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171.
讨论了四元数体上广义正定矩阵的Kronecker积和Hadamard积的一些性质,推广了文[1]、[2]的结果。 相似文献
172.
Michael F. Driscoll 《The American statistician》2013,67(3):273-275
Results from the theory of linear models establish a particular idempotency condition as being necessary and sufficient for a quadratic form in a nonsingular normal vector to follow a chi-square distribution. We give a theorem that is somewhat stronger than the standard ones, and provide a proof that is more accessible than those usually given, in that it uses only linear algebra and calculus. 相似文献
173.
Edith Cohen 《Journal of Combinatorial Optimization》1998,2(4):307-332
We consider1 the problem of predicting the nonzero structure of a product of two or more matrices. Prior knowledge of the nonzero structure can be applied to optimize memory allocation and to determine the optimal multiplication order for a chain product of sparse matrices. We adapt a recent algorithm by the author and show that the essence of the nonzero structure and hence, a near-optimal order of multiplications, can be determined in near-linear time in the number of nonzero entries, which is much smaller than the time required for the multiplications. An experimental evaluation of the algorithm demonstrates that it is practical for matrices of order 103 with 104 nonzeros (or larger). A relatively small pre-computation results in a large time saved in the computation-intensive multiplication. 相似文献
174.
Mariusz Grządziel 《Statistical Papers》2008,49(3):399-419
Gnot et al. (J Statist Plann Inference 30(1):223–236, 1992) have presented the formulae for computing Bayes invariant quadratic
estimators of variance components in normal mixed linear models of the form
where the matrices V
i
, 1 ≤ i ≤ k − 1, are symmetric and nonnegative definite and V
k
is an identity matrix. These formulae involve a basis of a quadratic subspace containing MV
1
M,...,MV
k-1
M,M, where M is an orthogonal projector on the null space of X′. In the paper we discuss methods of construction of such a basis. We survey Malley’s algorithms for finding the smallest
quadratic subspace including a given set of symmetric matrices of the same order and propose some modifications of these algorithms.
We also consider a class of matrices sharing some of the symmetries common to MV
1
M,...,MV
k-1
M,M. We show that the matrices from this class constitute a quadratic subspace and describe its explicit basis, which can be
directly used for computing Bayes invariant quadratic estimators of variance components. This basis can be also used for improving
the efficiency of Malley’s algorithms when applied to finding a basis of the smallest quadratic subspace containing the matrices
MV
1
M,...,MV
k-1
M,M. Finally, we present the results of a numerical experiment which confirm the potential usefulness of the proposed methods.
Dedicated to the memory of Professor Stanisław Gnot. 相似文献
175.
Summary Two quadratic formsS
H andS
E for a testable hypothesis and for an error in the multivariate Zyskind-Martin model with singular covariance matrix are expressed
by means of projector operators. Thus the results for the multivariate standard model with identity covariance matrix given
by Humak (1977) and Christensen (1987, 1991) are generalized for the case of Zyskind-Martin model. Special cases of our results
are formulae forS
H andS
E in Aitken's (1935) model. In the case of general Gauss-Markoff modelS
H andS
E can also be expressed by means of projector operators for some subclasses of testable hypotheses. For these hypotheses, testing
in Gauss-Markoff model is equivalent to testing in a Zyskind-Martin model. 相似文献
176.
The vec of a matrix X stacks columns of X one under another in a single column; the vech of a square matrix X does the same thing but starting each column at its diagonal element. The Jacobian of a one-to-one transformation X → Y is then ∣∣?(vecX)/?(vecY) ∣∣ when X and Y each have functionally independent elements; it is ∣∣ ?(vechX)/?(vechY) ∣∣ when X and Y are symmetric; and there is a general form for when X and Y are other patterned matrices. Kronecker product properties of vec(ABC) permit easy evaluation of this determinant in many cases. The vec and vech operators are also very convenient in developing results in multivariate statistics. 相似文献
177.
Summary A new class of matrix variate elliptically contoured distributions is defined. Properties of this class of distributions are
studied. Examples of distributions which belong to this class are also presented.
Rose-Hulman Institute of Technology
Research supported by the FRC Major Grant, Bowling Grant, Bowling Green State University. 相似文献
178.
Josefa Lopes Troya 《Journal of statistical planning and inference》1982,6(4):373-419
A linear model with one treatment at V levels and first order regression on K continuous covariates with values on a K-cube is considered. The D-criterion is used to judge the ‘goodness’ of any design for estimating the parameters of this model. Since this criterion is based on the determinant of the information matrix M(d) of a design d, upper bounds for |M(d)| yield lower bounds for the D-efficiency of any design d in estimating the vector of parameters in the model. We consider here only classes of designs d for which the number N of observations to be taken is a multiple of V, that is, there exists R≥2 such that N=V×R.Under these conditions, we determine the maximum of |M(d)|, and conditions under which the maximum is attained. These conditions include R being even, each treatment level being observed the same number of times, that is, R times, and N being a multiple of four. For the other cases of congruence of N (modulo 4) we further determine upper bounds on |M (d)| for equireplicated designs, i.e. for designs with equal number of observations per treatment level. These upper bounds are shown to depend also on the congruence of V (modulo 4). For some triples (N,V,K), the upper bounds determined are shown to be attained.Construction methods yielding families of designs which attain the upper bounds of |M(d)| are presented, for each of the sixteen cases of congruence of N and V.We also determine the upper bound for D-optimal designs for estimating only the treatment parameters, when first order regression on one continuous covariate is present. 相似文献
179.
Let EG(m, 2) denote the m-dimensional finite Euclidean space (or geometry) based on GF(2), the finite field with elements 0 and 1. Let T be a set of points in this space, then T is said to form a q-covering (where q is an integer satisfying 1?q?m) of EG(m, 2) if and only if T has a nonempty intersection with every (m-q)-flat of EG(m, 2). This problem first arose in the statistical context of factorial search designs where it is known to have very important and wide ranging applications. Evidently, it is also useful to study this from the purely combinatorial point of view. In this paper, certain fundamental studies have been made for the case when q=2. Let N denote the size of the set T. Given N, we study the maximal value of m. 相似文献
180.
Jun Hui 《统计学通讯:理论与方法》2013,42(6):935-941