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1.
Ann Cohen Brandwein 《Journal of multivariate analysis》1979,9(4):579-588
For X one observation on a p-dimensional (p ≥ 4) spherically symmetric (s.s.) distribution about θ, minimax estimators whose risks dominate the risk of X (the best invariant procedure) are found with respect to general quadratic loss, L(δ, θ) = (δ − θ)′ D(δ − θ) where D is a known p × p positive definite matrix. For C a p × p known positive definite matrix, conditions are given under which estimators of the form δa,r,C,D(X) = (I − (ar(|X|2)) D−1/2CD1/2 |X|−2)X are minimax with smaller risk than X. For the problem of estimating the mean when n observations X1, X2, …, Xn are taken on a p-dimensional s.s. distribution about θ, any spherically symmetric translation invariant estimator, δ(X1, X2, …, Xn), with have a s.s. distribution about θ. Among the estimators which have these properties are best invariant estimators, sample means and maximum likelihood estimators. Moreover, under certain conditions, improved robust estimators can be found. 相似文献
2.
When estimating, under quadratic loss, the location parameterθof a spherically symmetric distribution with known scale parameter, we show that it may be that the common practice of utilizing the residual vector as an estimate of the variance is preferable to using the known value of the variance. In the context of Stein-like shrinkage estimators, we exhibit sufficient conditions on the spherical distributions for which this paradox occurs. In particular, we show that it occurs fort-distributions when the dimension of the residual vector is sufficiently large. The main tools in the development are upper and lower bounds on the risks of the James–Stein estimators which are exact atθ=0. 相似文献
3.
In this paper we consider the problem of estimating the quadratic loss of point estimators of a location parameter for a family of spherically symmetric distributions. We compare the unbiased loss estimator of the minimax estimator with a new shrinkage type loss estimator. Conditions on the distributions for the domination of competing estimators are given. It is shown that, in addition to the class of scale mixtures of normal distributions, there exists a more general family for which the domination results hold. 相似文献
4.
Toyoaki Akai 《Annals of the Institute of Statistical Mathematics》1986,38(1):85-99
Summary LetX
i
,i=1,..., p be theith component of thep×1 vectorX=(X
1,X
2,...,X
p
)′. Suppose thatX
1,X
2,...,X
p
are independent and thatX
i
has a probability density which is positive on a finite interval, is symmetric about θ
i
and has the same variance. In estimation of the location vector θ=(θ1, θ2,...,θ
p
)′ under the squared error loss function explicit estimators which dominateX are obtained by using integration by parts to evaluate the risk function. Further, explicit dominating estimators are given
when the distributions ofX
i
′s are mixture of two uniform distributions. For the loss function
such an estimator is also given when the distributions ofX
i
′s are uniform distributions. 相似文献
5.
Dominique Fourdrinier William E. Strawderman Martin T. Wells 《Annals of the Institute of Statistical Mathematics》2006,58(1):73-92
In this article we consider estimating a location parameter of a spherically symmetric distribution under restrictions on
the parameter. First we consider a general theory for estimation on polyhedral cones which includes examples such as ordered
parameters and general linear inequality restrictions. Next, we extend the theory to cones with piecewise smooth boundaries.
Finally we consider shrinkage toward a closed convex set K where one has vague prior information that θ is in K but where θ is not restricted to be in K. In this latter case we give estimators which improve on the usual unbiased estimator while in the restricted parameter case
we give estimators which improve on the projection onto the cone of the unbiased estimator. The class of estimators is somewhat
non-standard as the nature of the constraint set may preclude weakly differentiable shrinkage functions. The technique of
proof is novel in the sense that we first deduce the improvement results for the normal location problem and then extend them
to the general spherically symmetric case by combining arguments about uniform distributions on the spheres, conditioning
and completeness. 相似文献
6.
We derive minimax generalized Bayes estimators of regression coefficients in the general linear model with spherically symmetric errors under invariant quadratic loss for the case of unknown scale. The class of estimators generalizes the class considered in Maruyama and Strawderman [Y. Maruyama, W.E. Strawderman, A new class of generalized Bayes minimax ridge regression estimators, Ann. Statist., 33 (2005) 1753–1770] to include non-monotone shrinkage functions. 相似文献
7.
Assume X = (X1, …, Xp)′ is a normal mixture distribution with density w.r.t. Lebesgue measure, , where Σ is a known positive definite matrix and F is any known c.d.f. on (0, ∞). Estimation of the mean vector under an arbitrary known quadratic loss function
Q(θ, a) = (a − θ)′ Q(a − θ), Q a positive definite matrix, is considered. An unbiased estimator of risk is obatined for an arbitrary estimator, and a sufficient condition for estimators to be minimax is then achieved. The result is applied to modifying all the Stein estimators for the means of independent normal random variables to be minimax estimators for the problem considered here. In particular the results apply to the Stein class of limited translation estimators. 相似文献
8.
We study existence of unbiased estimators of risk for estimators of the location parameter of a spherically symmetric distribution, when a residual vector is available to estimate scale, under invariant quadratic loss. We show such existence often characterizes normality. 相似文献
9.
This paper is primarily concerned with extending the results of Brandwein and Strawderman in the usual canonical setting of a general linear model when sampling from a spherically symmetric distribution. When the location parameter belongs to a proper linear subspace of the sampling space, we give an unbiased estimator of the difference of the risks between the least squares estimator φ0 and a general shrinkage estimator φ = φ0 − X − φ0 2 · g φ0. We obtain a general condition of domination for φ over φ0 which is weaker than that of Brandwein and Strawderman. We do not need any superharmonicity condition on g. Our results are valid for general quadratic loss. 相似文献
10.
Clyde D. Hardin Jr. 《Probability Theory and Related Fields》1982,61(3):293-302
Summary A stochastic process X={X
t
:tT| is called spherically generated if for each random vector
, there exist a random vector Y=(Y1,..., Y
m) with a spherical (radially symmetric) distribution and a matrix A such that X is distributed as AY. X is said to have the linear regression property if (X
0¦X
1,..., X
n) is a linear function of X
1,..., X
n
whenever the X
j's are elements of the linear span of X. It is shown that providing the linear span of X has dimension larger than two, then X has the linear regression property if and only if it is spherically generated. The class of symmetric stable processes which are spherically generated is shown to coincide with the class of socalled sub-Gaussian processes, characterizing those stable processes having the linear regression property.This research was supported by a grant from the University of Wisconsin-Milwaukee 相似文献
11.
The estimation of the location parameter of an ℓ1-symmetric distribution is considered. Specifically when a p-dimensional random vector has a distribution that is a mixture of uniform distributions on the ℓ1-sphere, we investigate a general class of estimators of the form δ=X+g. Under the usual quadratic loss, domination of δ over X is obtained through the partial differential inequality 4 div g+2X
∂2g+ g20 and a new superharmonicity-type-like notion adapted to the ℓ1-context. Specifically the condition of ℓ1-superharmonicity is that 2Δf+X
3f0 and div 3f0 as compared to the usual (ℓ2) condition Δf0. 相似文献
12.
We consider the estimation of the ratio of the scale parameters of two independent two-parameter exponential distributions with unknown location parameters. It is shown that the best affine equivariant estimator (BAEE) is inadmissible under any loss function from a large class of bowl-shaped loss functions. Two new classes of improved estimators are obtained. Some values of the risk functions of the BAEE and two improved estimators are evaluated for two particular loss functions. Our results are parallel to those of Zidek (1973, Ann. Statist., 1, 264–278), who derived a class of estimators that dominate the BAEE of the scale parameter of a two-parameter exponential distribution. 相似文献
13.
Michael Falk 《Journal of multivariate analysis》1998,67(2):306-317
The comedianCOM(X, Y) of random variablesX,Yis a median based robust alternative to the covariance ofXofY. For the bivariate normal case it is known thatCOM(X, Y), standardized by the median absolute deviations ofXandY, is a symmetric, strictly increasing and continuous function of the correlation coefficientρwith range [−1, 1] and can therefore serve as a robust alternative toρ. We show that this result, which is not true in general, extends to elliptical distributions even in the case where moments ofX,Ydo not exist. 相似文献
14.
Mohammad Jafari Jozani Éric Marchand William E. Strawderman 《Annals of the Institute of Statistical Mathematics》2014,66(4):811-832
For a vast array of general spherically symmetric location-scale models with a residual vector, we consider estimating the (univariate) location parameter when it is lower bounded. We provide conditions for estimators to dominate the benchmark minimax MRE estimator, and thus be minimax under scale invariant loss. These minimax estimators include the generalized Bayes estimator with respect to the truncation of the common non-informative prior onto the restricted parameter space for normal models under general convex symmetric loss, as well as non-normal models under scale invariant \(L^p\) loss with \(p>0\) . We cover many other situations when the loss is asymmetric, and where other generalized Bayes estimators, obtained with different powers of the scale parameter in the prior measure, are proven to be minimax. We rely on various novel representations, sharp sign change analyses, as well as capitalize on Kubokawa’s integral expression for risk difference technique. Several properties such as robustness of the generalized Bayes estimators under various loss functions are obtained. 相似文献
15.
The Riemann space whose elements are m × k (m k) matrices X, i.e., orientations, such that X′X = Ik is called the Stiefel manifold Vk,m. The matrix Langevin (or von Mises-Fisher) and matrix Bingham distributions have been suggested as distributions on Vk,m. In this paper, we present some distributional results on Vk,m. Two kinds of decomposition are given of the differential form for the invariant measure on Vk,m, and they are utilized to derive distributions on the component Stiefel manifolds and subspaces of Vk,m for the above-mentioned two distributions. The singular value decomposition of the sum of a random sample from the matrix Langevin distribution gives the maximum likelihood estimators of the population orientations and modal orientation. We derive sampling distributions of matrix statistics including these sample estimators. Furthermore, representations in terms of the Hankel transform and multi-sample distribution theory are briefly discussed. 相似文献
16.
Abhishek Banerjee 《Bulletin des Sciences Mathématiques》2011,(4):400-419
We consider schemes (X,OX) over an abelian closed symmetric monoidal category (C,⊗,1). Our aim is to extend a theorem of Kleiman on the relative Picard functor to schemes over (C,⊗,1). For this purpose, we also develop some basic theory on quasi-coherent modules on schemes (X,OX) over C. 相似文献
17.
J. Szenthe 《Central European Journal of Mathematics》2004,2(5):725-731
Spherically symmetric space-times have attained considerable attention ever since the early beginnings of the theory of general
relativity. In fact, they have appeared already in the papers of K. Schwarzschild [12] and W. De Sitter [5] which were published
in 1916 and 1917 respectively soon after Einstein's epoch-making work [7] in 1915. The present survey is concerned mainly
with recent results pertainig to the toplogy of spherically symmetric space-times.
Definition. By space-time a connected time-oriented 4-dimensional Lorentz manifold is meant. If (M,<,>) is a space-time, and Φ: SO(3)×M→M an isometric action such that the maximal dimension of its orbits is equal to 2, then the action Φ is said to be spherical and the space-time is said to be spherically symmetric [8]; [11]. Likewise, isometric actions Ψ: O(3)×M→M are also considered ([10], p. 365; [4]) which will be called quasi-spherical if the maximal dimension of its orbits is 2 and then the space-time is said to be quasi-spherically symmetric here. Each quasi-spherical action yields a spherical one by restricting it to the action of SO(3); the converse of this statement will be considered elsewhere.
The main results concerning spherically symmetric space-times are generally either of local character or pertaining to topologically
restricted simple situations [14], and earlier results of global character are scarce [1], [4], [6], [13]. A report on recent
results concerning the global geometry of spherically symmetric space-times [16] is presented below. 相似文献
18.
In this paper, we study the existence of the uniformly minimum risk equivariant (UMRE) estimators of parameters in a class
of normal linear models, which include the normal variance components model, the growth curve model, the extended growth curve
model, and the seemingly unrelated regression equations model, and so on. The necessary and sufficient conditions are given
for the existence of UMRE estimators of the estimable linear functions of regression coefficients, the covariance matrixV and (trV)α, where α > 0 is known, in the models under an affine group of transformations for quadratic losses and matrix losses, respectively.
Under the (extended) growth curve model and the seemingly unrelated regression equations model, the conclusions given in literature
for estimating regression coefficients can be derived by applying the general results in this paper, and the sufficient conditions
for non-existence of UMRE estimators ofV and tr(V) are expanded to be necessary and sufficient conditions. In addition, the necessary and sufficient conditions that there
exist UMRE estimators of parameters in the variance components model are obtained for the first time. 相似文献
19.
Wei-Min Huang 《Annals of the Institute of Statistical Mathematics》1986,38(1):137-144
Summary Let the random variablesX
1,X
2, ...,X
n
be generated by the first-order autoregressive modelX
i
=θX
i−1
+e
i
wheree
i
,i=1, 2, ...,n, are i.i.d. random variables with mean zero, variance σ2, and with unspecified density functiong(·). In the present paper we obtain a characterization of limiting distributions of nonparametric and parametric estimators
of θ as well as a local asymptotic minimax bound of the risks of estimators. 相似文献
20.
Liudas Giraitis Murad S. Taqqu Norma Terrin 《Probability Theory and Related Fields》1998,110(3):333-367
Summary. Let (X
t
,t∈Z) be a linear sequence with non-Gaussian innovations and a spectral density which varies regularly at low frequencies. This
includes situations, known as strong (or long-range) dependence, where the spectral density diverges at the origin. We study
quadratic forms of bivariate Appell polynomials of the sequence (X
t
) and provide general conditions for these quadratic forms, adequately normalized, to converge to a non-Gaussian distribution.
We consider, in particular, circumstances where strong and weak dependence interact. The limit is expressed in terms of multiple
Wiener-It? integrals involving correlated Gaussian measures.
Received: 22 August 1996 / In revised form: 30 August 1997 相似文献