Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Concentration inequalities for the sample correlation coefficient

Daniel Salnikov

Abstract

The sample correlation coefficient $R$ plays an important role in many statistical analyses. We study the moments of $R$ under the bivariate Gaussian model assumption, provide a novel approximation for its finite sample mean and connect it with known results for the variance. We exploit these approximations to present non-asymptotic concentration inequalities for $R$. Finally, we illustrate our results in a simulation experiment that further validates the approximations presented in this work.

Concentration inequalities for the sample correlation coefficient

Abstract

The sample correlation coefficient plays an important role in many statistical analyses. We study the moments of under the bivariate Gaussian model assumption, provide a novel approximation for its finite sample mean and connect it with known results for the variance. We exploit these approximations to present non-asymptotic concentration inequalities for . Finally, we illustrate our results in a simulation experiment that further validates the approximations presented in this work.
Paper Structure (10 sections, 5 theorems, 64 equations, 3 tables)

This paper contains 10 sections, 5 theorems, 64 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Mean and variance approximations
  3. Concentration inequalities
  4. Simulation Experiment
  5. Conclusion
  6. Preliminary approximations
  7. First moment
  8. Second moment
  9. Concentration inequalities
  10. Simulation experiment supplement

Key Result

Theorem 1

Let $R$ be the sample correlation coefficient computed from a sample of size $n \geq 3$ coming from a bivariate normal distribution with population correlation coefficient $\rho$. Then, we have that where $g_m (k) = \left \{ \Gamma \left (\frac{m + k + 1}{2} \right ) \Gamma \left (\frac{n - 2}{2} \right ) \right \} \left \{ \Gamma \left (\frac{n + m + k - 1}{2} \right ) \right \}^{-1} \mathbb{I}

Theorems & Definitions (11)

  • Theorem 1
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Lemma 1
  • proof
  • ...and 1 more