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. Dec 4, 2017 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, 2017 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 $n$. Dec 5, 2017 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. Dec 6, 2017 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
$$MSK=MS+|MK|^2$$