An estimation procedure for $(b, g)$ is developed for the transformation model $g(Y) = bz + \text{error, where} g$ is an unspecified strictly increasing function. The estimator for $b$ can be viewed as a hybrid between an $M$-estimator and an $R$-estimator. It differs from an $M$-estimator in that the dependent variable is replaced by a score based on ranks and from an $R$-estimator in that the ranks of dependent variable itself are used, not the ranks of the residuals. This…
Regression with the error measured by the supremum norm is considered. Analytic functions on [0, 1] and functions with a bounded $r$th derivative are considered as functions to be estimated. It is assumed that the experimenter chooses the points at which the observations are taken. Polynomial and piecewise polynomial estimates are considered. Asymptotic and nonasymptotic bounds for the error are obtained.
In a parametric counterpart of the nonparametric considerations of Neuhaus, a class of quadratic goodness of fit statistics is developed generalizing the usual Cramer-von Mises statistics.
Let $\{(X_i, \theta_i)\}$ be a sequence of independent random vectors where $X_i$ has a uniform density $U(0, \theta_i)$ for $0 < \theta_i < m (< \infty)$ and the unobservable $\theta_i$ are i.i.d. $G$ in some class $\mathscr{G}$ of prior distributions. In the $(n + 1)$st problem we estimate $\theta_{n + 1}$ by $t_n(X_1, \cdots, X_n, X_{n + 1}) \doteq t_n(\mathbf{X})$, incurring the risk $R_n \doteq \mathbf{E}(t_n(\mathbf{X}) - \theta_{n + 1})^2$, where $\mathbf{E}$ denotes…
This paper presents necessary and sufficient conditions for the stochastic ordering of absolute univariate Gaussian random variables. This result is based on the behavior of the particular Gaussian density at zero, which is equivalent to restrictions on the mean and variance.
A procedure for estimating the parameters of a time series is proposed. The estimate minimizes a criterion function which is the weighted sum of squares of the distances between the periodogram and the spectrum of the series. Under regularity conditions, the estimate is shown to be strongly consistent. The asymptotic distribution of the estimate is also obtained. It is shown that, for a Gaussian process, an asymptotically efficient estimate can be obtained by using an…
It is shown that in the stationary autoregressive case (Part I [Sections 1-5)] the distributions of least squares estimators and their close relatives such as sample autocovariances and the recently introduced general $M$-estimators converge to suitable Gaussian distributions uniformly over all Borel sets and uniformly over suitable neighborhoods of the parameter. Specifically, the notion of strongly asymptotically shift equivariance is introduced and it is shown that the…
We study the empirical Bayes decision theory with an $m$-truncated sequential statistical decision problem as the component. An empirical Bayes sequential decision procedure is constructed for the linear loss two-action problem. Asymptotic results are presented regarding the convergence of the Bayes risk of the empirical Bayes sequential decision procedure. With sequential components, an empirical Bayes sequential decision procedure selects both a stopping rule function and…
A class of proper scoring functions which combine the error in a decision problem and the precision of the statistical decision rule is introduced. The Bayesian procedures with respect to these loss functions are pairs formed by the usual Bayes decision and by the expected posterior loss. A necessary and sufficient condition for admissibility under the corresponding risk is given.
Yaakov Friedman · Alexander Gelman · Eswar Phadia·刊期 1988-09-01 · 卷 16（3）·The Annals of Statistics已编入正刊
Kolmogorov–Smirnov损失下分布函数的最优不变估计AI 解读
Given a random sample of size $n$ from an unknown continuous distribution function $F$, we consider the problem of estimating $F$ nonparametrically from a decision theoretic approach. In our treatment, we assume the Kolmogorov-Smirnov loss function and the group of all one-to-one monotone transformations of real numbers onto themselves which leave the sample values invariant. Under this setup, we obtain a best invariant estimator of $F$ which is shown to be unique. This…
Robust fixed size confidence procedures are derived for the location parameter $\theta$ of a sample of $N$ i.i.d. observations of a scalar random variable $Z$ with CDF $F(z - \theta)$. Here, $\theta$ is restricted to a closed interval $\Omega$ and the uncertainty in $F$ is modeled by an uncertainty class $\mathscr{F}$. These robust confidence procedures are, in turn, based on the solution of a related robust minimax decision problem that is characterized by a zero-one loss…
Polynomial regression in $q$ variables of degree $n$ on the $q$-cube is considered. Approximate $D$-optimal and approximate $D_s$-optimal designs for estimating higher degree terms are investigated. Numerical results are given for $n = 2$ with arbitrary $q$ and for $n = 3, 4, 5$ and $q = 2, 3$. Exact solutions are given within the class of product designs together with some efficiency calculations.
Let $(X_1, Y_1), \cdots, (X_n, Y_n)$ be a sample, denote the conditional density of $Y_i\mid X_i = x_i$ as $f(y\mid x_i, \theta(x_i))$ and $\theta$ an element of a metric space $(\Theta, d)$. A lower bound is provided for the $d$-error in estimating $\theta$. The order of the bound depends on the local behavior of the Kullback information of the conditional density. As an application, we consider the case where $\Theta$ is the space of $q$-smooth functions on $\lbrack 0, 1 \rbrack^d$…
We consider an arbitrary sequence of kernel density estimates $f_n$ with kernels $K_n$ possibly depending upon $n$. Under a mild restriction on the sequence $K_n$, we obtain inequalities of the type $E\big(\int|f_n - f|\big) \geq (1 + o(1))\Psi(n, f),$ where $f$ is the density being estimated and $\Psi(n, f)$ is a function of $n$ and $f$ only. The function $\psi$ can be considered as an indicator of the difficulty of estimating $f$ with any kernel estimate.
In a clinical trial to compare two treatments, suppose that patients are assigned to treatment according to a certain class of restricted randomization rules. Conditional on the difference in numbers between the two treatment groups, the asymptotic null distribution of a class of permutation tests is shown to be Gaussian.
A general version of the Cox regression model is studied as a discrete time process consisting of the embedded experiments at the jump times. Simple conditions are given for the consistency and asymptotic normality of the estimator for the regression parameters.
In this paper the asymptotic behaviour of the product limit estimator $F_n$ of an unknown distribution is investigated. We give an approximation of the difference $F_n(x) - F(x)$ by a Gaussian process and also by the average of i.i.d. processes. We get almost as good an approximation of the stochastic process $F_n(x) - F(x)$ as one can get for the analogous problem when the maximum likelihood estimator is approximated by a Gaussian random variable or by the average of i.i.d.…
When there exist nuisance parameters whose number increases in proportion to that of independent observations, it is in general difficult to get a consistent or efficient estimator of a common structural parameter. The present paper proposes a new theory based on a vector bundle consisting of certain random variables over the statistical model. Structures and properties of estimating functions are elucidated in the class of consistent estimators. A necessary and sufficient…
Let $g$ be a nonlinear function of the regression parameters $\beta$ in a heteroscedastic linear model and $\hat{\beta}$ be the least squares estimator of $\beta.$ We consider the estimation of the variance and bias of $g(\hat{\beta})$ [as an estimator of $g(\beta)$] by using three resampling methods: the weighted jackknife, the unweighted jackknife and the bootstrap. The asymptotic orders of the mean squared errors and biases of the resampling variance and bias estimators…
In this paper, for right censored competing risks data, a class of tests developed for comparing the cumulative incidence of a particular type of failure among different groups. The tests are based on comparing weighted averages of the hazards of the subdistribution for the failure type of interest. Asymptotic results are derived by expressing the statistics in terms of counting processes and using martingale central limit theory. It is proposed that weight functions very…
Richard D. Gill · Yehuda Vardi · Jon A. Wellner·刊期 1988-09-01 · 卷 16（3）·The Annals of Statistics已编入正刊
有偏抽样模型中经验分布的大样本理论AI 解读
Vardi (1985a) introduced an $s$-sample model for biased sampling, gave conditions which guarantee the existence and uniqueness of the nonparametric maximum likelihood estimator $\mathbb{G}_n$ of the common underlying distribution $G$ and discussed numerical methods for calculating the estimator. Here we examine the large sample behavior of the NPMLE $\mathbb{G}_n$, including results on uniform consistency of $\mathbb{G}_n$, convergence of $\sqrt n (\mathbb{G}_n - G)$ to a…
Let $\Omega$ denote the parameter space of a statistical model and let $\mathscr{K}$ be the domain of variation of the parameter of interest. Various differential-geometric structures on $\Omega$ are considered, including the expected information metric and the $\alpha$-connections studied by Chentsov and Amari, as well as the observed information metric and the observed $\alpha$-connections introduced by Barndorff-Nielsen. Under certain conditions these geometric objects on…
We develop a unified framework within which many commonly used bootstrap critical points and confidence intervals may be discussed and compared. In all, seven different bootstrap methods are examined, each being usable in both parametric and nonparametric contexts. Emphasis is on the way in which the methods cope with first- and second-order departures from normality. Percentile-$t$ and accelerated bias-correction emerge as the most promising of existing techniques. Certain…
Properties of a probability density estimator having the rational form of an ARMA spectrum are investigated. Under various conditions on the underlying density's Fourier coefficients, the ARMA estimator is shown to have asymptotically smaller mean integrated squared error (MISE) than the best tapered Fourier series estimator. The most interesting cases are those in which the Fourier coefficients $\phi_j$ are asymptotic to $Kj^{-p}$ as $j \rightarrow \infty$, where $\rho > 1/2$…
An estimation procedure for (b,g) is developed for the transformation model g(Y)=bz+error, whereg is an unspecified strictly increasing function. The estimator for b can be viewed as a hybrid between an M-estimator and an R-estimator. It differs from an M-estimator in that the dependent variable is replaced by a score based on ranks and from an R-estimator in that the ranks of dependent variable itself are used, not the ranks of the residuals. This…
Regression with the error measured by the supremum norm is considered. Analytic functions on [0, 1] and functions with a bounded rth derivative are considered as functions to be estimated. It is assumed that the experimenter chooses the points at which the observations are taken. Polynomial and piecewise polynomial estimates are considered. Asymptotic and nonasymptotic bounds for the error are obtained.
Convergence Rates for Empirical Bayes Estimation in the Uniform U(0,θ) Distribution
Let {(Xi,θi)} be a sequence of independent random vectors where Xi has a uniform density U(0,θi) for 0<θi<m(<∞) and the unobservable θi are i.i.d. G in some class G of prior distributions. In the (n+1)st problem we estimate θn+1 by tn(X1,⋯,Xn,Xn+1)≐tn(X), incurring the risk Rn≐E(tn(X)−θn+1)2, where E denotes…
It is shown that in the stationary autoregressive case (Part I [Sections 1-5)] the distributions of least squares estimators and their close relatives such as sample autocovariances and the recently introduced general M-estimators converge to suitable Gaussian distributions uniformly over all Borel sets and uniformly over suitable neighborhoods of the parameter. Specifically, the notion of strongly asymptotically shift equivariance is introduced and it is shown that the…
We study the empirical Bayes decision theory with an m-truncated sequential statistical decision problem as the component. An empirical Bayes sequential decision procedure is constructed for the linear loss two-action problem. Asymptotic results are presented regarding the convergence of the Bayes risk of the empirical Bayes sequential decision procedure. With sequential components, an empirical Bayes sequential decision procedure selects both a stopping rule function and…
Given a random sample of size n from an unknown continuous distribution function F, we consider the problem of estimating F nonparametrically from a decision theoretic approach. In our treatment, we assume the Kolmogorov-Smirnov loss function and the group of all one-to-one monotone transformations of real numbers onto themselves which leave the sample values invariant. Under this setup, we obtain a best invariant estimator of F which is shown to be unique. This…
Robust fixed size confidence procedures are derived for the location parameter θ of a sample of N i.i.d. observations of a scalar random variable Z with CDF F(z−θ). Here, θ is restricted to a closed interval Ω and the uncertainty in F is modeled by an uncertainty class F. These robust confidence procedures are, in turn, based on the solution of a related robust minimax decision problem that is characterized by a zero-one loss…
Efficient Ds-Optimal Designs for Multivariate Polynomial Regression on the q-Cube
Polynomial regression in q variables of degree n on the q-cube is considered. Approximate D-optimal and approximate Ds-optimal designs for estimating higher degree terms are investigated. Numerical results are given for n=2 with arbitrary q and for n=3,4,5 and q=2,3. Exact solutions are given within the class of product designs together with some efficiency calculations.
Let (X1,Y1),⋯,(Xn,Yn) be a sample, denote the conditional density of Yi∣Xi=xi as f(y∣xi,θ(xi)) and θ an element of a metric space (Θ,d). A lower bound is provided for the d-error in estimating θ. The order of the bound depends on the local behavior of the Kullback information of the conditional density. As an application, we consider the case where Θ is the space of q-smooth functions on [0,1]d…
We consider an arbitrary sequence of kernel density estimates fn with kernels Kn possibly depending upon n. Under a mild restriction on the sequence Kn, we obtain inequalities of the type E(∫∣fn−f∣)≥(1+o(1))Ψ(n,f), where f is the density being estimated and Ψ(n,f) is a function of n and f only. The function ψ can be considered as an indicator of the difficulty of estimating f with any kernel estimate.
In this paper the asymptotic behaviour of the product limit estimator Fn of an unknown distribution is investigated. We give an approximation of the difference Fn(x)−F(x) by a Gaussian process and also by the average of i.i.d. processes. We get almost as good an approximation of the stochastic process Fn(x)−F(x) as one can get for the analogous problem when the maximum likelihood estimator is approximated by a Gaussian random variable or by the average of i.i.d.…
Let g be a nonlinear function of the regression parameters β in a heteroscedastic linear model and β^ be the least squares estimator of β. We consider the estimation of the variance and bias of g(β^) [as an estimator of g(β)] by using three resampling methods: the weighted jackknife, the unweighted jackknife and the bootstrap. The asymptotic orders of the mean squared errors and biases of the resampling variance and bias estimators…
A Class of K-Sample Tests for Comparing the Cumulative Incidence of a Competing Risk
Vardi (1985a) introduced an s-sample model for biased sampling, gave conditions which guarantee the existence and uniqueness of the nonparametric maximum likelihood estimator Gn of the common underlying distribution G and discussed numerical methods for calculating the estimator. Here we examine the large sample behavior of the NPMLE Gn, including results on uniform consistency of Gn, convergence of n(Gn−G) to a…
Differential Geometry, Profile Likelihood, L-Sufficiency and Composite Transformation Models
Let Ω denote the parameter space of a statistical model and let K be the domain of variation of the parameter of interest. Various differential-geometric structures on Ω are considered, including the expected information metric and the α-connections studied by Chentsov and Amari, as well as the observed information metric and the observed α-connections introduced by Barndorff-Nielsen. Under certain conditions these geometric objects on…
We develop a unified framework within which many commonly used bootstrap critical points and confidence intervals may be discussed and compared. In all, seven different bootstrap methods are examined, each being usable in both parametric and nonparametric contexts. Emphasis is on the way in which the methods cope with first- and second-order departures from normality. Percentile-t and accelerated bias-correction emerge as the most promising of existing techniques. Certain…
Properties of a probability density estimator having the rational form of an ARMA spectrum are investigated. Under various conditions on the underlying density's Fourier coefficients, the ARMA estimator is shown to have asymptotically smaller mean integrated squared error (MISE) than the best tapered Fourier series estimator. The most interesting cases are those in which the Fourier coefficients ϕj are asymptotic to Kj−p as j→∞, where ρ>1/2…