Analytic, computational, and approximate forms for ratios of noncentral and central Gaussian quadratic forms
Analytic, computational, and approximate forms for ratios of noncentral and central Gaussian quadratic forms
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DOI：
10.1198/106186006x112954
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发表时间：
2006-06-01
影响因子：
2.4
通讯作者：
Taylor, DJ
中科院分区：
文献类型：
--
作者：
Kim, HY;Gribbin, MJ;Taylor, DJ
许多有用的统计量等于高斯变量中可能非中心卡方与二次型的比率，所有权重都为正。将密度和分布函数表示为相应F函数的正加权和具有许多优点。当嵌入到更复杂的问题中时，混合形式具有分析价值。混合形式也具有计算价值。这些展开式适用于分量少、自由度小的二次型。早期文献中的更一般的算法可能需要更长的时间或无法在相同的设置中收敛。对于这个问题，已经提出了许多近似方法。一个正加权的非中心二次型总是可以有两个矩与一个非中心卡方匹配。对于一个单一的二次型，非中心形式的表现既不一致，或多或少比旧的近似准确。该方法还给出了一个非中心F近似的任何比率的一个正加权的非中心形式的一个正加权的中心二次型。该方法提供了更好的精度比基于一个单一的卡方近似的非中心比率。精度足以满足许多实际应用，如功率分析，即使是很少的自由度。自然地，近似证明比任何精确方法更快和更简单地计算。在解析表达式中嵌入近似提供了简单的形式，正确地保证只有正值具有非零概率，并且当任一二次形式只有一项时，也自动减少到部分或完全精确的结果。
Many useful statistics equal the ratio of a possibly noncentral chi-square to a quadratic form in Gaussian variables with all positive weights. Expressing the density and distribution function as positively weighted sums of corresponding F functions has many advantages. The mixture forms have analytic value when embedded within a more complex problem. The mixture forms also have computational value. The expansions work well with quadratic forms having few components and small degrees of freedom. A more general algorithm from earlier literature can take longer or fail to converge in the same setting. Many approximations have been suggested for the problem. A positively weighted noncentral quadratic form can always have two moments matched to a noncentral chi-square. For a single quadratic form, the noncentral form performs neither uniformly more or less accurately than older approximations. The approach also gives a noncentral F approximation for any ratio of a positively weighted noncentral form to a positively weighted central quadratic form. The method provides better accuracy for noncentral ratios than approximations based on a single chi-square. The accuracy suffices for many practical applications, such as power analysis, even with few degrees of freedom. Naturally the approximation proves much faster and simpler to compute than any exact method. Embedding the approximation in analytic expressions provides simple forms which correctly guarantee only positive values have nonzero probabilities, and also automatically reduce to partially or fully exact results when either quadratic form has only one term.