GeoTestsupp_space: Parametric bootstrap test for spatial support (scale)

Description

Performs a parametric bootstrap likelihood-ratio test on the spatial scale parameter using the compactly supported GenWend correlation model. Two hypotheses are compared: H0 imposes a maximum admissible scale (or independence), while H1 estimates scale freely. The test statistic is the likelihood ratio and the p-value is obtained by parametric bootstrap under H0.

Usage

GeoTestsupp_space(data, coordx,
                  start, fixed,
                  model = "Gaussian",
                  h0 = NULL,
                  optimizer = "bobyqa",
                  lower = NULL, upper = NULL,
                  neighb = 5,
                  B = 1000,
                  likelihood = NULL,
                  type = NULL,
                  method = "Cholesky",
                  parallel = TRUE,
                  ncores = NULL,
                  progress = TRUE)

Value

An (invisible) list with components:

d_min: Minimum inter-point distance $d_{min}$.
h0: Threshold actually used under H0.
lambda_obs: Observed likelihood-ratio statistic $\Lambda_{obs}$.
pvalue: Bootstrap p-value.
B_rep: Vector of bootstrap statistics $\Lambda_b$ for valid replications.
fit_H0: GeoFit object fitted under H0.
fit_H1: GeoFit object fitted under H1.

Arguments

data: Numeric vector of observations of length n.
coordx: n x d matrix with spatial coordinates (at least 2 columns).
start: Named list of parameters to be estimated. Names must match CorrParam(corrmodel) and NuisParam(model).
fixed: Named list of parameters kept fixed. Together with start it must exactly cover the required set.
model: Marginal model for the data (default "Gaussian").
h0: Positive threshold for the scale parameter under H0. If NULL (default), H0 represents spatial independence with scale = h0_val set to the (adjusted) minimum inter-point distance.
optimizer: Optimization method passed to GeoFit (e.g., "bobyqa").
lower, upper: Named lists with bounds for parameters in start only. NULL means no extra bounds beyond defaults. Bounds are internally widened if too narrow.
neighb: Number of neighbors for composite likelihood when applicable. Ignored when the full likelihood is used.
B: Number of parametric bootstrap replications (default 1000).
likelihood: Type of (composite) likelihood to pass to GeoFit (e.g., "Marginal").
type: Composite-likelihood type (e.g., "Pairwise").
method: Simulation method for bootstrap under H0: "Cholesky" or "TB" (turning bands via GeoSimapprox).
parallel: If TRUE, bootstrap estimations are parallelized using future.
ncores: Number of cores for parallelization. If NULL, it is selected automatically up to B or parallel::detectCores()-1.
progress: If TRUE, progress bars are shown via progressr.

Author

Moreno Bevilacqua moreno.bevilacqua89@gmail.com https://sites.google.com/view/moreno-bevilacqua/home
Víctor Morales Oñate victor.morales@uv.cl https://sites.google.com/site/moralesonatevictor/
Christian Caamaño-Carrillo chcaaman@ubiobio.cl https://www.researchgate.net/profile/Christian-Caamano

Details

The function fixes the correlation model to corrmodel = "GenWend" (generalized Wendland, compact support). Input validation ensures consistency of parameter names between start and fixed w.r.t. CorrParam(corrmodel) and NuisParam(model).

Automatic likelihood selection. If $n > 10{,}000$, a composite-likelihood is used (likelihood = "Marginal", type = "Pairwise") with neighb neighbors; otherwise the full likelihood is used (likelihood = "Full", type = "Standard").

Threshold h0 and minimum distance. Let $d_{min}$ be the minimum inter-point distance, computed with nabor. If h0 = NULL, the test defaults to independence with h0\_val = max(1e-6, d_min - 1e-6). Under H0 one enforces scale <= h0 (or scale = h0\_val for independence), while under H1 the scale is estimated freely but capped above by a large, data-driven U\_big.

Likelihood-ratio statistic. Let $\Lambda_{obs} = \max\{0, 2(\ell_1 - \ell_0)\}$ where $\ell_0$ and $\ell_1$ are the log-(composite) likelihoods under H0 and H1. The null distribution of $\Lambda$ is approximated via parametric bootstrap with B datasets simulated under H0; the p-value is computed as $$(1 + \#\{\Lambda_b \ge \Lambda_{obs}\}) / (1 + B_{valid}).$$

Early exit. If the unconstrained fit under H1 already satisfies scale <= h0 (or <= h0\_val), the function returns lambda\_obs = 0 and pvalue = 1 without running the bootstrap.