GENERATING FUNCTIONS AND SHORT RECURSIONS, WITH APPLICATIONS TO THE MOMENTS OF QUADRATIC FORMS IN NONCENTRAL NORMAL VECTORS
生成函数与短递推：非中心正态向量二次型矩的应用AI 解读 ↓
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生成函数与短递推：非中心正态向量二次型矩的应用
- 研究问题
- 能否将依赖全部低阶项的递推改写为固定长度递推，从而提高非中心正态向量二次型矩及相关期望的计算效率？
- 方法与思路
- 作者通过生成函数关系推广短递推构造，并利用广义 Horner 方法进一步缩短递推，应用于矩、乘积矩和幂比期望。
- 材料陈述的贡献
- 摘要称短递推适用于比既有结果更广的对象和生成函数，并在相关二次型问题中得到计算效率更高的超短递推。
- 阅读关注点
- 应核对允许短递推的生成函数结构、递推长度与计算量的关系，以及二次型幂比期望的存在条件和数值稳定性。
- 阅读边界
- 仅依据所给 Crossref 摘要版本，未核验刊本变化；摘要未给出公式、复杂度界或运行数值，不能独立确认效率提升幅度。
原文摘要
Recursive relations for objects of statistical interest have long been important for computation, and they remain so even with hugely improved computing power. Such recursions are frequently derived by exploiting relations between generating functions. For example, the top-order zonal polynomials that occur in much distribution theory under normality can be recursively related to other (easily computed) symmetric functions (power-sum and elementary symmetric functions; Ruben, 1962, Annals of Mathematical Statistics 33, 542–570; Hillier, Kan, and Wang, 2009, Econometric Theory 25, 211–242). Typically, in a recursion of this type the k th object of interest, d k , say, is expressed in terms of all lower order d j ’s. In Hillier et al. (2009) we pointed out that, in the case of top-order zonal polynomials and other invariant polynomials of multiple matrix argument, a fixed length recursion can be deduced. We refer to this as a short recursion. The present paper shows that the main results in Hillier et al. (2009) can be generalized and that short recursions can be obtained for a much larger class of objects/generating functions. As applications, we show that short recursions can be obtained for various problems involving quadratic forms in noncentral normal vectors, including moments, product moments, and expectations of ratios of powers of quadratic forms. For this class of problems, we also show that the length of the recursion can be further reduced by an application of a generalization of Horner’s method (cf. Brown, 1986, SIAM Journal on Scientific and Statistical Computing 7, 689–695), producing a super-short recursion that is significantly more efficient than even the short recursion.