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1.
In this paper skewness and kurtosis characteristics of a multivariate p-dimensional distribution are introduced. The skewness measure is defined as a p-vector while the kurtosis is characterized by a p×p-matrix. The introduced notions are extensions of the corresponding measures of Mardia [K.V. Mardia, Measures of multivariate skewness and kurtosis with applications, Biometrika 57 (1970) 519–530] and Móri, Rohatgi & Székely [T.F. Móri, V.K. Rohatgi, G.J. Székely, On multivariate skewness and kurtosis, Theory Probab. Appl. 38 (1993) 547–551]. Basic properties of the characteristics are examined and compared with both the above-mentioned results in the literature. Expressions for the measures of skewness and kurtosis are derived for the multivariate Laplace distribution. The kurtosis matrix is used in Independent Component Analysis (ICA) where the solution of an eigenvalue problem of the kurtosis matrix determines the transformation matrix of interest [A. Hyvärinen, J. Karhunen, E. Oja, Independent Component Analysis, Wiley, New York, 2001].  相似文献   

2.
We study the distributions of the LASSO, SCAD, and thresholding estimators, in finite samples and in the large-sample limit. The asymptotic distributions are derived for both the case where the estimators are tuned to perform consistent model selection and for the case where the estimators are tuned to perform conservative model selection. Our findings complement those of Knight and Fu [K. Knight, W. Fu, Asymptotics for lasso-type estimators, Annals of Statistics 28 (2000) 1356–1378] and Fan and Li [J. Fan, R. Li, Variable selection via non-concave penalized likelihood and its oracle properties, Journal of the American Statistical Association 96 (2001) 1348–1360]. We show that the distributions are typically highly non-normal regardless of how the estimator is tuned, and that this property persists in large samples. The uniform convergence rate of these estimators is also obtained, and is shown to be slower than n−1/2 in case the estimator is tuned to perform consistent model selection. An impossibility result regarding estimation of the estimators’ distribution function is also provided.  相似文献   

3.
In the framework of generalized linear models, the nonrobustness of classical estimators and tests for the parameters is a well known problem, and alternative methods have been proposed in the literature. These methods are robust and can cope with deviations from the assumed distribution. However, they are based on first order asymptotic theory, and their accuracy in moderate to small samples is still an open question. In this paper, we propose a test statistic which combines robustness and good accuracy for moderate to small sample sizes. We combine results from Cantoni and Ronchetti [E. Cantoni, E. Ronchetti, Robust inference for generalized linear models, Journal of the American Statistical Association 96 (2001) 1022–1030] and Robinson, Ronchetti and Young [J. Robinson, E. Ronchetti, G.A. Young, Saddlepoint approximations and tests based on multivariate M-estimators, The Annals of Statistics 31 (2003) 1154–1169] to obtain a robust test statistic for hypothesis testing and variable selection, which is asymptotically χ2-distributed as the three classical tests but with a relative error of order O(n−1). This leads to reliable inference in the presence of small deviations from the assumed model distribution, and to accurate testing and variable selection, even in moderate to small samples.  相似文献   

4.
The formula for the blocking probability for the finite capacity M/G/1/K in terms of the steady state occupancy probability distribution of M/G/1 and the system utilization is known [Keilson, J. Royal Statistical Soc. Serie B, 28 (1966) 190–201]. The validity of this relationship is demonstrated for a broad class of state dependent M/G/1 vacation systems and priority systems. New methods are employed which may also be of interest in their own right.This research was conducted while J. Keilson was a Senior Staff Scientist at GTE Laboratories Incorporated.  相似文献   

5.
Although the theory of rank tests is rather complete in the one-sided case, it was not even known in 1959, whether the Wilcoxon two-sample test and other similar tests are unbiased against the two-sided alternatives (Lehmann (1959,Testing Statistical Hypotheses, p. 240, Wiley, New York)). A partial answer to this question was given by Sugiura in 1965, who found, that the test named above may be biased (Sugiura (1965,Ann. Inst. Statist. Math.,17, 261–263)). According to Lehmann (1986,Testing Statistical Hypotheses, 2nd ed., pp. 322–324, Wiley, New York) it seems to be still open, whether the same is true for the WILCOXON one-sample test, which is also known as WILCOXON signed rank test. This will be shown in the present paper.  相似文献   

6.
In this article, we prove the existence and uniqueness of a certain distribution function on the unit interval. This distribution appears in Brent's model of the analysis of the binary gcd algorithm. The existence and uniqueness of such a function was conjectured by Richard Brent in his original paper [R.P. Brent, Analysis of the binary Euclidean algorithm, in: J.F. Traub (Ed.), New Directions and Recent Results in Algorithms and Complexity, Academic Press, New York, 1976, pp. 321–355]. Donald Knuth also supposes its existence in [D.E. Knuth, The Art of Computer Programming, vol. 2, Seminumerical Algorithms, third ed., Addison-Wesley, Reading, MA, 1997] where developments of its properties lead to very good estimates in relation to the algorithm. We settle here the question of existence, giving a basis to these results, and study the relationship between this limiting function and the binary Euclidean operator B2, proving rigorously that its derivative is a fixed point of B2.  相似文献   

7.
New and simple proofs are given for the non-existence of unconditional bases in the spaces L1[0, 1] and C[0, 1].Translated from Matematicheskie Zametki, Vol. 11, No. 3, pp. 241–249, March, 1972.The author wishes to thank P. L. Ul'yanov for his interest in the problem under consideration.  相似文献   

8.
One obtains generalizations and refinements of ergodic and mixing theorems [Yu. V. Linnik, Ergodic Properties of Algebraic Fields, Springer-Verlag, New York (1968)] for the distribution of integral points on a two-sheeted hyperboloid. One gives a survey of the applications of the discrete ergodic method to indefinite ternary quadratic forms; in particular, one describes the results of the papers of A. M. Istamov, E. V. Podsypanin, A. V. Malyshev, and U. M. Pachev, published in this issue.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im V. A. Steklova AN SSSR, Vol. 93, pp. 5–24, 1980.  相似文献   

9.
In [1, 2], V. A. Trenogin and the author proposed a method for constructing regularizing equations (R.E.) for ill-posed problems in the theory of branching [3, 4]. Here we consider the problem of choosing initial approximations, that are optimal in a certain sense, to simple solutions of R.E. in a neighborhood of a branching point.Translated from Matematicheskie Zametki, Vol. 20, No. 2, pp. 273–278, August, 1976.The author thanks V. A. Trenogin for his interest in this research and his valuable advice.  相似文献   

10.
We show how classical modular forms and functions appear as tau-functions for a certain integrable reduction of the self-dual Yang-Mills equations obtained by S. Chakravarty, M. Ablowitz, and P. Clarkson [6]. We discuss possible consequences of this novel phenomenon in integrable systems which indicate deep connections between integrable equations, group representations, modular forms, and moduli spaces.In Memory of Prof. M. C. PolivanovDepartment of Mathematics, State University of New York at Stony Brook, Stony Brook, NY 11794-3651. Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 93, No. 2, pp. 330–341, November, 1992.  相似文献   

11.
Inference on the largest mean of a multivariate normal distribution is a surprisingly difficult and unexplored topic. Difficulties arise when two or more of the means are simultaneously the largest mean. Our proposed solution is based on an extension of R.A. Fisher’s fiducial inference methods termed generalized fiducial inference. We use a model selection technique along with the generalized fiducial distribution to allow for equal largest means and alleviate the overestimation that commonly occurs. Our proposed confidence intervals for the largest mean have asymptotically correct frequentist coverage and simulation results suggest that they possess promising small sample empirical properties. In addition to the theoretical calculations and simulations we also applied this approach to the air quality index of the four largest cities in the northeastern United States (Baltimore, Boston, New York, and Philadelphia).  相似文献   

12.
The asymptotic normality of U-statistics has so far been proved for iid data and under various mixing conditions such as absolute regularity, but not for strong mixing. We use a coupling technique introduced in 1983 by Bradley [R.C. Bradley, Approximation theorems for strongly mixing random variables, Michigan Math. J. 30 (1983),69–81] to prove a new generalized covariance inequality similar to Yoshihara’s [K. Yoshihara, Limiting behavior of U-statistics for stationary, absolutely regular processes, Z. Wahrsch. Verw. Gebiete 35 (1976), 237–252]. It follows from the Hoeffding-decomposition and this inequality that U-statistics of strongly mixing observations converge to a normal limit if the kernel of the U-statistic fulfills some moment and continuity conditions.The validity of the bootstrap for U-statistics has until now only been established in the case of iid data (see [P.J. Bickel, D.A. Freedman, Some asymptotic theory for the bootstrap, Ann. Statist. 9 (1981), 1196–1217]. For mixing data, Politis and Romano [D.N. Politis, J.P. Romano, A circular block resampling procedure for stationary data, in: R. Lepage, L. Billard (Eds.), Exploring the Limits of Bootstrap, Wiley, New York, 1992, pp. 263–270] proposed the circular block bootstrap, which leads to a consistent estimation of the sample mean’s distribution. We extend these results to U-statistics of weakly dependent data and prove a CLT for the circular block bootstrap version of U-statistics under absolute regularity and strong mixing. We also calculate a rate of convergence for the bootstrap variance estimator of a U-statistic and give some simulation results.  相似文献   

13.
The purpose of this paper is, in multivariate linear regression model (Part I) and GMANOVA model (Part II), to investigate the effect of nonnormality upon the nonnull distributions of some multivariate test statistics under normality. It is shown that whatever the underlying distributions, the difference of local powers up to order N−1 after either Bartlett’s type adjustment or Cornish-Fisher’s type size adjustment under nonnormality coincides with that in Anderson [An Introduction to Multivariate Statistical Analysis, 2nd ed. and 3rd ed., Wiley, New York, 1984, 2003] under normality. The derivation of asymptotic expansions is based on the differential operator associated with the multivariate linear regression model under general distributions. The performance of higher-order results in finite samples, including monotone Bartlett’s type adjustment and monotone Cornish-Fisher’s type size adjustment, is examined using simulation studies.  相似文献   

14.
In the article we discuss the question of reducing the Arrow-Debreu model [1, 2] to a problem of mathematical programming; we also study conditions under which the problem is convex. The interest to the question is aroused by the circumstance that the equilibrium problems turn rather difficult for numeric solution. Two approaches are used most frequently. One is based on the monotonicity property (formulated somehow) of the excessive demand [2–5]. If this property is satisfied, then the corresponding differential system, which has the sought equilibrium at the rest point, appears stable. The other approach consists in constructing the so-called heuristic methods (see, for instance, [6]) which have more or less reasonable grounds but in general do not guarantee that a solution is obtainable.The research was supported by the Russian Foundation for Fundamental Research (Grant 93-012-842).Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 35, No. 5, pp. 990–999, September–October, 1994.  相似文献   

15.
We investigate multivariate subordination of Lévy processes which was first introduced by Barndorff-Nielsen et al. [O.E. Barndorff-Nielsen, F.E. Benth, and A. Veraart, Modelling electricity forward markets by ambit fields, J. Adv. Appl. Probab. (2010)], in a Hilbert space valued setting which has been introduced in Pérez-Abreu and Rocha-Arteaga [V. Pérez-Abreu and A. Rocha-Arteaga, Covariance-parameter Lévy processes in the space of trace-class operators, Infin. Dimens. Anal. Quantum Probab. Related Top. 8(1) (2005), pp. 33–54]. The processes are explicitly characterized and conditions for integrability and martingale properties are derived under various assumptions of the Lévy process and subordinator. As an application of our theory we construct explicitly some Hilbert space valued versions of Lévy processes which are popular in the univariate and multivariate case. In particular, we define a normal inverse Gaussian Lévy process in Hilbert space. The resulting process has the property that at each time all its finite dimensional projections are multivariate normal inverse Gaussian distributed as introduced in Rydberg [T. Rydberg, The normal inverse Gaussian Lévy process: Simulation and approximation, Commun. Stat. Stochastic Models 13 (1997), pp. 887–910].  相似文献   

16.
In a regression analysis, it is assumed that the observations come from a single class in a data cluster and the simple functional relationship between the dependent and independent variables can be expressed using the general model; Y=f(X)+ε. However; a data cluster may consist of a combination of observations that have different distributions that are derived from different clusters. When faced with issues of estimating a regression model for fuzzy inputs that have been derived from different distributions, this regression model has been termed the ‘switching regression model’ and it is expressed with . Here li indicates the class number of each independent variable and p is indicative of the number of independent variables [J.R. Jang, ANFIS: Adaptive-network-based fuzzy inference system, IEEE Transaction on Systems, Man and Cybernetics 23 (3) (1993) 665–685; M. Michel, Fuzzy clustering and switching regression models using ambiguity and distance rejects, Fuzzy Sets and Systems 122 (2001) 363–399; E.Q. Richard, A new approach to estimating switching regressions, Journal of the American Statistical Association 67 (338) (1972) 306–310].In this study, adaptive networks have been used to construct a model that has been formed by gathering obtained models. There are methods that suggest the class numbers of independent variables heuristically. Alternatively, in defining the optimal class number of independent variables, the use of suggested validity criterion for fuzzy clustering has been aimed. In the case that independent variables have an exponential distribution, an algorithm has been suggested for defining the unknown parameter of the switching regression model and for obtaining the estimated values after obtaining an optimal membership function, which is suitable for exponential distribution.  相似文献   

17.
The problem of detection of a change in distribution is considered. Shiryayev (1963, Theory Probab. Appl., 8, pp. 22–46, 247–264 and 402–413; 1978, Optimal Stopping Rules, Springer, New York) solved the problem in a Bayesian framework assuming that the prior on the change point is Geometric (p). Shiryayev showed that the Bayes solution prescribes stopping as soon as the posterior probability of the change having occurred exceeds a fixed level. In this paper, a myopic policy is studied. An empirical Bayes stopping time is investigated for detecting a change in distribution when the prior is not completely known.Research was supported in part by the Natural Sciences and Engineering Research Council of Canada under grant GP 7987.  相似文献   

18.
Polynomial moments are often used for the computation of Gauss quadrature to stabilize the numerical calculation of the orthogonal polynomials, see [W. Gautschi, Computational aspects of orthogonal polynomials, in: P. Nevai (Ed.), Orthogonal Polynomials-Theory and Practice, NATO ASI Series, Series C: Mathematical and Physical Sciences, vol. 294. Kluwer, Dordrecht, 1990, pp. 181–216 [6]; W. Gautschi, On the sensitivity of orthogonal polynomials to perturbations in the moments, Numer. Math. 48(4) (1986) 369–382 [5]; W. Gautschi, On generating orthogonal polynomials, SIAM J. Sci. Statist. Comput. 3(3) (1982) 289–317 [4]] or numerical resolution of linear systems [C. Brezinski, Padé-type approximation and general orthogonal polynomials, ISNM, vol. 50, Basel, Boston, Stuttgart, Birkhäuser, 1980 [3]]. These modified moments can also be used to accelerate the convergence of sequences to a real or complex numbers if the error satisfies some properties as done in [C. Brezinski, Accélération de la convergence en analyse numérique, Lecture Notes in Mathematics, vol. 584. Springer, Berlin, New York, 1977; M. Prévost, Padé-type approximants with orthogonal generating polynomials, J. Comput. Appl. Math. 9(4) (1983) 333–346]. In this paper, we use Legendre modified moments to accelerate the convergence of the sequence Hn-log(n+1) to the Euler's constant γ. A formula for the error is given. It is proved that it is a totally monotonic sequence. At last, we give applications to the arithmetic property of γ.  相似文献   

19.
The relationship between the breaking strength and the temperature has been studied for butadiene/styrene block copolymers of the three-block configuration styrene-butadiene-styrene. The studies were carried out over the temperature range –20 to 60° at crosshead speeds of 5, 50, 250, 500, and 1000 mm/min. An analysis was carried out on the values of the parameter U in the equation linking the breaking strength with the temperature.For Communication 2 see [3].Voronezh Technological Institute. Translated from Mekhanika Polimerov, No. 3, pp. 392–398, May–June, 1974.  相似文献   

20.
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