This function performs highly intriguing test for normality using L-skew ($\tau_3$) and L-kurtosis ($\tau_4$) computed from an input vector of data. The test is simultaneously focused on L-skew and L-kurtosis. Harri and Coble (2011) presented two types of normality tests based on these two L-moment ratios. Their first test is dubbed the $\tau_3\tau_4$ test. Those authors however conclude that a second test dubbed the $\tau^2_{3,4}$ test in particular shows consistently high power against [sic] symmetric distributions and also against [sic] skewed distributions and is a powerful test that can be applied against a variety of distributions.A sample-size transformed quantity of the sample L-skew ($\hat\tau_3$) is
$$Z(\tau_3) = \hat\tau_3 \times \frac{1}{\sqrt{0.1866/n + 0.8/n^2}}\mathrm{,}$$
which has an approximate standard normal distribution. A sample-sized transformation of the sample L-kurtosis ($\hat\tau_4$) is
$$Z(\tau_4)' = \hat\tau_4 \times \frac{1}{\sqrt{0.0883/n}}\mathrm{,}$$
which also has an approximate standard normal distribution. A superior approximation to the standard normal distribution however is
$$Z(\tau_4) = \hat\tau_4 \times \frac{1}{\sqrt{0.0883/n + 0.68/n^2 + 4.9/n^3}}\mathrm{,}$$
and this expression is highly preferred for the algorithms in the tau34sq.normtest() function.
The $\tau_3\tau_4$ (not implemented in tau34sq.normtest()) test by Harri and Coble (2011) can be constructed from the $Z(\tau_3)$ and $Z(\tau_4)$ statistics as shown, and a square rejection region constructed on an L-moment ratio diagram of L-skew versus L-kurtosis. However, the preferred method is the Tau34-squared test $\tau^2_{3,4}$ that can be developed by expressing an ellipse on the L-moment ratio diagram of L-skew versus L-kurtosis. The $\tau^2_{3,4}$ test statistic is defined as
$$\tau^2_{3,4} = Z(\tau_3)^2 + Z(\tau_4)^2\mathrm{,}$$
which is approximately distributed as a $\chi^2$ distribution with two degrees of freedom. The $\tau^2_{3,4}$ also is the expression of the ellipical region on the L-moment ratio diagram of L-skew versus L-kurtosis.