The tetrachoric series is a technique for evaluating multivariate normal probabilities frequently cited in the statistical literature. In this paper we have examined the convergence properties of the tetrachoric series and have established the following. For orthant probabilities, the tetrachoric series converges if , 1 ≤ i < j ≤ k, where ?ij are the correlation coefficients of a k-variate normal distribution. The tetrachoric series for orthant probabilities diverges whenever k is even and or k is odd and , 1 ≤ i < j ≤ k. Other specific results concerning the convergence or divergence of this series are also given. The principal point is that the assertion that the tetrachoric series converges for all k ≥ 2 and all ?ij such that the correlation matrix is positive definite is false. 相似文献
The problem of estimating a mean vector of scale mixtures of multivariate normal distributions with the quadratic loss function is considered. For a certain class of these distributions, which includes at least multivariate-t distributions, admissible minimax estimators are given. 相似文献
This paper considers a bidimensional continuous-time renewal risk model of insurance business with different claim-number processes and strongly subexponential claims. For the finite-time ruin probability defined as the probability for the aggregate surplus process to break down the horizontal line at the level zero within a given time, an uniform asymptotic formula is established, which provides new insights into the solvency ability of the insurance company. 相似文献
The Fréchet distance between two multivariate normal distributions having means μX, μY and covariance matrices ΣX, ΣY is shown to be given by . The quantity d0 given by is a natural metric on the space of real covariance matrices of given order. 相似文献
Herman Chernoff used Hermite polynomials to prove an inequality for the normal distribution. This inequality is useful in solving a variation of the classical isoperimetric problem which, in turn, is relevant to data compression in the theory of element identification. As the inequality is of interest in itself, we prove a multivariate generalization of it using a different argument. 相似文献
Let α(n1, n2) be the probability of classifying an observation from population Π1 into population Π2 using Fisher's linear discriminant function based on samples of size n1 and n2. A standard estimator of α, denoted by T1, is the proportion of observations in the first sample misclassified by the discriminant function. A modification of T1, denoted by T2, is obtained by eliminating the observation being classified from the calculation of the discriminant function. The UMVU estimators, and , of ET1 = τ1(n1, n2) and ET2 = τ2(n1, n2) = α(n1 ? 1, n2) are derived for the case when the populations have multivariate normal distributions with common dispersion matrix. It is shown that and are nonincreasing functions of D2, the Mahalanobis sample distance. This result is used to derive the sampling distributions and moments of and . It is also shown that α is a decreasing function of Δ2 = (μ1 ? μ2)′Σ?1(μ1 ? μ2). Hence, by truncating and (or any estimator) at the value of α for Σ = 0, new estimators are obtained which, for all samples, are as close or closer to α. 相似文献
Let X1, X2,… be idd random vectors with a multivariate normal distribution N(μ, Σ). A sequence of subsets {Rn(a1, a2,…, an), n ≥ m} of the space of μ is said to be a (1 − α)-level sequence of confidence sets for μ if P(μ Rn(X1, X2,…, Xn) for every n ≥ m) ≥ 1 − α. In this note we use the ideas of Robbins Ann. Math. Statist.41 (1970) to construct confidence sequences for the mean vector μ when Σ is either known or unknown. The constructed sequence Rn(X1, X2, …, Xn) depends on Mahalanobis'
or Hotelling's
according as Σ is known or unknown. Confidence sequences for the vector-valued parameter in the general linear model are also given. 相似文献
In this paper we solve the problem of unique factorization of products ofn-variate nonsingular normal distributions with covariance matrices of the form , ij =pij forij, =
i2
,j=j,p0. 相似文献
In three or more dimensions it is well known that the usual point estimator for the mean of a multivariate normal distribution is minimax but not admissible with respect to squared Euclidean distance loss. This paper gives sufficient conditions on the prior distribution under which the Bayes estimator has strictly lower risk than the usual estimator. Examples are given for which the posterior density is useful in the formation of confidence sets. 相似文献
Sampling from a truncated multivariate normal distribution (TMVND) constitutes the core computational module in fitting many statistical and econometric models. We propose two efficient
methods, an iterative data augmentation (DA) algorithm and a non-iterative inverse Bayes formulae (IBF) sampler, to simulate TMVND and generalize them to multivariate normal distributions with linear inequality constraints.
By creating a Bayesian incomplete-data structure, the posterior step of the DA algorithm directly generates random vector
draws as opposed to single element draws, resulting obvious computational advantage and easy coding with common statistical
software packages such as S-PLUS, MATLAB and GAUSS. Furthermore, the DA provides a ready structure for implementing a fast
EM algorithm to identify the mode of TMVND, which has many potential applications in statistical inference of constrained
parameter problems. In addition, utilizing this mode as an intermediate result, the IBF sampling provides a novel alternative
to Gibbs sampling and eliminates problems with convergence and possible slow convergence due to the high correlation between
components of a TMVND. The DA algorithm is applied to a linear regression model with constrained parameters and is illustrated
with a published data set. Numerical comparisons show that the proposed DA algorithm and IBF sampler are more efficient than
the Gibbs sampler and the accept-reject algorithm. 相似文献
Logistic regression techniques can be used to restrict the conditional probabilities of a Bayesian network for discrete variables. More specifically, each variable of the network can be modeled through a logistic regression model, in which the parents of the variable define the covariates. When all main effects and interactions between the parent variables are incorporated as covariates, the conditional probabilities are estimated without restrictions, as in a traditional Bayesian network. By incorporating interaction terms up to a specific order only, the number of parameters can be drastically reduced. Furthermore, ordered logistic regression can be used when the categories of a variable are ordered, resulting in even more parsimonious models. Parameters are estimated by a modified junction tree algorithm. The approach is illustrated with the Alarm network. 相似文献
A procedure for selecting the t largest of k multivariate normal populations on the basis of distance is reviewed. Computation of integrals of products of non-central Beta distribution and density functions, required for implementing the procedure, is described. A table of minimum sample sizes needed to guarantee a specified probability of correct selection is given (Table 1). 相似文献
We consider estimation of a multivariate normal mean vector under sum of squared error loss.We propose a new class of minimax admissible estimator which are generalized Bayes with respect to a prior distribution which is a mixture of a point prior at the origin and a continuous hierarchical type prior. We also study conditions under which these generalized Bayes minimax estimators improve on the James–Stein estimator and on the positive-part James–Stein estimator. 相似文献
In this paper a procedure of construction of β-expectation tolerance regions in the framework of the structural method of inference has been developed. The procedure has been applied to the generalized multivariate model and the β-expectation tolerance region for this case has been constructed assuming the normal distribution for the error variables of the model. 相似文献
Let Ui = (Xi, Yi), i = 1, 2,…, n, be a random sample from a bivariate normal distribution with mean μ = (μx, μy) and covariance matrix . Let Xi, i = n + 1,…, N represent additional independent observations on the X population. Consider the hypothesis testing problem H0 : μ = 0 vs. H1 : μ ≠ 0. We prove that Hotelling's T2 test, which uses (Xi, Yi), i = 1, 2,…, n (and discards Xi, i = n + 1,…, N) is an admissible test. In addition, and from a practical point of view, the proof will enable us to identify the region of the parameter space where the T2-test cannot be beaten. A similar result is also proved for the problem of testing μx ? μy = 0. A Bayes test and other competitors which are similar tests are discussed. 相似文献
Bayesian model averaging (BMA) is the state of the art approach for overcoming model uncertainty. Yet, especially on small data sets, the results yielded by BMA might be sensitive to the prior over the models. Credal model averaging (CMA) addresses this problem by substituting the single prior over the models by a set of priors (credal set). Such approach solves the problem of how to choose the prior over the models and automates sensitivity analysis. We discuss various CMA algorithms for building an ensemble of logistic regressors characterized by different sets of covariates. We show how CMA can be appropriately tuned to the case in which one is prior-ignorant and to the case in which instead domain knowledge is available. CMA detects prior-dependent instances, namely instances in which a different class is more probable depending on the prior over the models. On such instances CMA suspends the judgment, returning multiple classes. We thoroughly compare different BMA and CMA variants on a real case study, predicting presence of Alpine marmot burrows in an Alpine valley. We find that BMA is almost a random guesser on the instances recognized as prior-dependent by CMA. 相似文献
The maximum likelihood estimators are uniquely obtained in a multivariate normal distribution with AR(1) covariance structure for monotone data. The maximum likelihood estimator of mean is unbiased. 相似文献
The multivariate model, where not only parameters of the mean value of the observation matrix, but also some other parameters
occur in constraints, is considered in the paper. Some basic inference is presented under the condition that the covariance
matrix is either unknown, or partially unknown, or known.
Supported by the grant of the Council of Czech Republic MSM 6 198 959 214. 相似文献
A procedure is developed that enables the encoding of a subjectiven-dimensional joint normal probability density function through the assessment of its marginal means and variances andn(n–1)/2 conditional means. The new method is based on the theory of conjugate directions for quadratic forms, and it exploits the fact that normal distributions have quadratic equal-likelihood surfaces. Unlike previous approaches, this new method enables easy detection and resolution of inconsistencies in the assessments that could lead to an indefinite estimate of the covariance matrix. 相似文献