Pearson's Correlation Matrix--Fully Analytical Inference Under the (Gaussian) Identity Matrix

p-value for the entire matrix

These results are obtainable under fully generalized data conditions and correlation values via

CDF matrix

the NAbC algorithm from Opdyke, JD, (2022), "Beating the Correlation Breakdown, for Pearson's

Inverse CDF ('quantile') Correlation Matrix

and Beyond: Robust Inference and Flexible Scenarios and Stress Testing for Financial Portfolios"

Upper/Lower 95% Correlation Matrices (i.e. Simultaneous Confidence Interval matrices,

(on SSRN, ResearchGate, www.DataMineit.com)

as well as Individual Cell Confidence Intervals)

This is a fully analytic solution, requiring no sampling ('rejection' methods or otherwise). The null hypothesis is the Gaussian identity matrix. Green cells require valid user input.

For 1. & 2.:

Insert a (positive definite) correlation matrix in the green cells of the top matrix (rows 17-20) to obtain the unique corresponding CDF matrix in the orange cells (as well as the 2-sided p-value of the matrix).

For 1. & 3.:

Conversely, insert a matrix of CDF values in the green cells of the next matrix (rows 27-30) to obtain the unique corresponding correlation matrix in the orange cells (as well as the 2-sided p-value of the matrix).

For 1. & 4.:

Insert the desired α (cell E36) to obtain, via Simultaneous Confidence Intervals, the Upper/Lower (1-α)% C.I. Correlation Matrices in the orange cells in rows 36-48 (as well as the 1-sided p-value each of the two matrices).

(user-specified input)

(user-specified input)

(user-specified input)

for CDF values near 0 or 1

N sample size =

estimated/observed Correlation Matrix

Cholesky factorization

Spherical Angles

corresponding CDF matrix

(recommended value=1.00E-307)

"'distance" = LNP =

LNP = ln( product of all 2-sided p-values )

1. & 2.:

matrix (2-sided) p-value =

= SUM [ ln(each p-value) ]

Null = Gaussian identity matrix

(user-specified input)

CDF matrix

Spherical Angles

Cholesky factorization

corresponding ('quantile') Correlation Matrix

"'distance" = LNP =

LNP = ln( product of all 2-sided p-values )

matrix (2-sided) p-value =

= SUM [ ln(each p-value) ]

1. & 3.:

SIMULTANEOUS CONFIDENCE INTERVALS (lower CDF values are associated with higher Correlation values, and vice versa)

(Upper) CDF matrix (α/2)

Spherical Angles

Cholesky factorization

UPPER (1-α/2)% Confidence Interval Correlation Matrix

(user-specified input)

"'distance" = LNP =

1. & 4.:

LNP = ln( product of all 1-sided p-values )

CI: 1-α

matrix (1-sided) p-value =

= SUM [ ln(each p-value) ]

('back into' individual rather

than simultaneous CI's; for

example, α=0.44734 gives 95%

CI's for all cells individually

(Lower) CDF matrix (1-α/2)%

Spherical Angles

Cholesky factorization

LOWER (α/2)% Confidence Interval Correlation Matrix

for N=252)

"'distance" = LNP =

1. & 4.:

LNP = ln( product of all 1-sided p-values )

matrix (1-sided) p-value =

= SUM [ ln(each p-value) ]