A note on the variance of the square components of a normal multivariate within a Euclidean ball

JOURNAL OF MULTIVARIATE ANALYSIS

卷:122

Article

Palombi, Filippo¹  Toti, Simona¹ 

[1] Ist Nazl Stat ISTAT, I-00184 Rome, Italy

关键词: Distributional truncation;    Covariance matrix reconstruction;    Fixed point iteration;   

DOI  :  10.1016/j.jmva.2013.08.011

来源: Elsevier

【 摘 要 】

We present arguments in favor of the inequalities var(X-n(2) vertical bar X is an element of B-v(rho)) <= 2 lambda E-n[X-n(2)] X is an element of B-v(rho)], where X similar to N-v(0, Lambda) is a normal vector in v >= 1 dimensions, with zero mean and covariance matrix Lambda = diag(lambda), and B-v(rho) is a centered v-dimensional Euclidean ball of square radius rho. Such relations lie at the heart of an iterative algorithm, proposed by Palombi et al. (2012) [6] to perform a reconstruction of Lambda from the covariance matrix of X conditioned to B-v(rho). In the regime of strong truncation, i.e. for rho less than or similar to lambda(n), the above inequality is easily proved, whereas it becomes harder for rho >> lambda(n). Here, we expand both sides in a function series controlled by powers of lambda(n)/p and show that the coefficient functions of the series fulfill the inequality order by order if rho is sufficiently large. The intermediate region remains at present an open challenge. (C) 2013 Elsevier Inc. All rights reserved.

【 授权许可】

Free