These are internal functions to calculate the coefficients in polynomial expansion of joint generating functions for two or three quadratic forms in potentially noncentral multivariate normal variables, \(\mathbf{x} \sim N_n(\bm{\mu}, \mathbf{I}_n)\). They are primarily used to calculate moments of a product of two or three quadratic forms.
dtil2_pq_m(A1, A2, mu = rep.int(0, n), p = 1L, q = 1L)dtil2_1q_m(A1, A2, mu = rep.int(0, n), q = 1L)
dtil2_pq_v(L1, L2, mu = rep.int(0, n), p = 1L, q = 1L)
dtil2_1q_v(L1, L2, mu = rep.int(0, n), q = 1L)
dtil3_pqr_m(A1, A2, A3, mu = rep.int(0, n), p = 1L, q = 1L, r = 1L)
dtil3_pqr_v(L1, L2, L3, mu = rep.int(0, n), p = 1L, q = 1L, r = 1L)
A (p + 1) * (q + 1) matrix for the 2D functions,
or a (p + 1) * (q + 1) * (r + 1) array for the 3D functions.
The 1st, 2nd, and 3rd dimensions correspond to increasing orders for
\(\mathbf{A}_1\), \(\mathbf{A}_2\), and
\(\mathbf{A}_3\), respectively. And the 1st row/column of each
dimension corresponds to the 0th order (hence [p + 1, q + 1] for
the \((p,q)\)-th moment).
Argument matrices. Assumed to be symmetric and of the same order.
Mean vector \(\bm{\mu}\) for \(\mathbf{x}\)
Integer-alikes to specify the order along the three argument matrices
Eigenvalues of the argument matrices
dtil2_pq_m() and dtil2_pq_v() calculate
\(\tilde{d}_{p,q}(\mathbf{A}_1, \mathbf{A}_2)\)
in Hillier et al. (2014). dtil2_1q_m() and dtil2_1q_v() are
fast versions for the commonly used case where \(p = 1\). Similarly,
dtil3_pqr_m() and dtil3_pqr_v() are for
\(\tilde{d}_{p,q,r}(\mathbf{A}_1, \mathbf{A}_2, \mathbf{A}_3)
\).
Those ending with _m take matrices as arguments, whereas
those with _v take eigenvalues. The latter can be used only when
the argument matrices share the same eigenvectors, to which the eigenvalues
correspond in the orders given, but is substantially faster.
These functions calculate the coefficients based on the super-short recursion algorithm described in Hillier et al. (2014: sec. 4.2).
Hillier, G., Kan, R. and Wang, X. (2014) Generating functions and short recursions, with applications to the moments of quadratic forms in noncentral normal vectors. Econometric Theory, 30, 436--473. tools:::Rd_expr_doi("10.1017/S0266466613000364").
qfpm is a front-end functions that utilizes these functions
d1_i for a single-matrix equivalent of \(\tilde{d}\)