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Reviving pseudo-inverses: Asymptotic properties of large dimensional Moore-Penrose and Ridge-type inverses with applications

Taras Bodnar, Nestor Parolya

Abstract

In this paper, we derive high-dimensional asymptotic properties of the Moore-Penrose inverse and, as a byproduct, of various ridge-type inverses of the sample covariance matrix. In particular, the analytical expressions of the asymptotic behavior of the weighted sample trace moments of generalized inverse matrices are deduced in terms of the partial exponential Bell polynomials which can be easily computed in practice. The existent results for pseudo-inverses are extended in several directions: (i) First, the population covariance matrix is not assumed to be a multiple of the identity matrix; (ii) Second, the assumption of normality is not used in the derivation; (iii) Third, the asymptotic results are derived under the high-dimensional asymptotic regime. Our findings provide universal methodology for construction of fully data-driven improved shrinkage estimators of the precision matrix, optimal portfolio weights and beyond. It is found that the Moore-Penrose inverse acts asymptotically as a certain regularizer of the true covariance matrix and it seems that its proper transformation (shrinkage) performs similarly to or even outperforms the existing benchmarks in many applications, while keeping the computational time as minimal as possible.

Reviving pseudo-inverses: Asymptotic properties of large dimensional Moore-Penrose and Ridge-type inverses with applications

Abstract

In this paper, we derive high-dimensional asymptotic properties of the Moore-Penrose inverse and, as a byproduct, of various ridge-type inverses of the sample covariance matrix. In particular, the analytical expressions of the asymptotic behavior of the weighted sample trace moments of generalized inverse matrices are deduced in terms of the partial exponential Bell polynomials which can be easily computed in practice. The existent results for pseudo-inverses are extended in several directions: (i) First, the population covariance matrix is not assumed to be a multiple of the identity matrix; (ii) Second, the assumption of normality is not used in the derivation; (iii) Third, the asymptotic results are derived under the high-dimensional asymptotic regime. Our findings provide universal methodology for construction of fully data-driven improved shrinkage estimators of the precision matrix, optimal portfolio weights and beyond. It is found that the Moore-Penrose inverse acts asymptotically as a certain regularizer of the true covariance matrix and it seems that its proper transformation (shrinkage) performs similarly to or even outperforms the existing benchmarks in many applications, while keeping the computational time as minimal as possible.
Paper Structure (46 sections, 36 theorems, 242 equations, 9 figures)

This paper contains 46 sections, 36 theorems, 242 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Asymptotic properties of pseudo-inverse matrices
  3. Weighted moments of the sample Moore-Penrose inverse
  4. Weighted moments of the sample ridge-type inverse
  5. Weighted moments of the sample Moore-Penrose-ridge inverse
  6. Applications
  7. Shrinkage estimator of the high-dimensional precision matrix
  8. Results for the Moore-Penrose inverse
  9. Results for the ridge-type inverse
  10. Results for the Moore-Penrose-ridge inverse
  11. Shrinkage estimator for minimum variance portfolio
  12. Finite sample performance
  13. Results for the estimation of precision matrix
  14. Results for the estimation of the global minimum variance portfolio
  15. Proofs of the main theorems and corollaries
  16. ...and 31 more sections

Key Result

Proposition 2.1

Let $\mathbf{Y}_n$ fulfill the stochastic representation obs. Then, under Assumptions (A1)-(A3) , $v(t)$ is strictly decreasing for $t \ge 0$ and $c_n >1$.

Figures (9)

  • Figure 1: Finite-sample performance of the estimators for $v(0)$ and $v^{(1)}(0)$ when $c_n \in (1,5]$, $n\in\{100,250,500\}$, and the elements of $\mathbf{X}_n$ drawn from the normal (first column) and scale $t$-distribution (second column).
  • Figure 2: Finite-sample performance of the estimators for $v(t)$ and $v^{(1)}(t)$ when $t \in [0.1,5]$, $c_n \in \{1.2,1.5,2,3.5,5\}$, $n=100$, and the elements of $\mathbf{X}_n$ drawn from the normal (first column) and scale $t$-distribution (second column).
  • Figure 3: PRIAL for $c_n \in (1,5]$, $n=100$ (first row) and $n=250$ (second row) when the elements of $\mathbf{X}_n$ are drawn from the normal distribution (first column) and scale $t$-distribution (second column).
  • Figure 4: rOSV for $c_n \in (1,5]$, $n=100$ (first row) and $n=250$ (second row) when the elements of $\mathbf{X}_n$ are drawn from the normal distribution (first column) and scale $t$-distribution (second column).
  • Figure S.5: PRIAL for $c_n \in (1,5]$, $n=100$ (first row) and $n=250$ (second row) when the elements of $\mathbf{X}_n$ are drawn from the normal distribution (first column) and scale $t$-distribution (second column).
  • ...and 4 more figures

Theorems & Definitions (64)

  • Remark 2.1: Discussion on Assumption (A3)
  • Proposition 2.1
  • Theorem 2.1
  • Corollary 2.1
  • Corollary 2.2
  • Corollary 2.3
  • Theorem 2.2
  • Corollary 2.4
  • Corollary 2.5
  • Corollary 2.6
  • ...and 54 more