Copulas, Outliers, and Rogue States of Nonelliptic Fields and Energy in Electromagnetic Reverberation

Excess Power, Energy, and Intensity of Stochastic Fields in Quasi-Static and Dynamic Environments

“…4 shows the numerically computed PDF (21) for selected u th , indicating a transition from a χ 2 2 PDF to a χ 2 3 PDF when u th decreases from +∞ to 0. Both these limit PDFs are easily retrieved analytically from (21). Note that the PDF of H+ |0+ ≡ Ũ + |0+ differs from the PDF of the unconditioned U .…”

“…The approximation in (29) reflects the departure from ideal full statistical dependence between L + and H + in writing A + 2(H + ) 3/2 /3. When u th → +∞, the actual dependence can be represented by a copula density linking f H + (h + ) and f L + ( + ) [21]. The departure of nonasymptotic f L + ( + ) from a χ 2 PDF can be evaluated numerically using a high-order Rice series for multiple crossings of u th [22, sec.…”

Statistical Anisotropy in Imperfect Electromagnetic Reverberation

“…Near or below the LUF [1], this assumption may be unsustainable. In that case, a characterization based on sampling circular, elliptic [17], or even more general [18] joint pdfs for E i and E j may be appropriate. A complete generalization involves additional parameters of the distribution, each requiring individual estimation (e.g., using Bayesian inference), and the characterization of the dependence structure between E i and E j , e.g., using copulas [18].…”

“…In that case, a characterization based on sampling circular, elliptic [17], or even more general [18] joint pdfs for E i and E j may be appropriate. A complete generalization involves additional parameters of the distribution, each requiring individual estimation (e.g., using Bayesian inference), and the characterization of the dependence structure between E i and E j , e.g., using copulas [18]. Here, we simply show how small sample sizes affect the cdf of A ij and A by using sampling distributions 6 of |E i | 2 for circular Gaussian E i [19] and manifested by a finite number of degrees of freedom N i for E i .…”