Sobolev_coefs function

Transformation between different coefficients in Sobolev statistics

Given a Sobolev statistic [REMOVE_ME]$S_{n, p} = \sum_{i, j = 1}^n \psi(\cos^{-1}({\bf X}_i'{\bf X}_j)),S_{n, p} = \sum_{i, j = 1}^n \psi(\cos^{-1}(X_i'X_j)), [REMOVE_ME_2]$

for a sample c("$X_1, \\ldots, X_n \\in S^{p - 1} :=\n$", "${x \\in R^p : ||x|| = 1}$"), $p\ge 2$, three important sequences are related to $S_{n, p}$.

• Gegenbauer coefficients $\{b_{k, p}\}$ of $\psi_p$ (see, e.g., the projected-ecdf statistics ), given by [REMOVE_ME]$b_{k, p} := \frac{1}{c_{k, p}}\int_0^\pi \psi_p(\theta)C_k^{p / 2 - 1}(\cos\theta)\,\mathrm{d}\theta.b_{k, p} := \frac{1}{c_{k, p}} \int_0^\pi \psi_p(\theta)C_k^(p / 2 - 1)(\cos\theta) d\theta. [REMOVE_ME_2]$
• Weights $\{v_{k, p}^2\}$ of the asymptotic distribution of the Sobolev statistic, $\sum_{k = 1}^\infty v_k^2 \chi^2_{d_{p, k}}$, given by [REMOVE_ME]$v_{k, p}^2 = \left(1 + \frac{2k}{p - 2}\right)^{-1} b_{k, p},\quad p \ge 3.v_{k, p}^2 = (1 + 2k / (p - 2))^{-1} b_{k, p}, p \ge 3. [REMOVE_ME_2]$
• Gegenbauer coefficients $\{u_{k, p}\}$ of the local projected alternative associated to $S_{n, p}$, given by [REMOVE_ME]$u_{k, p} = \left(1 + \frac{2k}{p - 2}\right) v_{k, p},\quad p \ge 3.u_{k, p} = (1 + 2k / (p - 2)) b_{k, p}, p \ge 3. [REMOVE_ME_2]$

For $p = 2$, the factor $(1 + 2k / (p - 2))$ is replaced by $2$.

bk_to_vk2(bk, p, log = FALSE)

bk_to_uk(bk, p, signs = 1)

vk2_to_bk(vk2, p, log = FALSE)

vk2_to_uk(vk2, p, signs = 1)

uk_to_vk2(uk, p)

uk_to_bk(uk, p)

Arguments

• bk: coefficients $b_{k, p}$ associated to the indexes 1:length(bk), a vector.
• p: integer giving the dimension of the ambient space $R^p$ that contains $S^{p-1}$.
• log: do operations in log scale (log-in, log-out)? Defaults to FALSE.
• signs: signs of the coefficients $u_{k, p}$, a vector of the same size as vk2 or bk, or a scalar. Defaults to 1.
• vk2: squared coefficients $v_{k, p}^2$ associated to the indexes 1:length(vk2), a vector.
• uk: coefficients $u_{k, p}$ associated to the indexes 1:length(uk), a vector.

Returns

The corresponding vectors of coefficients vk2, bk, or uk, depending on the call.

Description

Given a Sobolev statistic

for a sample c("$X_1, \\ldots, X_n \\in S^{p - 1} :=\n$", "${x \\in R^p : ||x|| = 1}$"), $p\ge 2$, three important sequences are related to $S_{n, p}$.

• Gegenbauer coefficients $\{b_{k, p}\}$ of $\psi_p$ (see, e.g., the projected-ecdf statistics ), given by

• Weights $\{v_{k, p}^2\}$ of the asymptotic distribution of the Sobolev statistic, $\sum_{k = 1}^\infty v_k^2 \chi^2_{d_{p, k}}$, given by

• Gegenbauer coefficients $\{u_{k, p}\}$ of the local projected alternative associated to $S_{n, p}$, given by

For $p = 2$, the factor $(1 + 2k / (p - 2))$ is replaced by $2$.

Details

See more details in Prentice (1978) and García-Portugués et al. (2023). The adequate signs of uk for the "PRt" Rothman test

can be retrieved with akx and sqr = TRUE, see the examples.

Examples

# bk, vk2, and uk for the PCvM test in p = 3
(bk <- Gegen_coefs_Pn(k = 1:5, type = "PCvM", p = 3))
(vk2 <- bk_to_vk2(bk = bk, p = 3))
(uk <- bk_to_uk(bk = bk, p = 3))

# vk2 is the same as
weights_dfs_Sobolev(K_max = 10, thre = 0, p = 3, type = "PCvM")$weights

# bk and uk for the Rothman test in p = 3, with adequate signs
t <- 1 / 3
(bk <- Gegen_coefs_Pn(k = 1:5, type = "PRt", p = 3, Rothman_t = t))
(ak <- akx(x = drop(q_proj_unif(t, p = 3)), p = 3, k = 1:5, sqr = TRUE))
(uk <- bk_to_uk(bk = bk, p = 3, signs = ak))

References

García-Portugués, E., Navarro-Esteban, P., Cuesta-Albertos, J. A. (2023) On a projection-based class of uniformity tests on the hypersphere. Bernoulli, 29(1):181--204. tools:::Rd_expr_doi("10.3150/21-BEJ1454") .

Prentice, M. J. (1978). On invariant tests of uniformity for directions and orientations. The Annals of Statistics, 6(1):169--176. tools:::Rd_expr_doi("10.1214/aos/1176344075")