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1.
In Flak/Schmid (1993) an outlier test for linear processes was introduced. The test statistic bases on a comparison of each observation with a one-step predictor. It was assumed that an upper bound for the total number of outlierss n is known, wheren denotes the sample size. The asymptotic distribution of the test statistic was derived under the assumption thats n/n → 0 ands n → ∞ asn → ∞. This note deals with the asymptotic behaviour of this quantity, ifs n/np 0 ∈ (0, 1).  相似文献   

2.
Summary Letx 1,x 2,x 3, ... be a sequence of independent identically distributed random variables andτ an estimable parameter of their distribution. We want to estimateτ by the correspondingU-statisticu n with loss function (u n τ)2 +cn. We derive a stopping time and prove its risk-efficiency in the sense of Starr (1966) without any assumption on the nature of the distribution function other than the existence of some moments. Research supported by the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 72, at the Universit?t Bonn.  相似文献   

3.
Let P = {F,G,…} be the set of all probability distribution functions with support (0, ). An unrestricted stochastic dominance relation> is defined on P for each real 1, where F > G means that xy = 0 (x - y) - 1 dG(y) xn = 0(xy)−1 dG(y) for all 0, with < for some x. These relations are partial orders that increase as increases with limit relation>. A class U of utility functions u on (0, ∞) is defined in such a way that F > G iff udF > udG for all u ε U. The U decrease as increases and have a non-empty intersection U. Each u ε U is an increasing function that has derivatives of all orders that alternate in sign. Criteria which tell when F eventually dominates G in the sense of F > G are noted. Comparisons with bounded stochastic dominance results are made in several places.  相似文献   

4.
Summary Let (X,A) be a measurable space andP ϑη |A (ϑη) ∈ Θ x H, ∥A, (θ, η) ∈ Θ×H, a parametrized family of probability measures (for short:p-measures). This paper is concerned with the problem of consistently estimatingθ from realizations governed by , where ηu ∈ H, v ∈ ℕ, are unknown.  相似文献   

5.
W. Stadje 《Metrika》1988,35(1):93-97
LetP be a probability measure on ℝ andI x be the set of alln-dimensional rectangles containingx. If for allx ∈ ℝn and θ ∈ ℝ the inequality holds,P is a normal distributioin with mean 0 or the unit mass at 0. The result generalizes Teicher’s (1961) maximum likelihood characterization of the normal density to a characterization ofN(0, σ2) amongall distributions (including those without density). The m.l. principle used is that of Scholz (1980).  相似文献   

6.
V. D. Naik  P. C. Gupta 《Metrika》1991,38(1):11-17
Summary A general class of estimators for estimating the population mean of the character under study which make use of auxiliary information is proposed. Under simple random sampling without replacement (SRSWOR), the expressions of Bias and Mean Square Error (MSE), up to the first and the second degrees of approximation are derived. General conditions, up to the first order approximation, are also obtained under which any member of this class performs more efficiently than the mean per unit estimator, the ratio estimator and the product estimator. The class of estimators in its optimum case, under the first degree approximation, is discussed. It is shown that it is not possible to obtain optimum values of parameters “a”, “b” and “p”, that are independent of each other. However, the optimum relation among them is given by (ba)p=ρ C y/C x. Under this condition, the expression of MSE of the class is that of the linear regression estimator.  相似文献   

7.
The center of a univariate data set {x 1,…,x n} can be defined as the point μ that minimizes the norm of the vector of distances y′=(|x 1−μ|,…,|x n−μ|). As the median and the mean are the minimizers of respectively the L 1- and the L 2-norm of y, they are two alternatives to describe the center of a univariate data set. The center μ of a multivariate data set {x 1,…,x n} can also be defined as minimizer of the norm of a vector of distances. In multivariate situations however, there are several kinds of distances. In this note, we consider the vector of L 1-distances y1=(∥x 1- μ1,…,∥x n- μ1) and the vector of L 2-distances y2=(∥x 1- μ2,…,∥x n-μ2). We define the L 1-median and the L 1-mean as the minimizers of respectively the L 1- and the L 2-norm of y 1; and then the L 2-median and the L 2-mean as the minimizers of respectively the L 1- and the L 2-norm of y 2. In doing so, we obtain four alternatives to describe the center of a multivariate data set. While three of them have been already investigated in the statistical literature, the L 1-mean appears to be a new concept. Received January 1999  相似文献   

8.
Consider the heteroscedastic regression model Y (j)(x in , t in ) = t in βg(x in ) + σ in e (j)(x in ), 1 ≤ j ≤ m, 1 ≤ i ≤ n, where sin2=f(uin){\sigma_{in}^{2}=f(u_{in})}, (x in , t in , u in ) are fixed design points, β is an unknown parameter, g(·) and f(·) are unknown functions, and the errors {e (j)(x in )} are mean zero NA random variables. The moment consistency for least-squares estimators and weighted least-squares estimators of β is studied. In addition, the moment consistency for estimators of g(·) and f(·) is investigated.  相似文献   

9.
Let P={F,G,…} be the set of probability distribution functions on [0,b]. For each αε[1, ∞), FαG means that ∫xo(xyα−1dF(y)∫xo(xy)α−1dG(y) for all xε[0, b], and F>αG means that FαG and FG. Each α is reflexive and transitive and each>α is asymmetric and transitive. Both α and>α increase as α increases but their limits are not complete. A class Uα of utility functions is defined to give F>αG iffudF>∫udG for all uεUα. These classes decrease as α increases, and their limit is empty. Similar decreasing classes are defined for each α, and their limit is essentially the constant functions on (0, b].  相似文献   

10.
Holger Dette 《Metrika》1993,40(1):37-50
The optimal design problem for the estimation of several linear combinationsc′ l ϑ (l=1, …,m) is considered in the usual linear regression modely=f′(x)ϑ (f(x) ∈ ℝ k ,ϑ ∈ ℝ k ). An optimal design minimizes a (weighted)p-norm of the variances of the least squares estimates for the different linear combinationsc′ l ϑ. A generalized Elfving theorem is used to derive the relation of the new optimality criterion to theE-optimal design problem. It is shown that theE-optimal design for the parameterϑ minimizes such a (weighted)p-norm whenever the vectorc=(c′ 1, …, c′k)′ is an inball vector of a symmetric convex and compact “Elfving set” in.  相似文献   

11.
A minimal characterization of the covariance matrix   总被引:1,自引:0,他引:1  
R. Grübel 《Metrika》1988,35(1):49-52
Summary LetX be ak-dimensional random vector with mean vectorμ and non-singular covariance matrix Σ. We show that among all pairs (a, Δ),a ∈ IR k , Δ ∈ IR k×k positive definite and symmetric andE(X−a)′ Δ−1(Xa)=k, (μ, Σ) is the unique pair which minimizes det Δ. This motivates certain robust estimators of location and scale. Research supported by the Nuffield Foundation.  相似文献   

12.
Dr. Klaus Abt 《Metrika》1960,3(1):26-45
Zusammenfassung Wenn aus Versuchen oder Beobachtungen stammende, nach einer oder mehreren Richtungen gruppierte Zahlenpaare (x;y) analysiert werden sollen, deren Gliederx undy Messungen des gleichen und daher i.a. an verschiedenen Zeitpunkten registrierten Merkmals sind, so stehen dazu zwei Verfahren zur Verfügung. Das eine ist dasjenige der Kovarianzanalyse; es beruht somit wesentlich auf der Regressionsrechnung. Dabei wird die Ver?nderlichey als stochastisch abh?ngig von der ihrerseits als unabh?ngig betrachteten Ver?nderlichenx angesehen. Das andere Verfahren ist die Varianzanalyse der Differenzen aus den beiden Gliedernx undy dieser Paare, wobeix im allgemeinen Fall noch mit dem dimensionslosen Koeffizientenβ 0 behaftet sei:d=yβ 0 x.β 0=1 beschreibt den wichtigsten Fall der gew?hnlichen Differenzen ausy undx. Dieses zweite Verfahren wird kurz mit “Differenzenanalyse” bezeichnet. Das Ziel der vorliegenden Arbeit ist es, die beiden Verfahren miteinander zu vergleichen.  相似文献   

13.
Given X,, where X is a topologically connected space and is an asymmetric binary relation, necessary and sufficient conditions are presented for the existence of a continuous representation of the form, u, δ; that is, for x,yX, xy if and only if u(x)>u(y)+δ where u:X→ is continuous and δ is a strictly positive real number. The results are related to existing results for numerical representations of interval orders and semiorders.  相似文献   

14.
We propose a class of nonparametric tests for testing non-stochasticity of the regression parameterβ in the regression modely i =βx i +ɛ i ,i=1, ...,n. We prove that the test statistics are asymptotically normally distributed both underH 0 and under contiguous alternatives. The asymptotic relative efficiencies (in the Pitman sense) with respect to the best parametric test have also been computed and they are quite high. Some simulation studies are carried out to illustrate the results. Research was supported by the University Grants Commission, India.  相似文献   

15.
LetX 1,X 2, ...,X n (n≥3) be a random sample on a random variableX with distribution functionF having a unique continuous inverseF −1 over (a,b), −∞≤a<b≤∞ the support ofF. LetX 1:n <X 2:n <...<X n:n be the corresponding order statistics. Letg be a nonconstant continuous function over (a,b). Then for some functionG over (a, b) and for some positive integersr ands, 1<r+1<sn
  相似文献   

16.
Summary For a linear modelY =ϑ + Z,ϑV,V ⊂ ℝ n a linear space, the following theorem is proved under simple conditions on the subspaceV: The projection onV (i.e. the least squares estimate forϑ) is a sufficient statistic iffZ is normally distributed. Further, this result is extended to the case of a multivariate linear model.  相似文献   

17.
Hira L. Koul 《Metrika》2002,55(1-2):75-90
Often in the robust analysis of regression and time series models there is a need for having a robust scale estimator of a scale parameter of the errors. One often used scale estimator is the median of the absolute residuals s 1. It is of interest to know its limiting distribution and the consistency rate. Its limiting distribution generally depends on the estimator of the regression and/or autoregressive parameter vector unless the errors are symmetrically distributed around zero. To overcome this difficulty it is then natural to use the median of the absolute differences of pairwise residuals, s 2, as a scale estimator. This paper derives the asymptotic distributions of these two estimators for a large class of nonlinear regression and autoregressive models when the errors are independent and identically distributed. It is found that the asymptotic distribution of a suitably standardizes s 2 is free of the initial estimator of the regression/autoregressive parameters. A similar conclusion also holds for s 1 in linear regression models through the origin and with centered designs, and in linear autoregressive models with zero mean errors.  This paper also investigates the limiting distributions of these estimators in nonlinear regression models with long memory moving average errors. An interesting finding is that if the errors are symmetric around zero, then not only is the limiting distribution of a suitably standardized s 1 free of the regression estimator, but it is degenerate at zero. On the other hand a similarly standardized s 2 converges in distribution to a normal distribution, regardless of the errors being symmetric or not. One clear conclusion is that under the symmetry of the long memory moving average errors, the rate of consistency for s 1 is faster than that of s 2.  相似文献   

18.
Let U 1, U 2, . . . , U n–1 be an ordered sample from a Uniform [0,1] distribution. The non-overlapping uniform spacings of order s are defined as \({G_{i}^{(s)} =U_{is} -U_{(i-1)s}, i=1,2,\ldots,N^\prime, G_{N^\prime+1}^{(s)} =1-U_{N^\prime s}}\) with notation U 0 = 0, U n = 1, where \({N^\prime=\left\lfloor n/s\right\rfloor}\) is the integer part of n/s. Let \({ N=\left\lceil n/s\right\rceil}\) be the smallest integer greater than or equal to n/s, f m (u), m = 1, 2, . . . , N, be a sequence of real-valued Borel-measurable functions. In this article a Cramér type large deviation theorem for the statistic \({f_{1,n} (nG_{1}^{(s)})+\cdots+f_{N,n} (nG_{N}^{(s)} )}\) is proved.  相似文献   

19.
Let be an interval order on a topological space (X, τ), and let x ˜* y if and only if [y z x z], and x ˜** y if and only if [z x z y]. Then ˜* and ˜** are complete preorders. In the particular case when is a semiorder, let x ˜0 y if and only if x ˜* y and x ˜** y. Then ˜0 is a complete preorder, too. We present sufficient conditions for the existence of continuous utility functions representing ˜*, ˜** and ˜0, by using the notion of strong separability of a preference relation, which was introduced by Chateauneuf (Journal of Mathematical Economics, 1987, 16, 139–146). Finally, we discuss the existence of a pair of continuous functions u, υ representing a strongly separable interval order on a measurable topological space (X, τ, μ, ).  相似文献   

20.
A test statistic is considered for testing a hypothesis for the mean vector for multivariate data, when the dimension of the vector, p, may exceed the number of vectors, n, and the underlying distribution need not necessarily be normal. With n,p→∞, and under mild assumptions, but without assuming any relationship between n and p, the statistic is shown to asymptotically follow a chi‐square distribution. A by product of the paper is the approximate distribution of a quadratic form, based on the reformulation of the well‐known Box's approximation, under high‐dimensional set up. Using a classical limit theorem, the approximation is further extended to an asymptotic normal limit under the same high dimensional set up. The simulation results, generated under different parameter settings, are used to show the accuracy of the approximation for moderate n and large p.  相似文献   

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