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Let X Nv(0, {\Lambda}) be a normal vector in v dimensions, where {\Lambda} isdiagonal. With reference to the truncated distribution of X on the interior ofa v-dimensional Euclidean ball, we completely prove a variance inequality and acovariance inequality that were recently discussed by F. Palombi and S. Toti[J. Multivariate Anal. 122 (2013) 355-376]. These inequalities ensure theconvergence of an algorithm for the reconstruction of {\Lambda} only on thebasis of the covariance matrix of X truncated to the Euclidean ball. Theconcept of monotone likelihood ratio is useful in our proofs. Moreover, we alsoprove and utilize the fact that the cumulative distribution function of anypositive linear combination of independent chi-square variates is log-concave,even though the same may not be true for the corresponding density function.

Variance and Covariance Inequalities for Truncated Joint Normal Distribution via Monotone Likelihood Ratio and Log-concavity