Countless test statistics can be written as quadratic forms in certain random vectors, or ratios thereof. Consequently, their distribution has received considerable attention in the literature. Except for a few special cases, no closed-form expression for the cdf exists, and one resorts to numerical methods. Traditionally the problem is analyzed under the assumption of joint Gaussianity; the algorithm that is usually employed is that of Imhof (1961). The present manuscript generalizes this result to the case of multivariate generalized hyperbolic random vectors. This flexible distribution nests, among others, the multivariate t, Laplace, and variance gamma distributions. An expression for the first partial moment is also obtained, which plays a vital role in financial risk management. The proof involves a generalization of the classic inversion formula due to Gil-Pelaez (1951). Two numerical applications are considered: first, the finite-sample distribution of the two stage least squares estimator of a structural parameter. Second, the Value at Risk and expected shortfall of a quadratic portfolio with heavy-tailed risk factors. An empirical application is examined, in which a portfolio of Dow Jones Industrial Index stock options is optimized with respect to its expected shortfall. The results demonstrate the benefits of the analytical expression.

中文翻译：

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多元广义双曲随机向量的二次型

无数的测试统计量可以写成某些随机向量或其比率的二次形式。因此，它们的分布在文献中受到了相当大的关注。除少数特殊情况外，不存在 cdf 的闭式表达式，只能采用数值方法。传统上是在联合高斯假设下分析问题；通常采用的算法是 Imhof (1961) 的算法。本手稿将此结果推广到多元广义双曲随机向量的情况。这种灵活的分布嵌套了多元 t、拉普拉斯和方差伽玛分布等。还获得了第一部分矩的表达式，它在金融风险管理中起着至关重要的作用。该证明涉及对 Gil-Pelaez (1951) 提出的经典反演公式的推广。考虑了两个数值应用：首先，结构参数的两阶段最小二乘估计量的有限样本分布。其次，具有重尾风险因子的二次投资组合的风险价值和预期缺口。考察了一个实证应用，其中道琼斯工业指数股票期权的投资组合针对其预期短缺进行了优化。结果证明了分析表达式的好处。考察了一个实证应用，其中道琼斯工业指数股票期权的投资组合针对其预期短缺进行了优化。结果证明了分析表达式的好处。考察了一个实证应用，其中道琼斯工业指数股票期权的投资组合针对其预期短缺进行了优化。结果证明了分析表达式的好处。