Bootstrap-Calibrated Goodness-of-Fit Test for Simplex Regression

Performs the $U_n$ local-influence goodness-of-fit test for a simplex regression model. The null distribution is calibrated via a parametric bootstrap, which provides accurate size control even in finite samples.

Usage

simplex_gof(
  y,
  X,
  Z = NULL,
  B = 1000,
  alpha = c(0.01, 0.05, 0.1),
  ncores = 1,
  seed = NULL,
  verbose = TRUE
)

Arguments

y: Numeric vector of responses in (0, 1).
X: Design matrix for the mean sub-model (n x p, including intercept).
Z: Design matrix for the dispersion sub-model (n x q, including intercept). If NULL, a constant-dispersion model is fitted.
B: Integer; number of bootstrap replicates. Default 1000.
alpha: Numeric vector of significance levels. Default c(0.01, 0.05, 0.10).
ncores: Integer; number of parallel workers. Default 1 (sequential). Set to NULL to use all available cores minus 1.
seed: Integer random seed for reproducibility. Default NULL.
verbose: Logical; whether to print progress. Default TRUE.

Value

An object of class "simplexgof" with components:

fit: The "simplexfit" object for the original data.
diag: The simplex_diag() output for the original data.
Un: Observed test statistic.
Tn: Observed $T_n$.
Un_boot: Numeric vector of B bootstrap $U_n^*$ values.
results: Data frame summarising decisions at each level.
B, alpha: As input.

Details

The test statistic is $$U_n = \frac{\sqrt{n}\bigl[\sum_{t=1}^n C_{I_t} - 2(p+q)\bigr]}{s_{k,c}}$$ where $C_{I_t} = 2|\Delta_t^\top (-\ddot\ell)^{-1} \Delta_t|$ is the total local influence of observation $t$ under case-weight perturbation, and $s_{k,c}^2$ is the sample variance of $\{k(y_t;\hat\theta)\}$.

Because the normal approximation is severely liberal for the simplex class (empirical size 3–7x nominal even at $n = 1000$), critical values from the parametric bootstrap are preferred. Asymptotic $N(0,1)$ critical values are also reported for comparison.

References

Espinheira P.L., Silva F.C., Barros M., Ospina R. (2026). A Bootstrap-Calibrated Local Influence Goodness-of-Fit Procedure for Simplex Regression Models.

Zhu H., Zhang H. (2004). A diagnostic procedure based on local influence. Biometrika, 91(3), 579–589.

Examples

# \donttest{
data(ammonia)
X <- cbind(1, ammonia$corr_ar, ammonia$temp_agua,
           ammonia$corr_ar * ammonia$temp_agua)
Z <- cbind(1, ammonia$temp_agua,
           ammonia$corr_ar * ammonia$temp_agua)
set.seed(42)
gof <- simplex_gof(ammonia$perda, X, Z, B = 200)
#> =============================================================
#>   simplexgof: Bootstrap U_n Test for Simplex Regression
#> =============================================================
#>   n = 21, p = 4, q = 3, B = 200
#> 
#> Fitting original model...
#> 
#> Model estimates:
#> 
#> Simplex Regression  (n = 21 ; p = 4 ; q = 3 )
#> 
#>        Estimate Std.Error z.value      Pr
#> beta1  -12.9893    2.1038 -6.1742 < 0.001
#> beta2    0.1312    0.0363  3.6140 < 0.001
#> beta3    0.2705    0.1024  2.6408 0.00827
#> beta4   -0.0037    0.0017 -2.1473 0.03177
#> gamma1   3.8342    3.3908  1.1308 0.25815
#> gamma2  -0.4454    0.2882 -1.5456 0.12219
#> gamma3   0.0044    0.0024  1.8791 0.06024
#> 
#> Log-likelihood: 100.4159  |  converged: TRUE
#> 
#> mu: min = 0.0075, mean = 0.0181, max = 0.0408
#> Tn = 8.0447
#> Un = 0.0298
#> 
#> Starting 200 bootstrap replicates...
#>   50 / 200 done
#>   100 / 200 done
#>   150 / 200 done
#>   200 / 200 done
#> 
#> === RESULT: Un = 0.0298 ===
#> 
#> Bootstrap critical values:
#>  alpha boot_lo boot_hi    decision_boot
#>     1% -1.4125  0.0511 Do not reject H0
#>     5% -0.8367  0.0368 Do not reject H0
#>    10% -0.6685  0.0301 Do not reject H0
#> 
#> Asymptotic N(0,1) critical values:
#>  alpha norm_lo norm_hi    decision_norm
#>     1% -2.5758  2.5758 Do not reject H0
#>     5% -1.9600  1.9600 Do not reject H0
#>    10% -1.6449  1.6449 Do not reject H0
#> 
print(gof)
#> simplexgof: U_n = 0.0298  (Tn = 8.0447, B = 200)
#> 
#>  alpha boot_lo boot_hi    decision_boot norm_lo norm_hi    decision_norm
#>     1% -1.4125  0.0511 Do not reject H0 -2.5758  2.5758 Do not reject H0
#>     5% -0.8367  0.0368 Do not reject H0 -1.9600  1.9600 Do not reject H0
#>    10% -0.6685  0.0301 Do not reject H0 -1.6449  1.6449 Do not reject H0
# }