R: Coefficients in polynomial expansion of generating...

d2_ij {qfratio}

R Documentation

Coefficients in polynomial expansion of generating function—for ratios with two matrices

Description

These are internal functions to calculate the coefficients in polynomial expansion of joint generating functions for two 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 two or three quadratic forms.

Usage

d2_ij_m(
  A1,
  A2,
  m = 100L,
  p = m,
  q = m,
  thr_margin = 100,
  fill_all = !missing(p) || !missing(q)
)

d2_ij_v(
  L1,
  L2,
  m = 100L,
  p = m,
  q = m,
  thr_margin = 100,
  fill_all = !missing(p) || !missing(q)
)

d2_pj_m(A1, A2, m = 100L, p = 1L, thr_margin = 100)

d2_1j_m(A1, A2, m = 100L, thr_margin = 100)

d2_pj_v(L1, L2, m = 100L, p = 1L, thr_margin = 100)

d2_1j_v(L1, L2, m = 100L, thr_margin = 100)

h2_ij_m(
  A1,
  A2,
  mu = rep.int(0, n),
  m = 100L,
  p = m,
  q = m,
  thr_margin = 100,
  fill_all = !missing(p) || !missing(q)
)

h2_ij_v(
  L1,
  L2,
  mu = rep.int(0, n),
  m = 100L,
  p = m,
  q = m,
  thr_margin = 100,
  fill_all = !missing(p) || !missing(q)
)

htil2_pj_m(A1, A2, mu = rep.int(0, n), m = 100L, p = 1L, thr_margin = 100)

htil2_1j_m(A1, A2, mu = rep.int(0, n), m = 100L, thr_margin = 100)

htil2_pj_v(L1, L2, mu = rep.int(0, n), m = 100L, p = 1L, thr_margin = 100)

htil2_1j_v(L1, L2, mu = rep.int(0, n), m = 100L, thr_margin = 100)

hhat2_pj_m(A1, A2, mu = rep.int(0, n), m = 100L, p = 1L, thr_margin = 100)

hhat2_1j_m(A1, A2, mu = rep.int(0, n), m = 100L, thr_margin = 100)

hhat2_pj_v(L1, L2, mu = rep.int(0, n), m = 100L, p = 1L, thr_margin = 100)

hhat2_1j_v(L1, L2, mu = rep.int(0, n), m = 100L, thr_margin = 100)

Arguments

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

Details

d2_**_*() functions calculate d_{i,j}(\mathbf{A}_1, \mathbf{A}_2) in Hillier et al. (2009, 2014) and Bao and Kan (2013). These are also related to the top-order invariant polynomials C_{[k_1],[k_2]}(\mathbf{A}_1, \mathbf{A}_2) in the following way: d_{i,j}(\mathbf{A}_1, \mathbf{A}_2) = \frac{1}{k_1! k_2!} \left( \frac{1}{2} \right)_{k_1 + k_2} C_{[k_1],[k_2]}(\mathbf{A}_1, \mathbf{A}_2), where (x)_k = x (x + 1) \dots (x + k - 1) (Chikuse 1987; Hillier et al. 2009).

h2_ij_*() and htil2_pj_*() functions calculate h_{i,j}(\mathbf{A}_1, \mathbf{A}_2) and \tilde{h}_{i,j}(\mathbf{A}_1; \mathbf{A}_2), respectively, in Bao and Kan (2013). Note that the latter is denoted by the symbol h_{i,j} in Hillier et al. (2014). hhat2_pj_*() functions are for \hat{h}_{i,j}(\mathbf{A}_1; \mathbf{A}_2) in Hillier et al. (2014), used to calculate an error bound for truncated sum for moments of a ratio of quadratic forms. The mean vector \bm{\mu} is a parameter in all these.

There are two different situations in which these coefficients are used in calculation of moments of ratios of quadratic forms: 1) within an infinite series for one of the subscripts, with the other subscript fixed (when the exponent p of the numerator is integer); 2) within a double infinite series for both subscripts (when p is non-integer) (see Bao and Kan 2013). In this package, the situation 1 is handled by the *_pj_* (and *_1j_*) functions, and 2 is by the *_ij_* functions.

In particular, the *_pj_* functions always return a (p + 1) * (m + 1) matrix where all elements are filled with the relevant coefficients (e.g., d_{i,j}, \tilde{h}_{i,j}), from which, typically, the [p + 1, ]-th row is used for subsequent calculations. (Those with *_1q_* are simply fast versions for the commonly used case where p = 1.) On the other hand, the *_ij_* functions by default return a (m + 1) * (m + 1) matrix whose upper-left triangular part (including the diagonals) is filled with the coefficients (d_{i,j} or h_{i,j}), the rest being 0, and all the coefficients are used in subsequent calculations.

(At present, the *_ij_* functions also have the functionality to fill all coefficients of a potentially non-square output matrix, but this is less efficient than *_pj_* functions so may be omitted in the future development.)

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.

This package also involves C++ equivalents for most of these functions (which are suffixed by E for Eigen), but these are exclusively for internal use and not exposed to the user.

Value

(p + 1) * (m + 1) matrix for the *_pj_* functions.

(m + 1) * (m + 1) matrix for the *_ij_* functions.

The rows and columns correspond to increasing orders for \mathbf{A}_1 and \mathbf{A}_2, respectively. And the 1st row/column of each dimension corresponds to the 0th order (hence [p + 1, q + 1] for the (p,q)-th order).

Has the attribute "logscale" as described in the “Scaling” section in d1_i. This is a matrix of the same size as the return itself.

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. doi:10.1016/j.jmva.2013.03.002.

Chikuse, Y. (1987) Methods for constructing top order invariant polynomials. Econometric Theory, 3, 195–207. doi:10.1017/S026646660001029X.

Hillier, G., Kan, R. and Wang, X. (2009) Computationally efficient recursions for top-order invariant polynomials with applications. Econometric Theory, 25, 211–242. doi:10.1017/S0266466608090075.

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.