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1.
Let (X,Y) be a Rd×N0-valued random vector where the conditional distribution of Y given X=x is a Poisson distribution with mean m(x). We estimate m by a local polynomial kernel estimate defined by maximizing a localized log-likelihood function. We use this estimate of m(x) to estimate the conditional distribution of Y given X=x by a corresponding Poisson distribution and to construct confidence intervals of level α of Y given X=x. Under mild regularity conditions on m(x) and on the distribution of X we show strong convergence of the integrated L1 distance between Poisson distribution and its estimate. We also demonstrate that the corresponding confidence interval has asymptotically (i.e., for sample size tending to infinity) level α, and that the probability that the length of this confidence interval deviates from the optimal length by more than one converges to zero with the number of samples tending to infinity.  相似文献   

2.
Let (X, Y) have regression function m(x) = E(Y | X = x), and let X have a marginal density f1(x). We consider two nonparameteric estimates of m(x): the Watson estimate when f1 is known and the Yang estimate when f1 is known or unknown. For both estimates the asymptotic distribution of the maximal deviation from m(x) is proved, thus extending results of Bickel and Rosenblatt for the estimation of density functions.  相似文献   

3.
Consider the heteroscedastic model Y=m(X)+σ(X)?, where ? and X are independent, Y is subject to right censoring, m(·) is an unknown but smooth location function (like e.g. conditional mean, median, trimmed mean…) and σ(·) an unknown but smooth scale function. In this paper we consider the estimation of m(·) under this model. The estimator we propose is a Nadaraya-Watson type estimator, for which the censored observations are replaced by ‘synthetic’ data points estimated under the above model. The estimator offers an alternative for the completely nonparametric estimator of m(·), which cannot be estimated consistently in a completely nonparametric way, whenever high quantiles of the conditional distribution of Y given X=x are involved.We obtain the asymptotic properties of the proposed estimator of m(x) and study its finite sample behaviour in a simulation study. The method is also applied to a study of quasars in astronomy.  相似文献   

4.
Consider the model Y=m(X)+ε, where m(⋅)=med(Y|⋅) is unknown but smooth. It is often assumed that ε and X are independent. However, in practice this assumption is violated in many cases. In this paper we propose modeling the dependence between ε and X by means of a copula model, i.e. (ε,X)∼Cθ(Fε(⋅),FX(⋅)), where Cθ is a copula function depending on an unknown parameter θ, and Fε and FX are the marginals of ε and X. Since many parametric copula families contain the independent copula as a special case, the so-obtained regression model is more flexible than the ‘classical’ regression model.We estimate the parameter θ via a pseudo-likelihood method and prove the asymptotic normality of the estimator, based on delicate empirical process theory. We also study the estimation of the conditional distribution of Y given X. The procedure is illustrated by means of a simulation study, and the method is applied to data on food expenditures in households.  相似文献   

5.
A density f=f(x1,…,xd) on [0,∞)d is block decreasing if for each j∈{1,…,d}, it is a decreasing function of xj, when all other components are held fixed. Let us consider the class of all block decreasing densities on [0,1]d bounded by B. We shall study the minimax risk over this class using n i.i.d. observations, the loss being measured by the L1 distance between the estimate and the true density. We prove that if S=log(1+B), lower bounds for the risk are of the form C(Sd/n)1/(d+2), where C is a function of d only. We also prove that a suitable histogram with unequal bin widths as well as a variable kernel estimate achieve the optimal multivariate rate. We present a procedure for choosing all parameters in the kernel estimate automatically without loosing the minimax optimality, even if B and the support of f are unknown.  相似文献   

6.
On the basis of a random sample of size n on an m-dimensional random vector X, this note proposes a class of estimators fn(p) of f(p), where f is a density of X w.r.t. a σ-finite measure dominated by the Lebesgue measure on Rm, p = (p1,…,pm), pj ≥ 0, fixed integers, and for x = (x1,…,xm) in Rm, f(p)(x) = ?p1+…+pm f(x)/(?p1x1 … ?pmxm). Asymptotic unbiasedness as well as both almost sure and mean square consistencies of fn(p) are examined. Further, a necessary and sufficient condition for uniform asymptotic unbisedness or for uniform mean square consistency of fn(p) is given. Finally, applications of estimators of this note to certain statistical problems are pointed out.  相似文献   

7.
Characterizations of probability distributions is a topic of great popularity in applied probability and reliability literature for over last 30 years. Beside the intrinsic mathematical interest (often related to functional equations) the results in this area are helpful for probabilistic and statistical modelling, especially in engineering and biostatistical problems. A substantial number of characterizations has been devoted to a legion of variants of exponential distributions. The main reliability measures associated with a random vector X are the conditional moment function defined by mφ(x)=E(φ(X)|X?x) (which is equivalent to the mean residual life function e(x)=mφ(x)-x when φ(x)=x) and the hazard gradient function h(x)=-∇logR(x), where R(x) is the reliability (survival) function, R(x)=Pr(X?x), and ∇ is the operator . In this paper we study the consequences of a linear relationship between the hazard gradient and the conditional moment functions for continuous bivariate and multivariate distributions. We obtain a general characterization result which is the applied to characterize Arnold and Strauss’ bivariate exponential distribution and some related models.  相似文献   

8.
Nonparametric quantile regression with multivariate covariates is a difficult estimation problem due to the “curse of dimensionality”. To reduce the dimensionality while still retaining the flexibility of a nonparametric model, we propose modeling the conditional quantile by a single-index function , where a univariate link function g0(⋅) is applied to a linear combination of covariates , often called the single-index. We introduce a practical algorithm where the unknown link function g0(⋅) is estimated by local linear quantile regression and the parametric index is estimated through linear quantile regression. Large sample properties of estimators are studied, which facilitate further inference. Both the modeling and estimation approaches are demonstrated by simulation studies and real data applications.  相似文献   

9.
Consider the nonparametric regression model Yni=g(xni)+εni for i=1,…,n, where g is unknown, xni are fixed design points, and εni are negatively associated random errors. Nonparametric estimator gn(x) of g(x) will be introduced and its asymptotic properties are studied. In particular, the pointwise and uniform convergence of gn(x) and its asymptotic normality will be investigated. This extends the earlier work on independent random errors (e.g. see J. Multivariate Anal. 25(1) (1988) 100).  相似文献   

10.
We consider the estimation of the regression operator r in the functional model: Y=r(x)+ε, where the explanatory variable x is of functional fixed-design type, the response Y is a real random variable and the error process ε is a second order stationary process. We construct the kernel type estimate of r from functional data curves and correlated errors. Then we study their performances in terms of the mean square convergence and the convergence in probability. In particular, we consider the cases of short and long range error processes. When the errors are negatively correlated or come from a short memory process, the asymptotic normality of this estimate is derived. Finally, some simulation studies are conducted for a fractional autoregressive integrated moving average and for an Ornstein-Uhlenbeck error processes.  相似文献   

11.
Summary The objective in nonparametric regression is to infer a functiong(x) and itspth order derivativesg (g)(x),p≧1 fixed, on the basis of a finite collection of pairs {x i, g(xi)+Z i} i=1 n , where the noise componentsZ i satisfy certain modest assumptions and the domain pointsx i are selected non-randomly. This paper exhibits a new class of kernel estimatesg n (p) ,p≧0 fixed. The main theoretical results of this study are the rates of convergence obtained for mean square and strong consistency ofg n (p) each of them being uniform on the (0,1).  相似文献   

12.
I. N. Herstein [10] proved that a prime ring of characteristic not two with a nonzero derivation d satisfying d(x)d(y) = d(y)d(x) for all x, y must be commutative, and H. E. Bell and M. N. Daif [8] showed that a prime ring of arbitrary characteristic with nonzero derivation d satisfying d(xy) = d(yx) for all x, y in some nonzero ideal must also be commutative. For semiprime rings, we show that an inner derivation satisfying the condition of Bell and Daif on a nonzero ideal must be zero on that ideal, and for rings with identity, we generalize all three results to conditions on derivations of powers and powers of derivations. For example, let R be a prime ring with identity and nonzero derivation d, and let m and n be positive integers such that, when charR is finite, mn < charR. If d(x m y n ) = d(y n x m ) for all x, yR, then R is commutative. If, in addition, charR≠ 2 and the identity is in the image of an ideal I under d, then d(x) m d(y) n = d(y) n d(x) m for all x, yI also implies that R is commutative.  相似文献   

13.
This article is concerned with the Titchmarsh–Weyl mα(λ) function for the differential equation d2y/dx2+[λq(x)]y=0. The test potential q(x)=x2, for which the relevant mα(λ) functions are meromorphic, having simple poles at the points λ=4k+1 and λ=4k+3, is studied in detail. We are able to calculate the mα(λ) function both far from and near to these poles. The calculation is then extended to several other potentials, some of which do not have analytical solutions. Numerical data are given for the Titchmarsh–Weyl mα(λ) function for these potentials to illustrate the computational effectiveness of the method used.  相似文献   

14.
We construct a discrete analogue D m () of the differential operator d2m /dx 2m + 2ω 2d2m?2 /dx 2m?2 + ω 4d2m?4 /dx 2m?4 for any m ≥ 2. In the case m = 2, we apply in the Hilbert space K 2(P 2) the discrete analogue D 2() for construction of optimal quadrature formulas and interpolation splines minimizing the seminorm, which are exact for trigonometric functions sin ωx and cos ωx.  相似文献   

15.
Let X={X(s)}sS be an almost sure continuous stochastic process (S compact subset of Rd) in the domain of attraction of some max-stable process, with index function constant over S. We study the tail distribution of ∫SX(s)ds, which turns out to be of Generalized Pareto type with an extra ‘spatial’ parameter (the areal coefficient from Coles and Tawn (1996) [3]). Moreover, we discuss how to estimate the tail probability P(∫SX(s)ds>x) for some high value x, based on independent and identically distributed copies of X. In the course we also give an estimator for the areal coefficient. We prove consistency of the proposed estimators. Our methods are applied to the total rainfall in the North Holland area; i.e. X represents in this case the rainfall over the region for which we have observations, and its integral amounts to total rainfall.The paper has two main purposes: first to formalize and justify the results of Coles and Tawn (1996) [3]; further we treat the problem in a non-parametric way as opposed to their fully parametric methods.  相似文献   

16.
A quasi-metric space (X,d) is called sup-separable if (X,ds) is a separable metric space, where ds(x,y)=max{d(x,y),d(y,x)} for all x,yX. We characterize those preferences, defined on a sup-separable quasi-metric space, for which there is a semi-Lipschitz utility function. We deduce from our results that several interesting examples of quasi-metric spaces which appear in different fields of theoretical computer science admit semi-Lipschitz utility functions. We also apply our methods to the study of certain kinds of dynamical systems defined on quasi-metric spaces.  相似文献   

17.
In this paper we aim to estimate the direction in general single-index models and to select important variables simultaneously when a diverging number of predictors are involved in regressions. Towards this end, we propose the nonconcave penalized inverse regression method. Specifically, the resulting estimation with the SCAD penalty enjoys an oracle property in semi-parametric models even when the dimension, pn, of predictors goes to infinity. Under regularity conditions we also achieve the asymptotic normality when the dimension of predictor vector goes to infinity at the rate of pn=o(n1/3) where n is sample size, which enables us to construct confidence interval/region for the estimated index. The asymptotic results are augmented by simulations, and illustrated by analysis of an air pollution dataset.  相似文献   

18.
We present a new algebraic algorithmic scheme to solve convex integer maximization problems of the following form, where c is a convex function on Rd and w1x,…,wdx are linear forms on Rn,
max{c(w1x,…,wdx):Ax=b,xNn}.  相似文献   

19.
Consider the random vector (X, Y), where X is completely observed and Y is subject to random right censoring. It is well known that the completely nonparametric kernel estimator of the conditional distribution ${F(\cdot|x)}$ of Y given Xx suffers from inconsistency problems in the right tail (Beran 1981, Technical Report, University of California, Berkeley), and hence any location function m(x) that involves the right tail of ${F(\cdot|x)}$ (like the conditional mean) cannot be estimated consistently in a completely nonparametric way. In this paper, we propose an alternative estimator of m(x), that, under certain conditions, does not share the above inconsistency problems. The estimator is constructed under the model Y = m(X) + σ(X)ε, where ${\sigma(\cdot)}$ is an unknown scale function and ε (with location zero and scale one) is independent of X. We obtain the asymptotic properties of the proposed estimator of m(x), we compare it with the completely nonparametric estimator via simulations and apply it to a study of quasars in astronomy.  相似文献   

20.
Let m be a dynamical system on the space of probability measures M1(Rd), and let Λ + (?) be the positive limit set for ? ∈ M1(Rd), where ? has compact support K ?Rd. The main result of this paper states that support of Λ+(?) ?
,support of Λ + (δx), where δx is the Dirac measure at point x.  相似文献   

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