In a nonparametric regression setup where the covariates are continuous, the problem of estimating the number of covariates will be discussed in this paper. The kernel method is used to estimate the regression function and the selection criterion is based on minimizing the cross-validation estimate of the mean squared prediction error. We consider choosing both the bandwidth and the number of covariates based on the data. Unlike the case of linear regression, it turns out…
Consider a $\mathscr{Y}$-valued response variable having a density function $f(\cdot\mid x)$ that depends on an $\mathscr{X}$-valued input variable $x.$ It is assumed that $\mathscr{X}$ and $\mathscr{Y}$ are compact intervals and that $f(\cdot\mid\cdot)$ is continuous and positive on $\mathscr{X} \times \mathscr{Y}.$ Let $F(\cdot\mid x)$ denote the distribution function of $f(\cdot\mid x)$ and let $Q(\cdot\mid x)$ denote its quantile function. A finite-parameter exponential…
Miklos Csorgo · Edit Gombay · Lajos Horvath·刊期 1991-12-01 · 卷 19（4）·The Annals of Statistics已编入正刊
随机删失下核密度估计Lp距离的中心极限定理AI 解读
A sequence of independent nonnegative random variables with common distribution function $F$ is censored on the right by another sequence of independent identically distributed random variables. These two sequences are also assumed to be independent. We estimate the density function $f$ of $F$ by a sequence of kernel estimators $f_n(t) = (\int^\infty_{-\infty}K((t - x)/h(n))d\hat{F}_n(x))/h(n),$ where $h(n)$ is a sequence of numbers, $K$ is kernel density function and $\hat{F}_n$…
The estimation of the treatment effect in the two-sample problem with right censoring is of interest in survival analysis. In this article we consider both the location shift model and the scale change model. We establish the large-sample properties of a generalized Hodges-Lehmann type estimator. The strong consistency is established under the minimal possible conditions. The asymptotic normality is also obtained without imposing any conditions on the censoring mechanisms.…
We present a general procedure for obtaining an exact confidence set for the variance components in a mixed linear model. The procedure can be viewed as a generalization of the ANOVA method used with balanced models. Our procedure uses, as pivotal quantities, quadratic forms that are distributed independently as chi-squared variables. These quadratic forms are constructed with reference to spaces that are orthogonal with respect to the covariance matrix of the observation…
This paper introduces a hierarchical Bayes (HB) approach for prediction in general mixed linear models. The results find application in small area estimation. Our model unifies and extends a number of models previously considered in this area. Computational formulas for obtaining the Bayes predictors and their standard errors are given in the general case. The methods are applied to two actual data sets. Also, in a special case, the HB predictors are shown to possess some…
Empirical likelihood is a nonparametric method of inference. It has sampling properties similar to the bootstrap, but where the bootstrap uses resampling, it profiles a multinomial likelihood supported on the sample. Its properties in i.i.d. settings have been investigated in works by Owen, by Hall and by DiCiccio, Hall and Romano. This article extends the method to regression problems. Fixed and random regressors are considered, as are robust and heteroscedastic…
We present a general statistical model for data coarsening, which includes as special cases rounded, heaped, censored, partially categorized and missing data. Formally, with coarse data, observations are made not in the sample space of the random variable of interest, but rather in its power set. Grouping is a special case in which the degree of coarsening is known and nonstochastic. We establish simple conditions under which the possible stochastic nature of the coarsening…
An attempt is made to determine the logically consistent rules for selecting a vector from any feasible set defined by linear constraints, when either all $n$-vectors or those with positive components or the probability vectors are permissible. Some basic postulates are satisfied if and only if the selection rule is to minimize a certain function which, if a "prior guess" is available, is a measure of distance from the prior guess. Two further natural postulates restrict the…
We consider interaction splines which model a multivariate regression function $f$ as a constant plus the sum of functions of one variable (main effects), plus the sum of functions of two variables (two-factor interactions), and so on. The estimation of $f$ by the penalized least squares method and the asymptotic properties of the models are studied in this article. It is shown that, under some regularity conditions on the data points, the expected squared error averaged…
The effect of a linear filter with monotone gain on the first-order autocorrelation of a weakly stationary time series is discussed. When the gain is monotone increasing, the first-order autocorrelation cannot increase. Otherwise, when the gain is monotone decreasing, the correlation cannot decrease. Further, when the gain is strictly monotone, the first-order autocorrelation is unchanged if and only if the process is a pure sinusoid with probability 1. Under the Gaussian…
This paper shows that Cox's partial score function is the projection of the score function on the (locally) $E$-ancillary subspace for the nuisance parameter (Small and McLeish). This is done by adapting the concepts of (locally) $E$-ancillarity and (locally) $E$-sufficiency for inference functions (McLeish and Small) to an extended Cox's regression model, where the baseline function is allowed to be a predictable process.
Ann Cohen Brandwein · William E. Strawderman·刊期 1991-09-01 · 卷 19（3）·The Annals of Statistics已编入正刊
球对称分布下 James–Stein 估计量的推广AI 解读
This paper is primarily concerned with extending the results of Stein to spherically symmetric distributions. Specifically, when $X \sim f(\|X - \theta\|^2)$, we investigate conditions under which estimators of the form $X + ag(X)$ dominate $X$ for loss functions $\|\delta - \theta\|^2$ and loss functions which are concave in $\|\delta - \theta\|^2$. Additionally, if the scale is unknown we investigate estimators of the location parameter of the form $X + aVg(X)$ in two…
Let $\mathscr{F}$ denote the set of cdf's on $\mathbb{R}$ with density everywhere positive. Let $C_A = \{(F,G) \in \mathscr{F} \times \mathscr{F}$: there exists a unique $x^\ast \in \mathbb{R}$ such that $F(x) > G(x) \text{for} x << x^\ast \text{and} F(x) < G(x) \text{for} x > x^\ast\}, C_B - \{(F,G) \in \mathscr{F} \times \mathscr{F}: (G,F) \in C_A\}$. Based on independent random samples from $F$ and $G$ (assumed unknown), we give distribution-free tests of $H_0: F = G$ vers…
A total of $nb$ judges rank $t$ objects $k$ at a time according to $n$ replications of a BIBD with $b$ blocks. The Durbin statistic is commonly used in this context and is equivalent to the usual analysis of variance on the rankings. The approach considered here is to introduce the notion of compatibility so as to define distances between incomplete rankings based on metrics on the space of complete rankings. Through this device we define a class of test statistics which…
Let $X_1, X_2,\ldots$ be possibly dependent random variables having the same marginal distribution. Consider the situation where $\bar{F}(x) := P\lbrack X_1 > x\rbrack$ is regularly varying at $\infty$ with an unknown index $- \alpha < 0$ which is to be estimated. In the i.i.d. setting, it is well known that Hill's estimator is consistent for $\alpha^{-1}$, and is asymptotically normally distributed. It is the purpose of this paper to demonstrate that such properties of…
The problem of bandwidth selection for nonparametric kernel regression is considered. It is well recognized that the classical bandwidth selectors are subject to large sample variation. Due to the large variation, these selectors might not be very useful in practice. Based on the frequency-domain representation of the residual sum of squares (RSS), the source of the variation is pointed out. The observation leads to consideration of a procedure which stabilizes the RSS by…
Some relationships have been established between unbiased linear predictors of processes, in signal and noise models, minimizing the predictive mean square error and some smoothing spline functions. We construct a new family of multidimensional splines adapted to the prediction of locally homogeneous random fields, whose "$m$-spectral measure" (to be defined) is absolutely continuous with respect to Lebesgue measure and satisfies some minor assumptions. By considering…
The Kolmogorov distance between the empirical $\operatorname{cdf} F_n$ and its symmetrization $sF_n$ with respect to an adequate estimator of the center of symmetry of $P$ is a natural statistic for testing symmetry. However, its limiting distribution depends on $P$. Using critical values from the symmetrically bootstrapped statistic (where the resampling is made from $sF_n$) produces tests that can be easily implemented and have asymptotically the correct levels as well as…
Consider a sequence $X_1, X_2,\ldots, X_n$ of independent random variables, where $X_1, X_2,\ldots, X_{n\theta}$ have distribution $P,$ and $X_{n\theta + 1}, X_{n\theta + 2},\ldots, X_n$ have distribution $Q$. The change-point $\theta \in (0,1)$ is an unknown parameter to be estimated, and $P$ and $Q$ are two unknown probability distributions. The nonparametric estimators of Darkhovskh and Carlstein are imbedded in a more general framework, where random seminorms are applied…
This paper deals with censored data estimation of a general class of von Mises-type functionals of the survival time distribution $F$. Conditions are given under which an almost sure asymptotic representation holds for the estimator, obtained by applying the same functional to $\hat{F}_n$, the product-limit estimator of Kaplan and Meier.
The present paper derives various survival tests and their optimality results for randomly censored lifetime data by an extension of familiar rank tests arguments. The approach is based on local asymptotic normal models which have a natural interpretation in terms of hazard rates. In particular, the description of classical rank tests by hazard rates may be of separate interest. As an application of the new methods, a justification of conditional survival tests is given also…
A general weak convergence theory is developed for time-sequential censored rank statistics in the two-sample problem of comparing time to failure between two treatment groups, such as in the case of a clinical trial in which patients enter serially and, after being randomly allocated to one of two treatments, are followed until they fail or withdraw from the study or until the study is terminated. Applications of the theory to time-sequential tests based on these censored…
Consider a Y-valued response variable having a density function f(⋅∣x) that depends on an X-valued input variable x. It is assumed that X and Y are compact intervals and that f(⋅∣⋅) is continuous and positive on X×Y. Let F(⋅∣x) denote the distribution function of f(⋅∣x) and let Q(⋅∣x) denote its quantile function. A finite-parameter exponential…
Central Limit Theorems for Lp Distances of Kernel Estimators of Densities Under Random Censorship
A sequence of independent nonnegative random variables with common distribution function F is censored on the right by another sequence of independent identically distributed random variables. These two sequences are also assumed to be independent. We estimate the density function f of F by a sequence of kernel estimators fn(t)=(∫−∞∞K((t−x)/h(n))dF^n(x))/h(n), where h(n) is a sequence of numbers, K is kernel density function and F^n…
An attempt is made to determine the logically consistent rules for selecting a vector from any feasible set defined by linear constraints, when either all n-vectors or those with positive components or the probability vectors are permissible. Some basic postulates are satisfied if and only if the selection rule is to minimize a certain function which, if a "prior guess" is available, is a measure of distance from the prior guess. Two further natural postulates restrict the…
We consider interaction splines which model a multivariate regression function f as a constant plus the sum of functions of one variable (main effects), plus the sum of functions of two variables (two-factor interactions), and so on. The estimation of f by the penalized least squares method and the asymptotic properties of the models are studied in this article. It is shown that, under some regularity conditions on the data points, the expected squared error averaged…
An E-Ancillarity Projection Property of Cox's Partial Score Function
This paper shows that Cox's partial score function is the projection of the score function on the (locally) E-ancillary subspace for the nuisance parameter (Small and McLeish). This is done by adapting the concepts of (locally) E-ancillarity and (locally) E-sufficiency for inference functions (McLeish and Small) to an extended Cox's regression model, where the baseline function is allowed to be a predictable process.
This paper is primarily concerned with extending the results of Stein to spherically symmetric distributions. Specifically, when X∼f(∥X−θ∥2), we investigate conditions under which estimators of the form X+ag(X) dominate X for loss functions ∥δ−θ∥2 and loss functions which are concave in ∥δ−θ∥2. Additionally, if the scale is unknown we investigate estimators of the location parameter of the form X+aVg(X) in two…
Let F denote the set of cdf's on R with density everywhere positive. Let CA={(F,G)∈F×F: there exists a unique x∗∈R such that F(x)>G(x)forx<<x∗andF(x)<G(x)forx>x∗},CB−{(F,G)∈F×F:(G,F)∈CA}. Based on independent random samples from F and G (assumed unknown), we give distribution-free tests of H0:F=G vers…
A total of nb judges rank t objects k at a time according to n replications of a BIBD with b blocks. The Durbin statistic is commonly used in this context and is equivalent to the usual analysis of variance on the rankings. The approach considered here is to introduce the notion of compatibility so as to define distances between incomplete rankings based on metrics on the space of complete rankings. Through this device we define a class of test statistics which…
Let X1,X2,… be possibly dependent random variables having the same marginal distribution. Consider the situation where Fˉ(x):=P[X1>x] is regularly varying at ∞ with an unknown index −α<0 which is to be estimated. In the i.i.d. setting, it is well known that Hill's estimator is consistent for α−1, and is asymptotically normally distributed. It is the purpose of this paper to demonstrate that such properties of…
Some relationships have been established between unbiased linear predictors of processes, in signal and noise models, minimizing the predictive mean square error and some smoothing spline functions. We construct a new family of multidimensional splines adapted to the prediction of locally homogeneous random fields, whose "m-spectral measure" (to be defined) is absolutely continuous with respect to Lebesgue measure and satisfies some minor assumptions. By considering…
The Kolmogorov distance between the empirical cdfFn and its symmetrization sFn with respect to an adequate estimator of the center of symmetry of P is a natural statistic for testing symmetry. However, its limiting distribution depends on P. Using critical values from the symmetrically bootstrapped statistic (where the resampling is made from sFn) produces tests that can be easily implemented and have asymptotically the correct levels as well as…
Consider a sequence X1,X2,…,Xn of independent random variables, where X1,X2,…,Xnθ have distribution P, and Xnθ+1,Xnθ+2,…,Xn have distribution Q. The change-point θ∈(0,1) is an unknown parameter to be estimated, and P and Q are two unknown probability distributions. The nonparametric estimators of Darkhovskh and Carlstein are imbedded in a more general framework, where random seminorms are applied…
This paper deals with censored data estimation of a general class of von Mises-type functionals of the survival time distribution F. Conditions are given under which an almost sure asymptotic representation holds for the estimator, obtained by applying the same functional to F^n, the product-limit estimator of Kaplan and Meier.