On an equivariant estimate of the density of a matrix normal distribution

1995 ◽  
Vol 75 (1) ◽  
pp. 1394-1400 ◽  
Author(s):  
V. M. Kondakov
Keyword(s):  
Normal Distribution ◽  
Matrix Normal ◽  
Matrix Normal Distribution

scholarly journals On n/p-Asymptotic Distribution Of Vector Of Weighted Traces Of Powers Of Wishart Matrices

2018 ◽  
Vol 33 ◽  
pp. 24-40 ◽  
Author(s):  
Jolanta Pielaszkiewicz ◽  
Dietrich Von Rosen ◽  
Martin Singull
Keyword(s):  
Normal Distribution ◽  
Covariance Matrix ◽  
Asymptotic Distribution ◽  
Joint Distribution ◽  
Recursive Formula ◽  
Multivariate Normal ◽  
Wishart Matrices ◽  
The Matrix ◽  
Matrix Normal ◽  
Matrix Normal Distribution

The joint distribution of standardized traces of $\frac{1}{n}XX'$ and of $\Big(\frac{1}{n}XX'\Big)^2$, where the matrix $X:p\times n$ follows a matrix normal distribution is proved asymptotically to be multivariate normal under condition $\frac{{n}}{p}\overset{n,p\rightarrow\infty}{\rightarrow}c>0$. Proof relies on calculations of asymptotic moments and cumulants obtained using a recursive formula derived in Pielaszkiewicz et al. (2015). The covariance matrix of the underlying vector is explicitely given as a function of $n$ and $p$.

scholarly journals A Bayesian Multiple-Trait and Multiple-Environment Model Using the Matrix Normal Distribution

2018 ◽  
Author(s):  
Osval A. Montesinos-López ◽  
Abelardo Montesinos-López ◽  
José Cricelio Montesinos-López ◽  
José Crossa ◽  
Francisco Javier Luna-Vázquez ◽  
...  
Keyword(s):  
Normal Distribution ◽  
Multiple Trait ◽  
Environment Model ◽  
The Matrix ◽  
Matrix Normal ◽  
Matrix Normal Distribution

scholarly journals Dynamic Brain Connectivity Alternation Detection via Matrix-variate Differential Network Model

2018 ◽  
Author(s):  
Jiadong Ji ◽  
Yong He ◽  
Lei Xie
Keyword(s):  
Normal Distribution ◽  
Partial Correlation ◽  
Graphical Model ◽  
Brain Connectivity ◽  
Connectivity Analysis ◽  
Differential Network ◽  
The Matrix ◽  
Matrix Normal ◽  
Matrix Normal Distribution ◽  
The Brain

AbstractMotivationNowadays brain connectivity analysis has attracted tremendous attention and has been at the foreground of neuroscience research. Brain functional connectivity reveals the synchronization of brain systems through correlations in neurophysiological measures of brain activity. Growing evidence now suggests that the brain connectivity network experiences alternations with the presence of numerous neurological disorders, thus differential brain network analysis may provides new insights into disease pathologies. For the matrix-valued data in brain connectivity analysis, existing graphical model estimation methods assume a vector normal distribution that in essence requires the columns of the matrix data to be independent. It is obviously not true, they have limited applications. Among the few solutions on graphical model estimation under a matrix normal distribution, none of them tackle the estimation of differential graphs across different populations. This motivates us to consider the differential network for matrix-variate data to detect the brain connectivity alternation.ResultsThe primary interest is to detect spatial locations where the connectivity, in terms of the spatial partial correlation, differ across the two groups. To detect the brain connectivity alternation, we innovatively propose a Matrix-Variate Differential Network (MVDN) model. MVDN assumes that the matrix-variate data follows a matrix-normal distribution. We exploit the D-trace loss function and a Lasso-type penalty to directly estimate the spatial differential partial correlation matrix where the temporal information is fully excavated. We propose an ADMM algorithm for the Lasso penalized D-trace loss optimization problem. We investigate theoretical properties of the estimator. We show that under mild and regular conditions, the proposed method can identify all differential edges accurately with probability tending to 1 in high-dimensional setting where dimensions of matrix-valued data p, q and sample size n are all allowed to go to infinity. Simulation studies demonstrate that MVDN provides more accurate differential network estimation than that achieved by other state-of-the-art methods. We apply MVDN to Electroencephalography (EEG) dataset, which consists of 77 alcoholic individuals and 45 controls. The hub genes and differential interaction patterns identified are consistent with existing experimental [email protected] informationSupplementary data are available online.

scholarly journals Robust Quadratic Programming for Price Optimization

2017 ◽  
Author(s):  
Akihiro Yabe ◽  
Shinji Ito ◽  
Ryohei Fujimaki
Keyword(s):  
Quadratic Programming ◽  
Pricing Strategies ◽  
Demand Modeling ◽  
Risk Return ◽  
Price Optimization ◽  
Trade Offs ◽  
Matrix Normal ◽  
Matrix Normal Distribution ◽  
Programming Algorithms ◽  
Actual Price

The goal of price optimization is to maximize total revenue by adjusting the prices of products, on the basis of predicted sales numbers that are functions of pricing strategies. Recent advances in demand modeling using machine learning raise a new challenge in price optimization, i.e., how to manage statistical errors in estimation. In this paper, we show that uncertainty in recently-proposed prescriptive price optimization frameworks can be represented by a matrix normal distribution. For this particular uncertainty, we propose novel robust quadratic programming algorithms for conservative lower-bound maximization. We offer an asymptotic probabilistic guarantee of conservativeness of our formulation. Our experiments on both artificial and actual price data show that our robust price optimization allows users to determine best risk-return trade-offs and to explore safe, profitable price strategies.

On the Mathematical Basis of Zelen’s Prerandomized Designs

1985 ◽  
Vol 24 (03) ◽  
pp. 120-130 ◽  
Author(s):  
E. Brunner ◽  
N. Neumann
Keyword(s):  
Clinical Trial ◽  
Normal Distribution ◽  
Random Variables ◽  
Small Samples ◽  
Selection Effects ◽  
Sample Sizes ◽  
Mathematical Basis ◽  
Additional Assumptions

SummaryThe mathematical basis of Zelen’s suggestion [4] of pre randomizing patients in a clinical trial and then asking them for their consent is investigated. The first problem is to estimate the therapy and selection effects. In the simple prerandomized design (PRD) this is possible without any problems. Similar observations have been made by Anbar [1] and McHugh [3]. However, for the double PRD additional assumptions are needed in order to render therapy and selection effects estimable. The second problem is to determine the distribution of the statistics. It has to be taken into consideration that the sample sizes are random variables in the PRDs. This is why the distribution of the statistics can only be determined asymptotically, even under the assumption of normal distribution. The behaviour of the statistics for small samples is investigated by means of simulations, where the statistics considered in the present paper are compared with the statistics suggested by Ihm [2]. It turns out that the statistics suggested in [2] may lead to anticonservative decisions, whereas the “canonical statistics” suggested by Zelen [4] and considered in the present paper keep the level quite well or may lead to slightly conservative decisions, if there are considerable selection effects.

Interrelations of Thrombo-Embolic Diseases and Blood-Group Distribution

1963 ◽  
Vol 09 (02) ◽  
pp. 472-474 ◽  
Author(s):  
W Dick ◽  
W Schneider ◽  
K Brockmüller ◽  
W Mayer
Keyword(s):  
Normal Distribution ◽  
Blood Group ◽  
Blood Groups ◽  
The Other ◽  
Other Hand ◽  
Group Distribution

SummaryA comparison between the repartition of the blood groups in 461 patients suffering from thromboembolic disorders and the normal distribution has shown a statistically ascertained predominance of the group A1. On the other hand the blood groups 0 and A2 are distinctly less frequent than in the normal distribution.

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