Limit theorems of Chatterjee's rank correlation
Establishing the limiting distribution of Chatterjee's rank correlation for a general, possibly non-independent, pair of random variables has been eagerly awaited by many. This paper shows that (a) Chatterjee's rank correlation is asymptotically normal as long as one variable is not a measurable function of the other, (b) the corresponding asymptotic variance is uniformly bounded by 36, and (c) a consistent variance estimator exists. Similar results also hold for Azadkia-Chatterjee's graph-based correlation coefficient, a multivariate analogue of Chatterjee's original proposal. The proof is given by appealing to H\'ajek representation and Chatterjee's nearest-neighbor CLT.
Code (1)
Similar Papers 제목 키워드 기반
Convolution, Product and Correlation Theorems for Simplified Fractional Fourier Transform: A Mathematical Investigation
The notion of fractional Fourier transform (FrFT) has been used and investigated for many years by various research communities, which finds widespread applications in many diverse fields of research study. The potential…
Convergence of stochastic gradient descent under a local Lojasiewicz condition for deep neural networks
We study the convergence of stochastic gradient descent (SGD) for non-convex objective functions. We establish the local convergence with positive probability under the local \L{}ojasiewicz condition introduced by Chatte…
validThe Implications of the No-Free-Lunch Theorems for Meta-induction
The important recent book by G. Schurz appreciates that the no-free-lunch theorems (NFL) have major implications for the problem of (meta) induction. Here I review the NFL theorems, emphasizing that they do not only conc…
Convolution and Correlation Theorems for Wigner-Ville Distribution Associated with the Quaternion Offset Linear Canonical Transform
The quaternion offset linear canonical transform(QOLCT) has gained much popularity in recent years because of its applications in many areas, including color image and signal processing. At the same time the applications…
Automated Search for Impossibility Theorems in Social Choice Theory: Ranking Sets of Objects
We present a method for using standard techniques from satisfiability checking to automatically verify and discover theorems in an area of economic theory known as ranking sets of objects. The key question in this area, …
Decision MakingDecision Making Under Uncertainty