Approximation of sign-regular kernels

• Published in: Journal of econometrics 2022-01, Vol.226 (1), p.171-191
• Main Author: Knox, Thomas A.
• Format: Article
• Language: English
• Subjects: Approximation; class; Density; domain; econometrics; Error analysis; Integrals; mathematical theory; Normal distribution; Probability; probability distribution
• ISSN: 0304-4076; 1872-6895
• DOI: 10.1016/j.jeconom.2021.07.009

Parameterized integrals, qy=∫Kx,ydPx, are common in economic applications. To optimize a parameterized integral, or to relate one such integral to others, it is helpful to reduce the rank of the kernel K while controlling approximation error. A bound on the uniform approximation error of Kx,y≈∑i=1nfixgiy also bounds the uniform approximation error of qy≈∑i=1ncigiy=∑i=1n∫fixdPxgiy over all y and all signed measures P whose total variation does not exceed a fixed bound (such as probability measures), which may be useful in optimizing q or in relating q to other parameterized integrals. Bounding the mean squared error or the local error in approximating K generally does not bound the uniform approximation error of q. Many economically interesting kernels of parameterized integrals, including expcxy, the Gaussian probability density function, and a wide class of Green’s functions, satisfy a condition known as strict sign-regularity. For Kx,y strictly sign-regular on a rectangular domain x∈xL,xU,y∈yL,yU, I introduce a new method to efficiently compute a lower bound on the uniform error achievable by any rank-n approximation. I also provide a novel method to construct a rank-n approximation that numerically achieves the lower bound in every example I have examined, so in each such example my new method solves, to within rounding error, (1)inffi,gii=1nsupx∈xL,xU,y∈yL,yUKx,y−∑i=1nfixgiy. My approach uses tools from the literature on n-widths in approximation theory (as summarized by Pinkus (1985)). I show that my new method’s uniform error can be orders of magnitude smaller than that of a Taylor series with the same rank. It also outperforms singular function approximations and the Chebfun2 approach of Townsend and Trefethen (2013) in uniform error, typically by wide margins. I describe several applications that demonstrate the practical utility of my approximation method.