How to get a single p-value from the two p-values of a Mardia's multinormality test

I am performing a Mardia's test to assess the multinormality of a multivariate distribution.

Mardia's test actually perform two tests, one on the skewness and another on the kurtosis, thus yielding two p-values.

Is there a valid way to combine those two p-values into one so that results are easier to share?

I've read about Fisher's method to combine several p-values, but this method assumes independence of the p-values, which is surely not the case here. Is it somehow still valid here?

• The two statistics are not independent; asymptotically they will be but you will require very large sample sizes before it's reasonable to treat them as close to independent. Fisher's method relies on independence. – Glen_b Dec 4 '17 at 23:36
• Thanks @Glen_b! It's what I understood from the link I give in the OP. do you know of any method that might apply in my case? – jrjd Dec 5 '17 at 8:07
• Well, you could (a) investigate the dependence in the null case, or (ii) do a each test at the $\alpha/2$ level (sometimes called a "rectangle test"; I wonder if that term is maybe in Bowman and SHenton's paper on the univariate case) -- though both the approximations to the components offered there are also large sample approximations that may not be so great if you don't have quite large $n$. – Glen_b Dec 5 '17 at 22:34
• If you want an omnibus (factotum, portmanteau) test that by Doornik and Hansen seems superior. (Looking at the data carefully could be even better. – Nick Cox Dec 6 '17 at 19:49

Two researchers did exactly what you are asking back in 1973 [ref 1]. If the skewness statistic is $$MS$$ and the kurtosis statistic is $$MK$$, the combined Mardia statistic is
$$MSK=MS+|MK|^2$$