Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userRevisiting Grusss inequality: covariance bounds,QDE but not QD copulas, and central moments
291
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Since the pioneering work of Gerhard Gruss dating back to 1935, Grusss inequality and, more generally, Gruss-type bounds for covariances have fascinated researchers and found numerous applications in areas such as economics, insurance, reliability, and, more generally, decision making under uncertainly. Gruss-type bounds for covariances have been established mainly under most general dependence structures, meaning no restrictions on the dependence structure between the two underlying random variables. Recent work in the area has revealed a potential for improving Gruss-type bounds, including the original Grusss bound, assuming dependence structures such as quadrant dependence (QD). In this paper we demonstrate that the relatively little explored notion of `quadrant dependence in expectation (QDE) is ideally suited in the context of bounding covariances, especially those that appear in the aforementioned areas of application. We explore this research avenue in detail, establish general Gruss-type bounds, and illustrate them with newly constructed examples of bivariate distributions, which are not QD but, nevertheless, are QDE. The examples rely on specially devised copulas. We supplement the examples with results concerning general copulas and their convex combinations. In the process of deriving Gruss-type bounds, we also establish new bounds for central moments, whose optimality is demonstrated.
rate research
Read More
We prove the large-dimensional Gaussian approximation of a sum of $n$ independent random vectors in $mathbb{R}^d$ together with fourth-moment error bounds on convex sets and Euclidean balls. We show that compared with classical third-moment bounds, our bounds have near-optimal dependence on $n$ and can achieve improved dependence on the dimension $d$. For centered balls, we obtain an additional error bound that has a sub-optimal dependence on $n$, but recovers the known result of the validity of the Gaussian approximation if and only if $d=o(n)$. We discuss an application to the bootstrap. We prove our main results using Steins method.
For extreme value copulas with a known upper tail dependence coefficient we find pointwise upper and lower bounds, which are used to establish upper and lower bounds of the Spearman and Kendall correlation coefficients. We shown that in all cases the lower bounds are attained on Marshall--Olkin copulas, and the upper ones, on copulas with piecewise linear dependence functions.
As an extension of a central limit theorem established by Svante Janson, we prove a Berry-Esseen inequality for a sum of independent and identically distributed random variables conditioned by a sum of independent and identically distributed integer-valued random variables.
For probability measures on a complete separable metric space, we present sufficient conditions for the existence of a solution to the Kantorovich transportation problem. We also obtain sufficient conditions (which sometimes also become necessary) for the convergence, in transportation, of probability measures when the cost function is continuous, non-decreasing and depends on the distance. As an application, the CLT in the transportation distance is proved for independent and some dependent stationary sequences.
In this paper we propose two schemes for the recovery of the spectrum of a covariance matrix from the empirical covariance matrix, in the case where the dimension of the matrix is a subunitary multiple of the number of observations. We test, compare and analyze these on simulated data and also on some data coming from the stock market.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
