PBGoF

PBGoF 0.2.0: Parametric Bootstrap Goodness-of-Fit Tests for Skew-Normal and Skew-t Distributions

Hongxiang Li, Chenglin Xu, and Tsung Fei Khang

PBGoF provides goodness-of-fit procedures, robust parameter fitting, and graphical diagnostics for univariate skew-normal and skew-t distributions. Version 0.1.0 focused on skew-normal models, including robust fitting, precomputed-quantile tests, and parametric-bootstrap KS and CvM tests. Version 0.2.0 extends the package to skew-t models through st.fit.robust(), skew-t parametric-bootstrap KS and CvM tests, and the unified para.bootstrap.test() interface, which allows users to select either the SN or ST model. It also adds st.plot.check() and the model-selectable cdf.plot.check() and QQ.plot() diagnostics for SN and ST models.

Installation

Install PBGoF 0.2.0 from a local source archive with:

install.packages("PBGoF_0.2.0.tar.gz", repos = NULL, type = "source")

The original PBGoF GitHub installation instructions remain applicable to PBGoF itself:

if (!"devtools" %in% rownames(installed.packages())) {
  install.packages("devtools")
}
devtools::install_github("Divo-Lee/PBGoF")

or with pak:

if (!"pak" %in% rownames(installed.packages())) {
  install.packages("pak")
}
pak::pkg_install("Divo-Lee/PBGoF")

Dependencies

PBGoF depends on the R packages sn and methods.

For skew-t models, PBGoF provides st.fit.robust(), para.bootstrap.test(), and st.plot.check().

Methods

Skew-normal model and composite null hypothesis

Let \(X_1,\ldots,X_n\) be an independent sample. Under the null hypothesis,

\[ H_0:\quad X_i \overset{\mathrm{iid}}{\sim} \mathrm{SN}(\xi,\omega,\alpha), \qquad \xi\in\mathbb{R},\quad \omega>0,\quad \alpha\in\mathbb{R}. \]

Writing \(z=(x-\xi)/\omega\), the skew-normal density is

\[ f_{\mathrm{SN}}(x;\xi,\omega,\alpha)=\frac{2}{\omega} \phi(z) \Phi(\alpha z), \]

where \(\phi\) and \(\Phi\) are the standard normal density and distribution functions. Because \(\boldsymbol\theta=(\xi,\omega,\alpha)^{\mathsf T}\) is estimated from the sample, this is a composite goodness-of-fit problem; the usual KS null distribution for a fully specified model is not applicable.

PBGoF estimates the model through sn.fit.robust(): MLE is attempted first, followed by increasingly stabilized penalized fits. A fit is accepted only if the estimates and standard errors are finite and the scale is positive.

Skew-t model

The skew-t family extends the skew-normal model by adding a positive degrees-of-freedom parameter \(\nu\). Its direct parameterization is \((\xi,\omega,\alpha,\nu)\). The extra parameter allows asymmetry and heavy tails to be represented together; as \(\nu\) becomes large, the skew-t model approaches the skew-normal model.

st.fit.robust() first attempts ordinary MLE and then uses Q-penalized MPLE as a fallback. Its DP output contains xi, omega, alpha, nu and their standard errors. CP output is available when the corresponding moments exist.

EDF statistics with estimated parameters

For order statistics \(X_{(1)}\leq\cdots\leq X_{(n)}\), define

\[ U_{(i)}=F_{\mathrm{SN}} \left(X_{(i)};\widehat{\boldsymbol\theta}\right), \qquad i=1,\ldots,n. \]

The two-sided Kolmogorov-Smirnov discrepancy and the statistic used by PBGoF are

\[ D_n=\max_{1\leq i\leq n}\left(\frac{i}{n}-U_{(i)},\;U_{(i)}-\frac{i-1}{n}\right), \qquad T_{\mathrm{KS}}=\sqrt{n}\,D_n. \]

The Cramer-von Mises statistic is

\[ W_n^2=\frac{1}{12n}+\sum_{i=1}^{n}\left[U_{(i)}-\frac{2i-1}{2n}\right]^2. \]

The parametric-bootstrap CvM test uses \(W_n^2\). Within the tabulated sample-size range, the precomputed-quantile CvM test uses

\[ T_{\mathrm{CvM}}=\sqrt{n}\,W_n^2. \]

Parametric-bootstrap calibration

The unified function para.bootstrap.test() selects the model through model = "SN" or model = "ST", and selects the statistic through statistic = "KS" or statistic = "CvM". For example, para.bootstrap.test(x, model = "SN", statistic = "KS") and para.bootstrap.test(x, model = "ST", statistic = "CvM") calibrate the corresponding tests for an estimated model. The previous model- and statistic-specific functions remain available for compatibility. Both model families use the same full-estimation bootstrap principle:

1. Fit the observed sample and compute \(T_{\mathrm{obs}}\).
2. Generate \(B\) samples of size \(n\) from the fitted SN or skew-t model.
3. Re-estimate all parameters independently in each bootstrap sample.
4. Compute the same statistic with that sample’s fitted parameters.

For \(B_{\mathrm{valid}}\) finite bootstrap statistics, PBGoF uses

\[ \widehat p_{\mathrm{boot}}=\frac{1+\displaystyle\sum_{b=1}^{B_{\mathrm{valid}}}\mathbf{1} \left(T_b^*\geq T_{\mathrm{obs}}\right)}{B_{\mathrm{valid}}+1}. \]

Failed fits are excluded and reported. Re-estimation in every replicate calibrates the statistic for the composite null rather than incorrectly treating the fitted distribution as fixed. For skew-t, all four parameters, including \(\nu\), are re-estimated in every valid replicate.

The precomputed skew-normal tests remain available through PBGoF_ks_test() and PBGoF_cvm_test().

Precomputed-quantile calibration for skew-normal data

PBGoF_ks_test() and PBGoF_cvm_test() use tables generated from 100,000 Monte Carlo replicates per available sample-size and centered-skewness combination. The data are fitted in two parameterizations:

• DP \((\xi,\omega,\alpha)\) evaluates the fitted CDF and EDF statistic.
• CP \((\mu,\sigma,\gamma_1)\) matches skewness to the simulation table.

The lookup value is

\[ \gamma_{1,\mathrm{used}}=\min \left(0.99,\max \left[0.01,\mathrm{round} \left(\left|\widehat\gamma_1\right|,2\right)\right]\right). \]

The absolute value follows reflection symmetry: if \(X\sim\mathrm{SN}(\xi,\omega,\alpha)\), then \(-X\) has shape \(-\alpha\). The signs of \(\alpha\) and \(\gamma_1\) reverse, but the null distributions of the reflection-invariant EDF statistics do not. Therefore, estimated skewness values with the same absolute magnitude but opposite signs use the same reference-table row. PBGoF retains the signed gamma1_hat and returns the non-negative lookup value as gamma1_used.

The bundled tables cover sample sizes through 500. For \(n>500\), all observations remain in the fit and EDF, while the \(n=500\) row is used for table lookup. The KS test compares \(\sqrt{n}D_n\) with the stored \(n=500\) quantiles. For CvM, the stored \(n=500\) quantiles are divided by \(\sqrt{500}\) and compared with the observed \(W_n^2\). This follows the approximation of using the \(n=500\) critical values for larger samples.

Let \(q_p^{\dagger}\) denote the selected quantiles expressed on the same scale as the observed statistic; for CvM with \(n>500\), \(q_p^{\dagger}=q_p/\sqrt{500}\). The table-based p-value is

\[ p^{\*}=\min_{q_p^{\dagger}\geq T_{\mathrm{obs}}}p,\qquad \widehat p_{\mathrm{table}}=1-p^{\*}. \]

The probability grid is 0.01 to 0.99 in increments of 0.01, so the result is a conservative step-function approximation. PBGoF does not interpolate across sample size or skewness.

Graphical checks and interpretation

A small p-value is evidence against the fitted model. A large p-value does not prove skew-normality or skew-t adequacy; it means that the test did not detect a departure at the available sample size and calibration resolution. Use sn.plot.check() for SN fits and st.plot.check() for skew-t fits alongside the corresponding formal tests. The precomputed-quantile functions remain specific to the skew-normal family.

References

• Azzalini, A. (1985). A class of distributions which includes the normal ones. Scandinavian Journal of Statistics, 12(2), 171-178.
• Azzalini, A., and Capitanio, A. (2014). The Skew-Normal and Related Families. Cambridge University Press.
• Babu, G. J., and Rao, C. R. (2004). Goodness-of-fit tests when parameters are estimated. Sankhya, 66, 63-74.
• Mateu-Figueras, G., Puig, P., and Pewsey, A. (2007). Goodness-of-fit tests for the skew-normal distribution when the parameters are estimated from the data. Communications in Statistics-Theory and Methods, 36(9), 1735-1755. https://doi.org/10.1080/03610920601126217