Asymptotics of Huber-Dutter Estimators for Partial Linear Model with Nonstochastic Designs

For partial linear model Y=X~τβ_0 _(g0)(T) εwith unknown β_0∈R~d and an unknown smooth function go, this paper considers the Huber-Dutter estimators of β_0, scale σfor the errors and the function go respectively, in which the smoothing B-spline function is used. Under some regular conditions, it is shown that the Huber-Dutter estimators of β_0 and σare asymptotically normal with convergence rate n~((-1)/2) and the B-spline Huber-Dutter estimator of go achieves the optimal convergence rate in nonparametric regression. A simulation study demonstrates that the Huber-Dutter estimator of β_0 is competitive with its M-estimator without scale parameter and the ordinary least square estimator. An example is presented after the simulation study.     

NS Condition of Admissibility for the Linear Estimator of Normal Mean with Unknown Variance

Suppose Y - N（β, σ^2 In）, where β ∈ R^n and σ^2 〉 0 are unknown. We study the admissibility of linear estimators of mean vector under a quadratic loss function. A necessary and sufficient condition of the admissible linear estimator is given.     

Testing Serial Correlation in Semiparametric Varying-Coefficient Partially Linear EV Models

This paper studies estimation and serial correlation test of a semiparametric varying-coefficient partially linear EV model of the form Y = X^Tβ ＋Z^Tα（T） ＋ε,ξ = X ＋ η with the identifying condition E[（ε,η^T）^T] =0, Cov[（ε,η^T）^T] = σ^2Ip＋1. The estimators of interested regression parameters /3 , and the model error variance σ2, as well as the nonparametric components α（T）, are constructed. Under some regular conditions, we show that the estimators of the unknown vector β and the unknown parameter σ2 are strongly consistent and asymptotically normal and that the estimator of α（T） achieves the optimal strong convergence rate of the usual nonparametric regression. Based on these estimators and asymptotic properties, we propose the VN,p test statistic and empirical log-likelihood ratio statistic for testing serial correlation in the model. The proposed statistics are shown to have asymptotic normal or chi-square distributions under the null hypothesis of no serial correlation. Some simulation studies are conducted to illustrate the finite sample performance of the proposed tests.     

Asymptotic properties for the semiparametric regression model with randomly censored data

Suppose that the patients’ survival times.Y, are random variables following the semiparametric regression modelY = Xβ +g(T) + ε, where (X,T) is a radom vector taking values inR×[0,1],βis an unknown parameter,g (*) is an unknown smooth regression function andE is the random error with zero mean and variance σ². It is assumed that (X,T) is independent of E. The estimators andg _n (*) of P andg(*) are defined, respectively, when the observations are randomly censored on the right and the censoring distribution is unknown. Moreover, it is shown that is asymptotically normal andg _n (*) is weak consistence with rateO _p(n-1/3). Project supported by China Postdoctoral Science Foundation and the National Natural Science Foundation of China.     

Polynomial Regression with Censored Data based on Preliminary Nonparametric Estimation

Consider the polynomial regression model , where σ²(X)=Var(Y|X) is unknown, and ε is independent of X and has zero mean. Suppose that Y is subject to random right censoring. A new estimation procedure for the parameters β₀,...,β _p is proposed, which extends the classical least squares procedure to censored data. The proposed method is inspired by the method of Buckley and James (1979, Biometrika, 66, 429–436), but is, unlike the latter method, a noniterative procedure due to nonparametric preliminary estimation of the conditional regression function. The asymptotic normality of the estimators is established. Simulations are carried out for both methods and they show that the proposed estimators have usually smaller variance and smaller mean squared error than the Buckley–James estimators. The two estimation procedures are also applied to a medical and an astronomical data set.     

WAVELET ESTIMATION IN NONPARAMETRIC MODEL UNDER MARTINGALE DIFFERENCE ERRORS

§1　IntroductionConsider the following heteroscedastic regression model:Yi =g(xi) +σiei,　1≤i≤n,(1.1)whereσ2i=f(ui) ,(xi,ui) are nonrandom design points,0≤x0 ≤x1 ≤...≤xn=1and0≤u0≤u1 ≤...≤un=1,Yi are the response variables,ei are random errors,and f(·) andg(·) are unknown functions defined on closed interval[0 ,1] .It is well known thatregression model has many applications in practical problems,sothe model (1.1) and its special cases have been studied extensively. For instance,…     

Gumbel statistics for the longest interval of identical spins in a one-dimensional Gibbs measure

We consider one-dimensional Gibbs measures on spin configurations σ ∈ {–1,+1}^ℤ. For N ∈ ℕ let l _N denote the length of the longest interval of consecutive spins of the same kind in the interval [0,N]. We show that the distribution of a suitable continuous modification l _c (N) of l _N converges to the Gumbel distribution, i.e., for some α, β ∈ (0, ∞) and γ ∈ ℝ, lim _{N →∞} ℙ(l _c (N) ≤ α log N + βx + γ) = e ^{–e –x} . Received: 2 September 2002     

On equalities for BLUEs under misspecified Gauss-Markov models

This paper studies relationships between the best linear unbiased estimators （BLUEs） of an estimable parametric functions Kβunder the Gauss-Markov model {y, Xβ, σ^2]E} and its misspecified model {y, X0β,σ^2∑0}. In addition, relationships between BLUEs under a restricted Gauss Markov model and its misspecified model are also investigated.     

Semi-parametric Estimation of the Hölder Exponent of a Stationary Gaussian Process with Minimax Rates

Let (X(t))_t∈[0,1] be a centered Gaussian process with stationary increments such that IE[(X_{u+t-X_u)²] = C|t|^s+r(t). Assume that there exists an extra parameter β > 0 and a polynomial P of degree smaller than s + β such that |r(t)-P(t)| is bounded with respect to |t|^s+β. We consider the problem of estimating the parameter s ∈ (0,2) in the asymptotic framework given by n equispaced observations in [0, 1]. Adding possibly stronger regularity conditions to r, we define classes of such processes over which we show that s cannot be estimated at a better rate than n^{min(1/2, β)}. Then, we study increment (or, more generally, discrete variation) estimators. We obtained precise bounds of the bias of the variance which show that the bias mainly depend on the parameter β and the variance on two terms, one depending on the parameter s and one on some regularity properties of r. A central limit theorem is given when the variance term relying on s dominates the bias and the other variance term. Eventually, we exhibit an estimator which achieves the minimax rate over a wide range of classes for which sufficient regularity conditions are assumed on r. This revised version was published online in August 2006 with corrections to the Cover Date.     

To the problem of a strong differentiability of integrals along different directions

It is proved that for any given sequence (σ _n ,n ∈ ℕ)=Γ₀ ⊂ Γ, where Γ is the set of all directions in ℝ² (i.e., pairs of orthogonal straight lines) there exists a locally integrable functionf on ℝ² such that: (1) for almost all directionsσ ∈ Γ\Γ₀ the integral ∫f is differentiable with respect to the familyB _2σ of open rectangles with sides parallel to the straight lines fromσ: (2) for every directionσ _n ∈ Γ₀ the upper derivative of ∫f with respect toB _{2σ n} equals +∞; (3) for every directionσ ∈ Γ the upper derivative of ∫ |f| with respect toB _2σ equals +∞.     

When the cone condition fails

Summary We study the class of convergence E∪L¹of a family of moving averages which does not satisfy the cone condition. We show that if E₀is a finite subset of Ewhich is (E)-stable for the multiplication operation: f,g∈E₀ →f·g∈E, then the supremum sup { f, f∈E₀} is dominated by sup{ g, g∈G₀}where G₀is a Gaussian family with same covariance function. This is used to derive a maximal inequality for families Fsuch that each finite subset is E-stable and Fis a GB set.     

THE ASYMPTOTICALLY OPTIMAL EMPIRICAL BAYES ESTIMATION IN MULTIPLE LINEAR REGRESSION MODEL

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Wavelet Estimation in Heteroscedastic Model Under Censored Samples

Consider the heteroscedastic regression model Yi = g（xi） ＋ σiei, 1 ≤ i ≤ n, where σi^2 = f（ui）, here （xi, ui） being fixed design points, g and f being unknown functions defined on [0, 1], ei being independent random errors with mean zero. Assuming that Yi are censored randomly and the censored distribution function is known or unknown, we discuss the rates of strong uniformly convergence for wavelet estimators of g and f, respectively. Also, the asymptotic normality for the wavelet estimators of g is investigated.     

Adaptive estimation in circular functional linear models

We consider the problem of estimating the slope parameter in circular functional linear regression, where scalar responses Y ₁, ..., Y _n are modeled in dependence of 1-periodic, second order stationary random functions X ₁, ...,X _n . We consider an orthogonal series estimator of the slope function β, by replacing the first m theoretical coefficients of its development in the trigonometric basis by adequate estimators. We propose a model selection procedure for m in a set of admissible values, by defining a contrast function minimized by our estimator and a theoretical penalty function; this first step assumes the degree of ill-posedness to be known. Then we generalize the procedure to a random set of admissible m’s and a random penalty function. The resulting estimator is completely data driven and reaches automatically what is known to be the optimal minimax rate of convergence, in terms of a general weighted L ²-risk. This means that we provide adaptive estimators of both β and its derivatives.     

Bootstrap choice of tuning parameters

Consider the problem of estimating θ=θ(P) based on datax _n from an unknown distributionP. Given a family of estimatorsT _{n, β} of θ(P), the goal is to choose β among β∈I so that the resulting estimator is as good as possible. Typically, β can be regarded as a tuning or smoothing parameter, and proper choice of β is essential for good performance ofT _{n, β} . In this paper, we discuss the theory of β being chosen by the bootstrap. Specifically, the bootstrap estimate of β, , is chosen to minimize an empirical bootstrap estimate of risk. A general theory is presented to establish the consistency and weak convergence properties of these estimators. Confidence intervals for θ(P) based on , are also asymptotically valid. Several applications of the theory are presented, including optimal choice of trimming proportion, bandwidth selection in density estimation and optimal combinations of estimates.     

Full likelihood inferences in the Cox model: an empirical likelihood approach

For the regression parameter β ₀ in the Cox model, there have been several estimators constructed based on various types of approximated likelihood, but none of them has demonstrated small-sample advantage over Cox’s partial likelihood estimator. In this article, we derive the full likelihood function for (β ₀, F ₀), where F ₀ is the baseline distribution in the Cox model. Using the empirical likelihood parameterization, we explicitly profile out nuisance parameter F ₀ to obtain the full-profile likelihood function for β ₀ and the maximum likelihood estimator (MLE) for (β ₀, F ₀). The relation between the MLE and Cox’s partial likelihood estimator for β ₀ is made clear by showing that Taylor’s expansion gives Cox’s partial likelihood estimating function as the leading term of the full-profile likelihood estimating function. We show that the log full-likelihood ratio has an asymptotic chi-squared distribution, while the simulation studies indicate that for small or moderate sample sizes, the MLE performs favorably over Cox’s partial likelihood estimator. In a real dataset example, our full likelihood ratio test and Cox’s partial likelihood ratio test lead to statistically different conclusions.     

Double stage estimation of population variance

Summary Consider a normal population with mean μ and variance σ². We are interested in the estimation of population variance with the help of guess value σ ₀ ² and a sample of observations. In this paper, a double stage shrinkage estimator based on the shrinkage estimatorks ₁ ² +(1-k)σ ₀ ² ifs ₁ ² ∈R and the usual estimator ifs ₁ ² ∋R, whereR is some specified region, have been proposed. The expressions for bias and mean squared error have been obtained. Comparison with the usual estimators ² have been made. It was found that though the largest gain is obtained fork=0, we can use with 0≦k≦1/2 even when σ² is very close to σ ₀ ²     

Precise asymptotics of error variance estimator in partially linear models

In this paper, we focus our attention on the precise asymptotics of error variance estimator in partially linear regression models, y _i = x _i ^τ β + g(t _i ) + ε _i , 1 ≤ i ≤ n, {ε _i , i = 1, ⋯ n} are i.i.d random errors with mean 0 and positive finite variance σ ². Following the ideas of Allan Gut and Aurel Spătaru^[7,8] and Zhang^[21], on precise asymptotics in the Baum-Katz and Davis laws of large numbers and precise rate in laws of the iterated logarithm, respectively, and subject to some regular conditions, we obtain the corresponding results in partially linear regression models.      

Asymptotic analysis and scaling of friction parameters

We consider an eigenvalue problem associated to the antiplane shearing on a system of collinear faults under a slip-dependent friction law. Firstly we consider a periodic system of faults in the whole plane. We prove that the first eigenvalues/eigenfunctions of different physical periodicity are all equal and that the other eigenvalues converge to this first common eigenvalue as their physical period becomes indefinitely large. Secondly we consider a large scale fault system composed on a small scale collinear faults periodically disposed. If β₀^* is the first eigenvalue of the periodic problem in the whole plane, we prove that the first eigenvalue of the microscopic problem behaves as β₀^*/∈ when ∈→ 0 regardless the geometry of the domain (here ∈ is the scale quotient). The geophysical implications of this result is that the macroscopic critical slip D_c scales with D_c^∈/∈ (here D_c^∈ is the small scale critical slip).     

Convolution properties of some classes of analytic functions

Let A denote the class of functions which are analytic in |z|<1 and normalized so that f(0)=0 and f′(0)=1, and let R(α, β)⊂A be the class of functions f such thatRe[f′(z)+αzf″(z)]>β,Re α>0, β<1. We determine conditions under which (i) f ∈ R(α₁, β₁), g ∈ R(α₂, β₂) implies that the convolution f×g of f and g is convex; (ii) f ∈ R(0, β₁), g ∈ R(0, β₂) implies that f×g is starlike; (iii) f≠A such that f′(z)[f(z)/z]^μ-1 ≺ 1 + λz, μ>0, 0<λ<1, is starlike, and (iv) f≠A such that f′(z)+αzf″(z) ≺ 1 + λz, α>0, δ>0, is convex or starlike. Bibliography: 16 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 138–154.