Bayesian Estimation of the Precision Matrix with Monotone Missing Data
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Lithuanian Mathematical Journal
Bayesian estimation of the precision matrix with monotone missing data Emna Ghorbel a , Kaouthar Kammoun a, and Mahdi Louati b a
b
Faculty of Sciences of Sfax B.P. 1171, Sfax University, Tunisia Sfax National School of Electronics and Telecommunications, B.P. 1163, CP 3018, Sfax University, Tunisia (e-mail: [email protected]; [email protected]; [email protected])
Received July 17, 2018; revised February 4, 2020
Abstract. This research paper stands for the estimation of the precision matrix of the normal matrix with monotone missing data. We explicitly provide maximum and expectation a posteriori estimators. For this purpose, we basically use an extension of the Wishart distribution, that is, the Riesz distribution on symmetric matrices. We prove that some of the latter distributions may be presented using Gaussian samples with missing data. An algorithm for generating this distribution is illustrated. Therefore we prove that the inverse Riesz model extends the conjugate property of the inverse Wishart one. This allows us to determine the desired Bayesian estimators. Besides, we propose an estimator of the precision matrix based on the notion of the Cholesky decomposition. Finally, we test the performance of the estimators by means of the mean squared error. MSC: primary 62H12; secondary 44A10 Keywords: Bayesian estimation, Cholesky decomposition, conjugate prior, maximum a posterior distribution, Riesz distribution, Wishart distribution
1 Introduction The precision matrix or the covariance matrix has been investigated for a long time. It corresponds to an interesting subject of research, and its applications are numerous. It is of special interest in graphical Gaussian model since it is closely related to the idea of partial correlation (see Lauritzen [17]), in classification procedures (see Anderson [1]), in exploratory data analysis (see Tukey [21]), in Bayesian analysis of the multivariate normal distribution (see Bernardo and Smith [3]), and in portfolio selection (see Ledoit et al. [18]). The precision matrix is of particular significance in multivariate analysis; the problem of estimating has attracted considerable attention, and a great deal of works have emerged. In this direction, Haff [12] highlighted that estimating the linear discriminant coefficient is related to estimating the multivariate normal precision matrix. Besides, a variety of loss functions and several methods were adapted to set forward different types of estimators (see Dey [4], Efron and Morris [6], Gupta and Krishnamoorthy [10], and Haff [11]). The main finding of the paper falls within this framework. A natural extension of the Wishart distribution, that is, the Riesz distribution on the space of symmetric matrices, is introduced to establish new estimators of the precision matrix. More precisely, we consider k1 > 0 observations of a multivariate centered normal N (0, σ) c 2020 Springer Science+Business Media, LLC 0363-1672/20/6004-0001
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E. Ghorbel, K. Kammou
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