Smoothing splines of a fixed order are commonly used as nonparametric regression estimates. The only parameter, then, that needs to be estimated is the smoothing parameter, which is often estimated using some form of cross validation. This work allows the order of the smoothing spline to be estimated using a model in which the order parameter is continuous. Within this setting, generalized cross validation and modified maximum likelihood estimates of the order and smoothing…
In this paper we consider the regression model with smooth regression function and smooth error and covariate distributions. We study how well one can estimate functionals of the regression function which may also depend on the distribution of the covariate. This is done by deriving the efficient influence functions of least dispersed regular estimators of such functionals under various assumptions on the parameters of our model. Then we demonstrate how efficient estimates…
In a recent paper, C. F. J. Wu showed that the jackknife estimator of a distribution function has optimal convergence rate $O(n^{-1/2})$, where $n$ denotes the sample size. This rate is achieved by retaining $O(n)$ data values from the original sample during the jackknife algorithm. Wu's result is particularly important since it permits a direct comparison of jackknife and bootstrap methods for distribution estimation. In the present paper we show that a very simple,…
This paper considers estimators of the distribution function of a variable over a finite population, when a sample of units is available and values of a related auxiliary variable are known for the whole population. Theory is offered for several estimators which rely on nonparametric regression, and for calibration estimators which require a parametric model.
We introduce a nonparametric estimator for the renewal function and discuss its properties, including consistency, asymptotic normality and asymptotic validity of bootstrap confidence regions. The underlying theme is that stochastic models can be regarded as functionals or nonlinear operators. This view leads to nonparametric estimators in a natural way and statistical properties of the estimators can be related to the local behaviour of the functionals.
Consistency is shown for the minimum covariance determinant (MCD) estimators of multivariate location and scale and asymptotic normality is shown for the former. The proofs are made possible by showing a separating ellipsoid property for the MCD subset of observations. An analogous property is shown for the MCD subset computed from the population distribution.
Let $\mathbf{H}$ and $\mathbf{G}$ be independently distributed according to the Wishart distributions $W_m(M,\Phi)$ and $W_m(N,\Psi)$, respectively. We derive the limiting null distributions of the likelihood ratio criteria for testing $H_0: \Phi = \Psi$ against $H_1 - H_0$ with $H_1: \Phi \geq \Psi$, and for testing $H^{(R)}_0: \Phi \geq \Psi, \operatorname{rank}(\Phi - \Psi) \leq R$ (for given $R$) against $H_1 - H^{(R)}_0$. They are particular cases of the chi-bar-squared…
The lattice conditional independence model $\mathbf{N}(\mathscr{K})$ is defined to be the set of all normal distributions on $\mathbb{R}^I$ such that for every pair $L, M \in \mathscr{K}, x_L$ and $x_M$ are conditionally independent given $x_{L \cap M}$. Here $\mathscr{K}$ is a ring of subsets of the finite index set $I$ and, for $K \in \mathscr{K}, x_K$ is the coordinate projection of $x \in \mathbb{R}^I$ onto $\mathbb{R}^K$. Statistical properties of $\mathbf{N}(\mathscr{K})$…
This paper introduces and investigates the notion of a hyper Markov law, which is a probability distribution over the set of probability measures on a multivariate space that (i) is concentrated on the set of Markov probabilities over some decomposable graph, and (ii) satisfies certain conditional independence restrictions related to that graph. A stronger version of this hyper Markov property is also studied. Our analysis starts by reconsidering the properties of Markov…
Frank Critchley · Paul Marriott · Mark Salmon·刊期 1993-09-01 · 卷 21（3）·The Annals of Statistics已编入正刊
优选点几何与统计流形AI 解读
A new mathematical object called a preferred point geometry is introduced in order to (a) provide a natural geometric framework in which to do statistical inference and (b) reflect the distinction between homogeneous aspects (e.g., any point $\theta$ may be the true parameter) and preferred point ones (e.g., when $\theta_0$ is the true parameter). Although preferred point geometry is applicable generally in statistics, we focus here on its relationship to statistical…
It is well known that we can use the likelihood ratio statistic to test hypotheses and to construct confidence intervals in full parametric models. Recently, Owen introduced the empirical likelihood method in nonparametric models. In this paper, we generalize his results to biased sample problems. A Wilks theorem leading to a likelihood ratio confidence interval for the mean is given. Some extensions, discussion and simulations are presented.
Standard empirical likelihood confidence intervals for quantiles are identical to sign-test intervals. They have relatively large coverage error, of size $n^{-1/2}$, even though they are two-sided intervals. We show that smoothed empirical likelihood confidence intervals for quantiles have coverage error of order $n^{-1}$, and may be Bartlett-corrected to produce intervals with an error of order only $n^{-2}$. Necessary and sufficient conditions on the smoothing parameter,…
David Assaf · Moshe Pollak · Ya'acov Ritov · Benjamin Yakir·刊期 1993-09-01 · 卷 21（3）·The Annals of Statistics已编入正刊
具有误报概率界的动态抽样正态均值变化检测AI 解读
We show that when dynamic sampling is feasible, there exist surveillance schemes for which the probability of a false alarm is bounded and which have a bounded expected delay when detecting a (true) change. In the case of detecting a change of a normal mean, we probe optimality and suggest procedures. These procedures compare favorably to those having a fixed sampling rate which have been developed for an expectation constraint on the average run length until a false alarm.
We investigate the problem of fiducial prediction for unobserved quantities within the framework of the functional model described previously by Dawid and Stone. It is supposed that these are related to a completely unknown parameter by means of a regular functional model, and that the observations are either given as known functions of the predictands, or are themselves related to them by means of a functional model. We develop algebraic conditions which allow the…
Let $X_1, X_2, \ldots$ be a sequence of i.i.d. random variables with d.f. $F$. We observe $Z_i = \min(X_i,Y_i)$ and $\delta_i = 1_{\{X_i \leq Y_i\}}$, where $Y_1, Y_2, \ldots$ is a sequence of i.i.d. censoring random variables. Denote by $\hat{F}_n$ the Kaplan-Meier estimator of $F$. We show that for any $F$-integrable function $\varphi, \int\varphi d\hat{F}_n$ converges almost surely and in the mean. The result may be applied to yield consistency of many estimators under…
For small sample random effects models, results are derived which show in certain cases, and indicate in general, that an estimated random effects variance matrix may be used in the weight matrices without causing undue error in the empirically weighted mean. Exact error variances are derived mathematically for the empirically weighted mean for the two sample case in one and two dimensions. Simulation is used to determine errors for a practical example of six five-variate…
In a heteroscedastic linear model, we establish the asymptotic normality of the iterative weighted least squares estimators with weights constructed by using the within-group residuals obtained from the previous model fitting. An adaptive procedure is proposed which ensures that the iterative process stops after a finite number of iterations and produces an estimator asymptotically equivalent to the best estimator that can be obtained by using the iterative procedure.…
The limiting distribution of the normalized autocorrelation coefficient in the case of a unit root is given in a closed form. It is found that high order transcendental functions such as the parabolic cylinder functions are indispensable to express this distribution, thus departing from the simple standard normal distribution that arises in the case of a stable root. Using the formulae derived in this paper, some numerical results available from previous studies are then…
Let $(\mathbf{X}_t, Y_t)$ be a stationary time series with $\mathbf{X}_t$ being $R^d$-valued and $Y_t$ real valued, and where $Y_t$ is not necessarily bounded. Let $E(Y_0 \mid \mathbf{X}_0)$ be the conditional mean function. Under appropriate regularity conditions, local average estimators of this function can be chosen to achieve the optimal rate of convergence $(n^{-1} \log n)^{1/(d + 2)}$ in $L_\infty$ norm restricted to a compact. The result answers a question raised by…
A variety of exponential models with affine dual foliations have been noted to possess certain rather similar statistical properties. To give a precise meaning to what has been conceived as "similar," we here propose a set of five conditions, of a differential geometric/statistical nature, that specify the class of what we term orthogeodesic models. It is discussed how these conditions capture the properties in question, and it is shown that some important nonexponential…
In this paper we consider the problem of robust estimation of the scale of the location residuals when the underlying distribution of the data belongs to a contamination neighborhood of a parametric location-scale family. We define the class of $M$-estimates of scale with general location, and show that under certain regularity assumptions, these scale estimates converge to their asymptotic functionals uniformly with respect to the underlying distribution, and with respect…
A new class of bias-robust estimates of multiple regression is introduced. If $y$ and $x$ are two real random variables, let $T(y, x)$ be a univariate robust estimate of regression of $y$ on $x$ through the origin. The regression estimate $\mathbf{T}(y, \mathbf{x})$ of a random variable $y$ on a random vector $\mathbf{x} = (x_1,\cdots, x_p)'$ is defined as the vector $\mathbf{t} \in \mathfrak{R}^p$ which minimizes $\sup_{\|\mathbf{\lambda}\| = 1} \mid T(y - \mathbf{t'x, \lambda' x}) \mid s(\mathbf{\lambda'x})$…
The background of the paper is the empirical observation from a variety of subject areas that long-range correlations appear to be much more frequent than has been previously assumed. This includes high-quality measurement series which are commonly treated as prototypes of "i.i.d." observations. Evidence is briefly cited in the paper. It has already been shown elsewhere that long-range dependence leads to results that can be qualitatively different from those obtained under…
Suppose that $(X_1, Y_1),\cdots, (X_n, Y_n)$ are i.i.d. bivariate random vectors and that $\xi_p(x)$ is the $p$-quantile of $Y_1$ given $X_1 = x$ for $0 < p < 1$. Estimation of $\xi_p(x)$, when it is monotone in $x$, has been studied in the literature. In the nonparametric conditional quantile estimation one uses only some smoothness assumptions. The asymptotic properties are superior in the latter case; however, monotonicity is not guaranteed. We introduce a new estimator…
In a recent paper, C. F. J. Wu showed that the jackknife estimator of a distribution function has optimal convergence rate O(n−1/2), where n denotes the sample size. This rate is achieved by retaining O(n) data values from the original sample during the jackknife algorithm. Wu's result is particularly important since it permits a direct comparison of jackknife and bootstrap methods for distribution estimation. In the present paper we show that a very simple,…
Let H and G be independently distributed according to the Wishart distributions Wm(M,Φ) and Wm(N,Ψ), respectively. We derive the limiting null distributions of the likelihood ratio criteria for testing H0:Φ=Ψ against H1−H0 with H1:Φ≥Ψ, and for testing H0(R):Φ≥Ψ,rank(Φ−Ψ)≤R (for given R) against H1−H0(R). They are particular cases of the chi-bar-squared…
The lattice conditional independence model N(K) is defined to be the set of all normal distributions on RI such that for every pair L,M∈K,xL and xM are conditionally independent given xL∩M. Here K is a ring of subsets of the finite index set I and, for K∈K,xK is the coordinate projection of x∈RI onto RK. Statistical properties of N(K)…
A new mathematical object called a preferred point geometry is introduced in order to (a) provide a natural geometric framework in which to do statistical inference and (b) reflect the distinction between homogeneous aspects (e.g., any point θ may be the true parameter) and preferred point ones (e.g., when θ0 is the true parameter). Although preferred point geometry is applicable generally in statistics, we focus here on its relationship to statistical…
Standard empirical likelihood confidence intervals for quantiles are identical to sign-test intervals. They have relatively large coverage error, of size n−1/2, even though they are two-sided intervals. We show that smoothed empirical likelihood confidence intervals for quantiles have coverage error of order n−1, and may be Bartlett-corrected to produce intervals with an error of order only n−2. Necessary and sufficient conditions on the smoothing parameter,…
Let X1,X2,… be a sequence of i.i.d. random variables with d.f. F. We observe Zi=min(Xi,Yi) and δi=1{Xi≤Yi}, where Y1,Y2,… is a sequence of i.i.d. censoring random variables. Denote by F^n the Kaplan-Meier estimator of F. We show that for any F-integrable function φ,∫φdF^n converges almost surely and in the mean. The result may be applied to yield consistency of many estimators under…
Let (Xt,Yt) be a stationary time series with Xt being Rd-valued and Yt real valued, and where Yt is not necessarily bounded. Let E(Y0∣X0) be the conditional mean function. Under appropriate regularity conditions, local average estimators of this function can be chosen to achieve the optimal rate of convergence (n−1logn)1/(d+2) in L∞ norm restricted to a compact. The result answers a question raised by…
In this paper we consider the problem of robust estimation of the scale of the location residuals when the underlying distribution of the data belongs to a contamination neighborhood of a parametric location-scale family. We define the class of M-estimates of scale with general location, and show that under certain regularity assumptions, these scale estimates converge to their asymptotic functionals uniformly with respect to the underlying distribution, and with respect…
A new class of bias-robust estimates of multiple regression is introduced. If y and x are two real random variables, let T(y,x) be a univariate robust estimate of regression of y on x through the origin. The regression estimate T(y,x) of a random variable y on a random vector x=(x1,⋯,xp)′ is defined as the vector t∈Rp which minimizes sup∥λ∥=1∣T(y−t′x,λ′x)∣s(λ′x)…
Suppose that (X1,Y1),⋯,(Xn,Yn) are i.i.d. bivariate random vectors and that ξp(x) is the p-quantile of Y1 given X1=x for 0<p<1. Estimation of ξp(x), when it is monotone in x, has been studied in the literature. In the nonparametric conditional quantile estimation one uses only some smoothness assumptions. The asymptotic properties are superior in the latter case; however, monotonicity is not guaranteed. We introduce a new estimator…