I've come across two different forms of a skewness-adjusted t-statistic, which was developed originally by Johnson (1978): $$J = t + \frac{gt^2}{3n} + \frac{g}{6n}$$

and $$J = t + \frac{gt^2}{3\sqrt{n}} + \frac{g}{6\sqrt{n}},$$

where $t$ is the conventional t-statistic, $n$ is the number of observations, and $g$ is the skewness estimate. The null hypothesis is zero mean.

Could you advise me what's difference between the two forms?

Many thanks, Dave

-
Hey Dave, and welcome to the site. I have fixed the formatting of your question, and just want to let you know that equation can be typed within \$\$ or \$\$ \$\$ to enable TeX. – Stefan Hansen Oct 30 '12 at 16:31
wow, excellent functionality! thanks Stefan. – Dave Oct 30 '12 at 16:32
Can you point to the sources where you found this, particularly the definition of $g$? I found Johnson's original paper, but his form doesn't match either of those given, as far as I can tell. – Jonathan Christensen Oct 30 '12 at 18:21
Hi Jonathan, $g$ is defined as follows: $$g = \frac{\sum_{i=1}^n (x_i-\overline{x})^3}{n\sigma(x)^3}$$ As far as I can understand Johnson's statistic given in his equation (2.5) is equivalent to the second form I have written. So $g$ corresponds to $\mu_3$ in his equation, and $\mu=0$. – Dave Oct 30 '12 at 18:36
The issue I'm having is that when $\overline{x}$ is very negative, and the data is highly skewed ($g$), the second form yields an adjusted t-statistic that is positive, while the conventional t-statistic is negative. This doesn't happen with the first form (the adjustment is smaller). – Dave Oct 30 '12 at 18:44