Package: qfratio 1.1.1.9000

Symbols used | Target | First examples | Mathematical details | Moment existence conditions | Expressions for moments | In these expressions, $\beta_{\cdot}$ are arbitrary scaling constants thatsatisfy $0 < \beta < 2 / \qfrlmax$, with $\qfrlmax$ beingthe largest eigenvalue of the argument matrix.In addition, $h_{k_1, \dots k_s}$ and $\tilde{h}_{k_1; k_2 \dots k_s}$ arethe coefficients of $t_1^{k_1} \dots t_s^{k_s}$ in the following power seriesexpansions:\begin{multline}\det{ \mathbf{I}_n - t_1 \mathbf{A}_1 - \dots - t_s \mathbf{A}_s }^{-\frac{1}{2}}\\cdot\exp \left( \frac{(1 - t_1 - \dots - t_s) \boldsymbol{\mu}^T\left( \mathbf{I}_n - t_1 \mathbf{A}_1 - \dots - t_s \mathbf{A}_s \right)^{-1}\boldsymbol{\mu}- \boldsymbol{\mu}^T \boldsymbol{\mu}}{2} \right)\ | \sum_{k_1 = 0}^{\infty} \dots \sum_{k_s = 0}^{\infty} \qfrhijk[k_1, \dots, k_s]{ \mathbf{A}_1 }{\dots}{ \mathbf{A}_s } t_1^{k_1} \dots t_s^{k_s}, \label{gfun_hij} \\end{multline}\begin{multline}\det{ \mathbf{I}_n - t_1 \mathbf{A}_1 - \dots - t_s \mathbf{A}_s }^{-\frac{1}{2}}\ \cdot\exp \left( \frac{(1 - t_2 - \dots - t_s) \boldsymbol{\mu}^T\left( \mathbf{I}_n - t_1 \mathbf{A}_1 - \dots - t_s \mathbf{A}_s \right)^{-1}\boldsymbol{\mu}- \boldsymbol{\mu}^T \boldsymbol{\mu}}{2} \right)\ | A common special case is when $\boldsymbol | Special cases | A notable special case for moments of simple ratios is when $p$ is a positiveinteger and $\mathbf{B} = \mathbf{I}_n$.In this case, (2) simplifies into [@HillierEtAl2014, theorem 4]\begin{equation}\qfrE \left( \frac{(\mathbf{x}^T \mathbf{A} \mathbf{x})^p}{(\mathbf{x}^T \mathbf{x})^q} \right) | 2^{p - q} p!\sum_{k=0}^{p}\frac{ \qfrGmf{ \frac{n}{2} + p - q + k } }{ 2^k k! \qfrGmf{ \frac{n}{2} + p + k } }{}1 F_1 \left( q ; \frac{n}{2} + p + k ; - \frac{\boldsymbol{\mu}^T \boldsymbol{\mu}}{2} \right) a{p, k} \left( \mathbf{A}, \boldsymbol{\mu} \right),\tag{5}\end{equation}where ${}1 F_1 \left( \cdot ; \cdot ; \cdot \right)$ is the confluenthypergeometric function, and $a{p, k}$ are the coefficients of $t^p$ in\begin{align}&\det{ \mathbf{I}_n - t \mathbf{A} }^{-\frac{1}{2}}\left(\boldsymbol{\mu}^T \left( \mathbf{I}_n - t \mathbf{A} \right)^{-1} \boldsymbol{\mu}- \boldsymbol{\mu}^T \boldsymbol{\mu}\right) ^ k | Numerical evaluation | Truncation error | A great advantage in the above expressions is that an error bound is availablefor a partial (truncated) sum for the simple ratio, provided that $p$ is apositive integer and $\mathbf{B}$ is positive definite.By denoting the expression (2) as $M \left( \mathbf{A}, \mathbf{B}, p, q \right)$and the partial sum of the same up to $j = m$ as$\hat{M}m \left( \mathbf{A}, \mathbf{B}, p, q \right)$, the error bound is[@HillierEtAl2014, theorem 7]\begin{multline}\lvert M \left( \mathbf{A}, \mathbf{B}, p, q \right) - \hat{M}m \left( \mathbf{A}, \mathbf{B}, p, q \right) \rvert \\leq\frac{ 2^{p - q} \beta{\mathbf{B}}^{q} p! \qfrGmf{ \frac{n}{2} + p - q } \qfrrf[m+1]{q} }{ \qfrGmf{ \frac{n}{2} + p + m + 1 } }\left[\frac{\exp \left( \frac{ \bar{\boldsymbol{\mu}}^T \bar{\boldsymbol{\mu}} - \boldsymbol{\mu}^T \boldsymbol{\mu} }{2} \right)\qfrdtk[p]{ \bar{A} }{ \bar{\boldsymbol{\mu}} }}{\det{\beta{\mathbf{B}} \mathbf{B}}^{\frac{1}{2}}}- \sum_{j=0}^{m}\qfrhhij[p;j]{ \mathbf{A}^+ }{ \mathbf{I}n - \beta{\mathbf{B}} \mathbf{B} }\right],\end{multline}where $\mathbf{A}^+$ is a symmetric matrix constructed from the eigenvectors and"positivized" eigenvalues of $\mathbf{A}$ (above),$\bar{\boldsymbol{\mu}} = \sqrt{2}\left( \beta_\mathbf{B} \mathbf{B} \right)^{-\frac{1}{2}} \boldsymbol{\mu}$,$\bar{\mathbf{A}} = \beta_{\mathbf{B}}^{-1} \mathbf{B}^{-\frac{1}{2}} \mathbf{A}^+ \mathbf{B}^{-\frac{1}{2}}$,and $\tilde{d}$ and $\hat{h}$ are coefficients in the following generating functions:\begin{equation}\det{ \mathbf{I}_n - t \mathbf{A} }^{-\frac{1}{2}}\exp \left( \frac{\boldsymbol{\mu}^T \left( \mathbf{I}_n - t \mathbf{A} \right)^{-1} \boldsymbol{\mu}- \boldsymbol{\mu}^T \boldsymbol{\mu}}{2} \right) | \sum_{k = 0}^{\infty} \qfrdtk[k]{ \mathbf{A} }{ \boldsymbol{\mu} } t^{k},\end{equation}\begin{multline}\det{ \mathbf{I}_n - t_1 \mathbf{A}_1 - \dots - t_s \mathbf{A}_s }^{-\frac{1}{2}}\\quad \cdot\exp \left( \frac{(1 + t_2 + \dots + t_s) \boldsymbol{\mu}^T\left( \mathbf{I}_n - t_1 \mathbf{A}_1 - \dots - t_s \mathbf{A}_s \right)^{-1}\boldsymbol{\mu}- \boldsymbol{\mu}^T \boldsymbol{\mu}}{2} \right) \ | Recursions | Functions | qfrm(): quadratic form ratio moment | Usage | Inside the function | qfmrm(): quadratic form multiple ratio moment | qfm_Ap_int(), qfpm_ABpq_int(), qfpm_ABDpqr_int(): quadratic form (product) moment | rqfr(), rqfmr(), rqfp(): random number generation | d1_i(), d2_*j_*(), d3_*jk_*(), etc.: recursion algorithms | Using the functionality | Recommended workflow | Checking for numerical convergence | Computational cost | References

Symbols used | Theory | Preliminaries | Series expression | Distribution function | Density function | Numerical inversion | Saddlepoint approximation | Let $M_{X_q}(s)$ be the moment generating function of $X_q$,\begin{equation}M_{X_q}(s) | A first-order saddlepoint approximation formula for the distribution function$F_Q$ is [@ButlerPaolella2007; @ButlerPaolella2008]:\begin{equation}\widehat{\Pr{}}_1 (Q < q) | A more accurate second-order approximation is [@ButlerPaolella2007]\begin{equation}\widehat{\Pr{}}_2 (Q < q) | A first-order saddlepoint approximation for the density function $f_Q$ is[@ButlerPaolella2007; @ButlerPaolella2008]\begin{equation}\hat{f_1}(q) | A second-order approximation is [@ButlerPaolella2007]\begin{equation}\hat{f_2}(q) | \hat{f_1}(q) (1 + O) ,\end{equation}where\begin{equation}O | Implementation details | Exported functions | Choosing a method | Use with ks.test() | Series expressions | Specifying integration error | autoscale_args | trim_values | Options | Error bound | pqfr() and dqfr() | qqfr() | Distribution of powers | When $\mathbf{A}$ is nonnegative definite or $p$ is an odd integer | When $\mathbf{A}$ is indefinite and $p$ is an even integer | When $\mathbf{A}$ is indefinite and $p$ is non-integer | Graphical examples | References