where the $x_i$'s are independent but not necessarily identically distributed random variables. The results apply readily to M-estimators of regression with nonstochastic designs. More specifically, we consider the minimum $L_p$ distance estimators, bounded influence GM-estimators and regression quantiles. Under appropriate design…
We wish to estimate the probability density $g(y)$ that produced an observed random sample of vectors $y_1, y_2, \dots, y_n$. Estimates of $g(y)$ are traditionally constructed in two quite different ways: by maximum likelihood fitting within some parametric family such as the normal or by nonparametric methods such as kernel density estimation. These two methods can be combined by putting an exponential family "through" a kernel estimator. These are the specially designed…
In model selection, usually a "best" predictor is chosen from a collection ${\hat{\mu}(\cdot, s)}$ of predictors where $\hat{\mu}(\cdot, s)$ is the minimum least-squares predictor in a collection $\mathsf{U}_s$ of predictors. Here s is a complexity parameter; that is, the smaller s, the lower dimensional/smoother the models in $\mathsf{U}_s$. If $\mathsf{L}$ is the data used to derive the sequence ${\hat{\mu}(\cdot, s)}$, the procedure is called unstable if a small change in…
This paper presents a lower bound, derived from the information inequality for the Bayes risk with respect to truncated priors under quadratic loss. It is discussed in cases where the regularity condition of Brown and Gajek is not always satisfied. A related result for the minimax risk is also given.
An upper bound for the tail probability $P_{\theta} (\log (L(x_{(n_1, \dots, n_q)}, \Theta)/L(x_{(n_1, \dots, n_q)}, \theta)) \geq t)$ is derived in the case of sampling from q populations. This estimate is used for establishing the Hodges-Lehmann optimality of a test statistic for a hypothesis on exponential distributions.
We show that the maximally selected standardized U-statistic goes in distribution to an infinite sum of weighted chi-square random variables in the degenerate case. The result is applied to the detection of possible changes in the distribution of a sequence observation.
In a partial linear model, the dependence of a response variate Y on covariates (W, X$ is given by Y=Wβ+η(X)+E where E is independent of (W,X) with densities g and f, respectively. In this paper an asymptotically efficient estimator of β is constructed solely under mild smoothness assumptions on the unknown η, f and g, thereby removing the assumption of finite residual variance on which all least-squares-type estimators…
Let F^n be the Kaplan-Meier estimator of a distribution function F computed from randomly censored data. It is known that, under certain integrability assumptions on a function φ, the Kaplan-Meier integral ∫φdF^n, when properly standardized, is asymptotically normal. In this paper it is shown that, with probability 1, the jackknife estimate of variance consistently estimates the (limit) variance.
Given data X1,…,Xn and a kernel h with m arguments, Serfling introduced the class of generalized L-statistics (GL-statistics), which is defined by taking linear combinations of the ordered h(Xi1,…,Xim) where (i1,…,im) ranges over all n!/(n−m)! distinct m-tuples of (1,…,n). In this paper we derive a class of incomplete generalized L-statistics (IGL-statistics) by taking linear combinations of the ordered elements from a subset…
A structured linear regression model is one in which there are permanent dependencies among some p row vectors of the n×p design matrix. To study structured linear regression, we introduce a new class of robust estimators, called D-estimators, which can be regarded as a generalization of the least median of squares and least trimmed squares estimators. They minimize a dispersion function of the ordered absolute residuals up to the rank h. We investigate their…
A general constrained minimum risk inequality is derived. Given two densities fθ and f0 we find a lower bound for the risk at the point θ given an upper bound for the risk at the point 0. The inequality sheds new light on superefficient estimators in the normal location problem and also on an adaptive estimation problem arising in nonparametric functional estimation.
We shall study the general regression model Y=g0(X)+ε, where X and varepsilon are independent. The available information about g0 can be expressed by g0ϵG for some class G. As an estimator of g0 we choose the least squares estimator. We shall give necessary and sufficient conditions for consistency of this estimator in terms of (basically) geometric properties of G. Our main tool will be the theory of…
We define a minimum distance estimate of the smoothing factor for kernel density estimates, based on a methodology first developed by Yatracos. It is shown that if fnh denotes the kernel density estimate on Rd for an i.i.d. sample of size n drawn from an unknown density f, where h is the smoothing factor, and if fn is the kernel estimate with the same kernel and with the proposed new data-based smoothing factor, then, under a regularity condition on the…
Let a function f be observed with a noise. We wish to test the null hypothesis that the function is identically zero, against a composite nonparametric alternative: functions from the alternative set are separated away from zero in an integral (e.g., L2) norm and also possess some smoothness properties. The minimax rate of testing for this problem was evaluated in earlier papers by Ingster and by Lepski and Spokoiny under different kinds of smoothness assumptions. It was…
Signal recovery in Gaussian white noise with variance tending to zero has served for some time as a representative model for nonparametric curve estimation, having all the essential traits in a pure form. The equivalence has mostly been stated informally, but an approximation in the sense of Le Cam's deficiency distance Δ would make it precise. The models are then asymptotically equivalent for all purposes of statistical decision with bounded loss. In nonparametrics,…
On the basis of N i.i.d. random variables with a common unknown distribution P we wish to estimate a functional τN(P). An obvious and very general approach to this problem is to find an estimator P^N of P first, and then construct a so-called substitution estimator τN(P^N) of τN(P). In this paper we investigate how to choose the estimator P^N so that the substitution estimator τN(P^N) will be consistent. Although our…
We obtain strong Bahadur representations for a general class of M-estimators that satisfies Σiψ(xi,θ)=o(δn), where the xi's are independent but not necessarily identically distributed random variables. The results apply readily to M-estimators of regression with nonstochastic designs. More specifically, we consider the minimum Lp distance estimators, bounded influence GM-estimators and regression quantiles. Under appropriate design…
We wish to estimate the probability density g(y) that produced an observed random sample of vectors y1,y2,…,yn. Estimates of g(y) are traditionally constructed in two quite different ways: by maximum likelihood fitting within some parametric family such as the normal or by nonparametric methods such as kernel density estimation. These two methods can be combined by putting an exponential family "through" a kernel estimator. These are the specially designed…
In model selection, usually a "best" predictor is chosen from a collection μ^(⋅,s) of predictors where μ^(⋅,s) is the minimum least-squares predictor in a collection Us of predictors. Here s is a complexity parameter; that is, the smaller s, the lower dimensional/smoother the models in Us. If L is the data used to derive the sequence μ^(⋅,s), the procedure is called unstable if a small change in…
An upper bound for the tail probability Pθ(log(L(x(n1,…,nq),Θ)/L(x(n1,…,nq),θ))≥t) is derived in the case of sampling from q populations. This estimate is used for establishing the Hodges-Lehmann optimality of a test statistic for a hypothesis on exponential distributions.