Upper bound for singular normal probability integrals by integration over a hypersphere

Abstract

Following an inequality of G. Pólya[1949], certain probability integrals, namely of the limiting singular normal distribution of a specified multinomial, are shown to be bounded above by integrals T_k(U_k) of the uncorrelated standard multivariate normal density over k-dimensional hyperspheres *U_k of radius U_k=U_k(y) and center (0,…,0). The hypersphere U_k is so chosen that its k-dimensional volume is equal to the k-dimensional volume of a simplex R_k, the image under a diagonalizing transformation, of the simplicial region of integration f_k in Φ_k(y,R). For a symmetric multinomial, R_k turns out to be a regular simplex and explicit formulas for the upper bound in the equicorrelated-equicoordinate case, with common correlation ρ= -1/k,- are derived in terms of (a) normal density and distribution functions and (b) incomplete gamma-function ratio.

Keywords:

• Pólya
• multinomial
• limiting singular normal
• regular simplex
• hypersphere
• equicorrelated-equicoordinate normal probability integral
• diagonalizing transformation
• uncorrelated standard multivariate normal distribution
• incomplete gamma-function ratio