Exact mean and covariance formulas after diagonal transformations of a multivariate normal

2024/06/28 by Rebecca Morrison, Morrison, Rebecca, Estelle Basor +1 · 1 citation
Mathematics · #Advanced Statistical Methods and Models #FOS: Mathematics #Statistical and numerical algorithms #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.2407.00240

openalex publication_date 2024/06/28 · openalex created_date 2024/07/03 · openalex updated_date 2026/07/28

Abstract

Consider \boldsymbol X ∼ N(\boldsymbol 0, \boldsymbol Σ) and \boldsymbol Y = (f₁(X₁), f₂(X₂),…, f_d(X_d)). We call this a diagonal transformation of a multivariate normal. In this paper we compute exactly the mean vector and covariance matrix of the random vector \boldsymbol Y. This is done two different ways: One approach uses a series expansion for the function f_i and the other a transform method. We compute several examples, show how the covariance entries can be estimated, and compare the theoretical results with numerical ones.

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