Deterministic evaluation and holonomic structure for moderate sample sizes

Determine, for each fixed sample size n, the order of a linear differential equation with polynomial coefficients satisfied by the Jarque–Bera density f_{JB_n}(x), and determine whether this equation is ever of hypergeometric (first-order recurrence) type, thereby enabling deterministic evaluation of the distribution function by the holonomic gradient method.

Background

The paper represents the finite-sample Jarque–Bera distribution through period integrals of algebraic families. This structure suggests that f_{JB_n} may be holonomic, meaning that it satisfies a linear differential equation in the statistic x with polynomial coefficients. If such equations were available, the corresponding moments would typically satisfy polynomial-coefficient recurrences in the moment order.

The authors propose determining the differential-equation order as a function of n and deciding whether a hypergeometric reduction occurs. These results could support deterministic computation of the Jarque–Bera distribution function through the holonomic gradient method, replacing simulation for moderate sample sizes and improving finite-sample calibration.