dtil2_pq {qfratio} | R Documentation |
Coefficients in polynomial expansion of generating function—for products
Description
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.
Usage
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)
Arguments
A1 , A2 , A3 |
Argument matrices. Assumed to be symmetric and of the same order. |
mu |
Mean vector |
p , q , r |
Integer-alikes to specify the order along the three argument matrices |
L1 , L2 , L3 |
Eigenvalues of the argument matrices |
Details
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).
Value
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).
References
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. doi:10.1017/S0266466613000364.
See Also
qfpm
is a front-end functions that utilizes these functions
d1_i
for a single-matrix equivalent of \tilde{d}