Variance and covariance inequalities for truncated joint normal distribution via monotone likelihood ratio and log-concavity

Let X∼Nv(0,Λ)X∼Nv(0,Λ) be a normal vector in v(≥1) dimensions, where ΛΛ is diagonal. With reference to the truncated distribution of XX on the interior of a vv-dimensional Euclidean ball, we completely prove a variance inequality and a covariance inequality that were recently discussed by Palombi and Toti (2013). These inequalities ensure the convergence of an algorithm for the reconstruction of ΛΛ only on the basis of the covariance matrix of XX truncated to the Euclidean ball. The concept of monotone likelihood ratio is useful in our proofs. Moreover, we also prove and utilize the fact that the cumulative distribution function of any positive linear combination of independent chi-square variates is log-concave, even though the same may not be true for the corresponding density function.