perform_GoF_test {BootstrapTests}R Documentation

Perform a univariate goodness-of-fit (GoF) hypothesis test via bootstrap resampling

Description

This function performs a bootstrap goodness-of-fit hypothesis test for a specific univariate parametric family. The null hypothesis corresponds to the sample coming from the specified parametric family, while the alternative hypothesis corresponds to the sample not coming from the specified parametric family. This function implements a parametric bootstrap and a non-parametric bootstrap. The test statistic is the Kolmogorov-Smirnov test statistic. To estimate the parameters of the parametric family, either a minimum distance estimator, or a MLE estimator (the sample mean and variance) is used. On the bootstrap sample, we have also implemented a centered MD estimator, as in the paper. For now, only a test of normality is implemented. This function gives the corresponding p-values, the true test statistic and the bootstrap-version test statistics. The default (and valid) method implemented in this function is the parametric bootstrap, together with the equivalent test statistic and the MLE parameter estimator. Via the bootstrapOptions argument, the user can specify other bootstrap resampling schemes, test statistics, and parameter estimators.

Usage

perform_GoF_test(
  X_data,
  parametric_fam = "normal",
  nBootstrap = 100,
  mygrid = NULL,
  show_progress = TRUE,
  bootstrapOptions = NULL,
  verbose = 0
)

Arguments

X_data

numerical input vector. Perform a GoF test whether or not this sample comes from "parametric_fam", a specified parametric distribution.

parametric_fam

name of the parametric family. For the moment, only "normal" is supported.

nBootstrap

numeric value of the number of bootstrap resamples. Defaults to 100.

mygrid

description of the grid used to compute the CDFs on. This must be one of

  • NULL: a regularly spaced grid from the minimum value to the maximum value with 100 points is used. This is the default.

  • A numeric of size 1. This is used at the length of the grid, replacing 100 in the above explanation.

  • A numeric vector of size larger than 1. This is directly used as the grid.

show_progress

logical value indicating whether to show a progress bar

bootstrapOptions

This can be one of

  • NULL. This uses the default options type_boot = "param", type_stat = "eq" and type_estimator_bootstrap = "MLE".

  • a list with at most 3 elements named:

    • type_boot type of bootstrap resampling scheme. It must be one of

      • "param" for the parametric bootstrap (i.e. under the null). This is the default.

      • "NP" for the non-parametric bootstrap (i.e. n out of n bootstrap).

    • type_stat type of test statistic to be used. It must be one of

      • "eq" for the equivalent test statistic T_n^* = \sqrt{n} || \hat{F}^* - F_{\hat\theta^*} ||

      • "cent" for the centered test statistic T_n^* = \sqrt{n} || \hat{F}^* - \hat{F} + F_{\hat\theta} - F_{\hat\theta^*} ||

      For each type_boot there is only one valid choice of type_stat to be made. If type_stat is not specified, the valid choice is automatically used.

    • type_estimator_bootstrap: the bootstrap parameter estimator to be used. It must be one of:

      • "MLE" for the MLE estimator (for the normal distribution, this corresponds to the usual empirical mean and variance).

        This is always a valid choice in the case that the combination (type_boot, type_stat) is valid (as defined above). Therefore, this is the default option. It is also the fastest type of estimator.

      • "MD-eq" for the Minimum Distance estimator. This is a valid choice if and only if type_stat = "eq". It is necessary in this case to use an equivalent bootstrap estimator to match the equivalent bootstrap test statistic. This bootstrap parameter estimator is given as: \theta_n^{*,MD}=\arg\min_{\theta} || \hat{F}^* - F_{\theta} ||

      • "MD-cent" for the centered Minimum Distance estimator. This is a valid choice if and only if type_stat = "cent". It is necessary in this case to perform a centering on the bootstrap estimator to match the centered bootstrap test statistic. This bootstrap parameter estimator is given as: \theta_n^{*,MD, cent}=\arg\min_{\theta} || \hat{F}^* - F_{\theta}- \hat{F} + F_{\hat\theta} ||

  • "all" this gives test results for all theoretically valid combinations of bootstrap resampling schemes.

  • "all and also invalid" this gives test results for all possible combinations of bootstrap resampling schemes and test statistics, including invalid ones.

A warning is raised if the given combination of type_boot, type_stat, and type_estimator_bootstrap is theoretically invalid.

verbose

If verbose = 0, this function is silent and does not print anything. Increasing values of verbose print more details about the progress of the computations.

Value

A class object with components

References

Derumigny, A., Galanis, M., Schipper, W., & van der Vaart, A. (2025). Bootstrapping not under the null? ArXiv preprint, doi:10.48550/arXiv.2512.10546

See Also

perform_regression_test,perform_independence_test. The print and plot methods, such as plot.bootstrapTest.

Examples

n <- 100
# Under H1
X_data <- rgamma(n,2,3)
result <- perform_GoF_test(X_data,
                          nBootstrap = 100,
                          bootstrapOptions = list(type_boot = "param",
                                                  type_stat = "eq",
                                                  type_estimator_bootstrap = "MLE")
                         )
print(result)
plot(result)

# Under H0
X_data <- rnorm(n)
result <- perform_GoF_test(X_data, nBootstrap = 100)
print(result)
plot(result)
[Package BootstrapTests version 0.1.0 Index]