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1.
Xia Chen 《Statistics》2013,47(5):687-696
Consider the nonparametric regression model with martingale difference errors. Nonparametric estimator g n (x) of regression function g(x) will be introduced, and its asymptotic properties are studied. In particular, the pointwise and uniform convergence of g n (x) and its asymptotic normality will be investigated. This extends the earlier work on independent random errors.  相似文献   

2.
In this work, we investigate a new class of skew-symmetric distributions, which includes the distributions with the probability density function (pdf) given by g α(x)=2f(x) Gx), introduced by Azzalini [A class of distributions which includes the normal ones, Scand. J. Statist. 12 (1985), pp. 171–178]. We call this new class as the symmetric-skew-symmetric family and it has the pdf proportional to f(x) G βx), where G β(x) is the cumulative distribution function of g β(x). We give some basic properties for the symmetric-skew-symmetric family and study the particular case obtained from the normal distribution.  相似文献   

3.
Consider observations (representing lifelengths) taken on a random field indexed by lattice points. Estimating the distribution function F(x) = P(X i  ≤ x) is an important problem in survival analysis. We propose to estimate F(x) by kernel estimators, which take into account the smoothness of the distribution function. Under some general mixing conditions, our estimators are shown to be asymptotically unbiased and consistent. In addition, the proposed estimator is shown to be strongly consistent and sharp rates of convergence are obtained.  相似文献   

4.
In this article, we derive exact expressions for the single and product moments of order statistics from Weibull distribution under the contamination model. We assume that X1, X2, …, Xn ? p are independent with density function f(x) while the remaining, p observations (outliers) Xn ? p + 1, …, Xn are independent with density function arises from some modified version of f(x), which is called g(x), in which the location and/or scale parameters have been shifted in value. Next, we investigate the effect of the outliers on the BLUE of the scale parameter. Finally, we deduce some special cases.  相似文献   

5.
Jump-detection and curve estimation methods for the discontinuous regression function are proposed in this article. First, two estimators of the regression function based on B-splines are considered. The first estimator is obtained when the knot sequence is quasi-uniform; by adding a knot with multiplicity p + 1 at a fixed point x0 on support [a, b], we can obtain the second estimator. Then, the jump locations are detected by the performance of the difference of the residual sum of squares DRSS(x0) (x0 ∈ (a, b)); subsequently the regression function with jumps can be fitted based on piecewise B-spline function. Asymptotic properties are established under some mild conditions. Several numerical examples using both simulated and real data are presented to evaluate the performance of the proposed method.  相似文献   

6.
Suppose that we have a nonparametric regression model Y = m(X) + ε with XRp, where X is a random design variable and is observed completely, and Y is the response variable and some Y-values are missing at random. Based on the “complete” data sets for Y after nonaprametric regression imputation and inverse probability weighted imputation, two estimators of the regression function m(x0) for fixed x0Rp are proposed. Asymptotic normality of two estimators is established, which is used to construct normal approximation-based confidence intervals for m(x0). We also construct an empirical likelihood (EL) statistic for m(x0) with limiting distribution of χ21, which is used to construct an EL confidence interval for m(x0).  相似文献   

7.
Consider the regression model Yi= g(xi) + ei, i = 1,…, n, where g is an unknown function defined on [0, 1], 0 = x0 < x1 < … < xn≤ 1 are chosen so that max1≤i≤n(xi-xi- 1) = 0(n-1), and where {ei} are i.i.d. with Ee1= 0 and Var e1 - s?2. In a previous paper, Cheng & Lin (1979) study three estimators of g, namely, g1n of Cheng & Lin (1979), g2n of Clark (1977), and g3n of Priestley & Chao (1972). Consistency results are established and rates of strong uniform convergence are obtained. In the current investigation the limiting distribution of &in, i = 1, 2, 3, and that of the isotonic estimator g**n are considered.  相似文献   

8.
There are many situations where the usual random sample from a population of interest is not available, due to the data having unequal probabilities of entering the sample. The method of weighted distributions models this ascertainment bias by adjusting the probabilities of actual occurrence of events to arrive at a specification of the probabilities of the events as observed and recorded. We consider two different classes of contaminated or mixture of weight functions, Γ a ={w(x):w(x)=(1−ε)w 0(x)+εq(x),qQ} and Γ g ={w(x):w(x)=w 0 1−ε (x)q ε(x),qQ} wherew 0(x) is the elicited weighted function,Q is a class of positive functions and 0≤ε≤1 is a small number. Also, we study the local variation of ϕ-divergence over classes Γ a and Γ g . We devote on measuring robustness using divergence measures which is based on the Bayesian approach. Two examples will be studied.  相似文献   

9.
ABSTRACT

Consider the heteroscedastic partially linear errors-in-variables (EV) model yi = xiβ + g(ti) + εi, ξi = xi + μi (1 ? i ? n), where εi = σiei are random errors with mean zero, σ2i = f(ui), (xi, ti, ui) are non random design points, xi are observed with measurement errors μi. When f( · ) is known, we derive the Berry–Esseen type bounds for estimators of β and g( · ) under {ei,?1 ? i ? n} is a sequence of stationary α-mixing random variables, when f( · ) is unknown, the Berry–Esseen type bounds for estimators of β, g( · ), and f( · ) are discussed under independent errors.  相似文献   

10.
This paper considers estimation of the function g in the model Yt = g(Xt ) + ?t when E(?t|Xt) ≠ 0 with nonzero probability. We assume the existence of an instrumental variable Zt that is independent of ?t, and of an innovation ηt = XtE(Xt|Zt). We use a nonparametric regression of Xt on Zt to obtain residuals ηt, which in turn are used to obtain a consistent estimator of g. The estimator was first analyzed by Newey, Powell & Vella (1999) under the assumption that the observations are independent and identically distributed. Here we derive a sample mean‐squared‐error convergence result for independent identically distributed observations as well as a uniform‐convergence result under time‐series dependence.  相似文献   

11.
Let H(x, y) be a continuous bivariate distribution function with known marginal distribution functions F(x) and G(y). Suppose the values of H are given at several points, H(x i , y i ) = θ i , i = 1, 2,…, n. We first discuss conditions for the existence of a distribution satisfying these conditions, and present a procedure for checking if such a distribution exists. We then consider finding lower and upper bounds for such distributions. These bounds may be used to establish bounds on the values of Spearman's ρ and Kendall's τ. For n = 2, we present necessary and sufficient conditions for existence of such a distribution function and derive best-possible upper and lower bounds for H(x, y). As shown by a counter-example, these bounds need not be proper distribution functions, and we find conditions for these bounds to be (proper) distribution functions. We also present some results for the general case, where the values of H(x, y) are known at more than two points. In view of the simplification in notation, our results are presented in terms of copulas, but they may easily be expressed in terms of distribution functions.  相似文献   

12.
Let f(x) and g(x) denote two probability density functions and g(x)≠0. There are two ways to estimate the density ratio f(x)/g(x). One is to estimate f(x) and g(x) first and then the ratio, the other is to estimate f(x)/g(x) directly. In this paper, we derive asymptotic mean square errors and central limit theorems for both estimators.  相似文献   

13.
14.
The authors propose a new nonparametric diagnostic test for checking the constancy of the conditional variance function σ2(x) in the regression model Yi = m(xi) + σ(xi)?i, i = 1,…, m. Their test, which does not assume a known parametric form for the conditional mean function m(x), is inspired by a recent asymptotic theory in the analysis of variance when the number of factor levels is large. The authors demonstrate through simulations the good finite‐sample properties of the test and illustrate its use in a study on the effect of drug utilization on health care costs.  相似文献   

15.
We study the construction of regression designs, when the random errors are autocorrelated. Our model of dependence assumes that the spectral density g(ω) of the error process is of the form g(ω) = (1 − α)g0(ω) + αg1(ω), where g0(ω) is uniform (corresponding to uncorrelated errors), α ϵ [0, 1) is fixed, and g1(ω) is arbitrary. We consider regression responses which are exactly, or only approximately, linear in the parameters. Our main results are that a design which is asymptotically (minimax) optimal for uncorrelated errors retains its optimality under autocorrelation if the design points are a random sample, or a random permutation, of points from this distribution. Our results are then a partial extension of those of Wu (Ann. Statist. 9 (1981), 1168–1177), on the robustness of randomized experimental designs, to the field of regression design.  相似文献   

16.
Let {X 1, …, X n } and {Y 1, …, Y m } be two samples of independent and identically distributed observations with common continuous cumulative distribution functions F(x)=P(Xx) and G(y)=P(Yy), respectively. In this article, we would like to test the no quantile treatment effect hypothesis H 0: F=G. We develop a bootstrap quantile-treatment-effect test procedure for testing H 0 under the location-scale shift model. Our test procedure avoids the calculation of the check function (which is non-differentiable at the origin and makes solving the quantile effects difficult in typical quantile regression analysis). The limiting null distribution of the test procedure is derived and the procedure is shown to be consistent against a broad family of alternatives. Simulation studies show that our proposed test procedure attains its type I error rate close to the pre-chosen significance level even for small sample sizes. Our test procedure is illustrated with two real data sets on the lifetimes of guinea pigs from a treatment-control experiment.  相似文献   

17.
Two recursive schemes are presented for the calculation of the probabilityP(g(x)S n (x)≤h(x) for allx∈®), whereS n is the empirical distribution function of a sample from a continuous distribution andh, g are continuous and isotone functions. The results are specialized for the calculation of the distribution and the corresponding percentage points of the test statistic of the two-sided Kolmogorov-Smirnov one sample test. The schemes allow the calculation of the power of the test too. Finally an extensive tabulation of percentage points for the Kolmogorov-Smirnov test is given.  相似文献   

18.
It is well known that when the true values of the independent variable are unobservable due to measurement error, the least squares estimator for a regression model is biased and inconsistent. When repeated observations on each xi are taken, consistent estimators for the linear-plateau model can be formed. The repeated observations are required to classify each observation to the appropriate line segment. Two cases of repeated observations are treated in detail. First, when a single value of yi is observed with the repeated observations of xi the least squares estimator using the mean of the repeated xi observations is consistent and asymptotically normal. Second, when repeated observations on the pair (xi, yi ) are taken the least squares estimator is inconsistent, but two consistent estimators are proposed: one that consistently estimates the bias of the least squares estimator and adjusts accordingly; the second is the least squares estimator using the mean of the repeated observations on each pair.  相似文献   

19.
Consider an inhomogeneous Poisson process X on [0, T] whose unk-nown intensity function “switches” from a lower function g* to an upper function h* at some unknown point ?* that has to be identified. We consider two known continuous functions g and h such that g*(t) ? g(t) < h(t) ? h*(t) for 0 ? t ? T. We describe the behavior of the generalized likelihood ratio and Wald’s tests constructed on the basis of a misspecified model in the asymptotics of large samples. The power functions are studied under local alternatives and compared numerically with help of simulations. We also show the following robustness result: the Type I error rate is preserved even though a misspecified model is used to construct tests.  相似文献   

20.
Consider the situation where measurements are taken at two different times and let Mj(x) be some conditional robust measure of location associated with the random variable Y at time j, given that some covariate X=x. The goal is to test H0: M1(x)=M2(x) for each xx1,?…?, xK such that the probability of one or more Type I errors is less than α, where x1,?…?, xK are K specified values of the covariate. The paper reports simulation results comparing two methods aimed at accomplishing this goal without specifying some parametric form for the regression line. The first method is based on a simple modification of the method in Wilcox [Introduction to robust estimation and hypothesis testing. 3rd ed. San Diego, CA: Academic Press; 2012, Section 11.11.1]. The main result here is that the second method, which has never been studied, can have higher power, sometimes substantially so. Data from the Well Elderly 2 study, which motivated this paper, are used to illustrate that the alternative approach can make a practical difference. Here, the estimate of Mj(x) is based in part on either a 20% trimmed mean or the Harrell–Davis quantile estimator, but in principle the more successful method can be used with any robust location estimator.  相似文献   

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