d3_ijk: Coefficients in polynomial expansion of generating function---for ratios with three matrices

Description

These are internal functions to calculate the coefficients in polynomial expansion of joint generating functions for three quadratic forms in potentially noncentral multivariate normal variables, \(\mathbf{x} \sim N_n(\bm{\mu}, \mathbf{I}_n)\). They are primarily used in calculations around moments of a ratio involving three quadratic forms.

Usage

d3_ijk_m(
  A1,
  A2,
  A3,
  m = 100L,
  p = m,
  q = m,
  r = m,
  thr_margin = 100,
  fill_across = c(!missing(p), !missing(q), !missing(r))
)
d3_ijk_v(
  L1,
  L2,
  L3,
  m = 100L,
  p = m,
  q = m,
  r = m,
  thr_margin = 100,
  fill_across = c(!missing(p), !missing(q), !missing(r))
)
d3_pjk_m(A1, A2, A3, m = 100L, p = 1L, thr_margin = 100)
d3_pjk_v(L1, L2, L3, m = 100L, p = 1L, thr_margin = 100)
h3_ijk_m(
  A1,
  A2,
  A3,
  mu = rep.int(0, n),
  m = 100L,
  p = m,
  q = m,
  r = m,
  thr_margin = 100,
  fill_across = c(!missing(p), !missing(q), !missing(r))
)
h3_ijk_v(
  L1,
  L2,
  L3,
  mu = rep.int(0, n),
  m = 100L,
  p = m,
  q = m,
  r = m,
  thr_margin = 100,
  fill_across = c(!missing(p), !missing(q), !missing(r))
)
htil3_pjk_m(A1, A2, A3, mu = rep.int(0, n), m = 100L, p = 1L, thr_margin = 100)
htil3_pjk_v(L1, L2, L3, mu = rep.int(0, n), m = 100L, p = 1L, thr_margin = 100)
hhat3_pjk_m(A1, A2, A3, mu = rep.int(0, n), m = 100L, p = 1L, thr_margin = 100)
hhat3_pjk_v(L1, L2, L3, mu = rep.int(0, n), m = 100L, p = 1L, thr_margin = 100)

Value

(p + 1) * (m + 1) * (m + 1) array for the *_pjk_* functions

(m + 1) * (m + 1) * (m + 1) array for the *_ijk_* functions (by default; see “Details”).

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, r + 1] for the \((p,q,r)\)-th order).

Has the attribute "logscale" as described in the “Scaling”

section in d1_i. This is an array of the same size as the return itself.

Arguments

A1, A2, A3: Argument matrices. Assumed to be symmetric and of the same order.
m: Integer-alike to specify the desired order along A2/L2 and A3/L3
p, q, r: Integer-alikes to specify the desired orders along A1/L1, A2/L2, and A3/L3, respectively.
thr_margin: Optional argument to adjust the threshold for scaling (see “Scaling” in d1_i)
fill_across: Logical vector of length 3, to specify whether each dimension of the output matrix should be filled.
L1, L2, L3: Eigenvalues of the argument matrices
mu: Mean vector \(\bm{\mu}\) for \(\mathbf{x}\)

Details

All these functions have equivalents for two-matrix cases (d2_ij), to which the user is referred for documentations. The primary difference of these functions from the latter is the addition of arguments for the third matrix A3/L3.

d3_*jk_*() functions calculate \(d_{i,j,k}(\mathbf{A}_1, \mathbf{A}_2, \mathbf{A}_3) \) in Hillier et al. (2009, 2014) and Bao and Kan (2013). These are also related to the top-order invariant polynomials as described in d2_ij.

h3_ijk_*(), htil3_pjk_*(), and hhat3_pjk_*() functions calculate \(h_{i,j,k}(\mathbf{A}_1, \mathbf{A}_2, \mathbf{A}_3) \), \(\tilde{h}_{i;j,k}(\mathbf{A}_1; \mathbf{A}_2, \mathbf{A}_3) \), and \(\hat{h}_{i;j,k}(\mathbf{A}_1; \mathbf{A}_2, \mathbf{A}_3) \), respectively, as described in the package vignette. These are equivalent to similar coefficients described in Bao and Kan (2013) and Hillier et al. (2014).

The difference between the *_pjk_* and *_ijk_* functions is as described for *_pj_* and *_ij_* (see “Details” in d2_ij). The only difference is that these functions return a 3D array. In the *_pjk_* functions, all the slices along the first dimension (i.e., [i, , ]) are an upper-left triangular matrix like what the *_ij_* functions return in the 2D case; in other words, the return has the coefficients for the terms that satisfy \(j + k \le m\) for all \(i = 0, 1, \dots, p\). Typically, the [p + 1, , ]-th slice is used for subsequent calculations. In the return of the *_ijk_* functions, only the triangular prism close to the [1, 1, 1] is filled with coefficients, which correspond to the terms satisfying \(i + j + k \le m\).

References

Bao, Y. and Kan, R. (2013) On the moments of ratios of quadratic forms in normal random variables. Journal of Multivariate Analysis, 117, 229--245. tools:::Rd_expr_doi("10.1016/j.jmva.2013.03.002").