SKNv0.2.0

Bootstrap tests for symmetry, kurtosis, and normality under dependence.

DOC 02 / 06Implementation guide

Test targets and the autoregressive sieve

The moment conditions, autoregressive-sieve construction, and fixed-order nested bootstrap used by the package.

Targets

Four related checks

  1. Sample skewness

    Tests the symmetry condition \(\beta_1=0\).

  2. Robust skewness

    Reports an additional symmetry check based on the package's robust skewness measure.

  3. Mesokurtosis

    Tests the condition \(\beta_2=3\).

  4. Normality

    Combines the skewness and kurtosis moment conditions; satisfying these moments alone does not characterize every distribution as Gaussian.

AR sieve

Dependence is reproduced through an autoregression

The implementation selects an autoregressive lag order with AIC or BIC over candidate orders up to

\[p_{\max}=\left\lceil\frac{(\log n)^2}{2}\right\rceil.\]

It then fits the selected autoregressive model and generates bootstrap series recursively. Symmetry is imposed through a symmetrized residual distribution; Gaussian innovations impose the kurtosis and normality null conditions.

\[X_t = c + \sum_{j=1}^{p} \phi_j X_{t-j} + \varepsilon_t.\]

The lag order p is selected from the observed input series. In the studentized path, the corresponding outer and inner bootstrap calculations retain that order while re-estimating the autoregressive coefficients.

Statistics

Studentized moment statistics

Let \(\widehat\beta_1\) and \(\widehat\beta_2\) denote sample skewness and kurtosis, with bootstrap standard errors \(\widehat\omega_1\) and \(\widehat\omega_2\).

\[S_n=\frac{\widehat\beta_1}{\widehat\omega_1},\qquad K_n=\frac{\widehat\beta_2-3}{\widehat\omega_2},\qquad G_n=S_n^2+K_n^2.\]

Large absolute skewness or kurtosis statistics, and large values of the joint statistic, provide evidence against the relevant null.