# Distribution of sample scatter matrix (mean-corrected SSCP matrix) in case of missingness (MCAR)/unbalanced design

The sample scatter matrix (or mean-corrected sum of squares and cross products [SSCP] matrix) $$S$$ of multivariate normally distributed vectors $$\boldsymbol x_{j}$$ ($$j=1,\ldots,n)$$ each of size $$m$$ (e.g., https://en.wikipedia.org/wiki/Scatter_matrix) has a Wishart distribution:

$$S\sim \mathcal{W}_m(\Sigma,n-1)$$

with $$(n-1)$$ degrees of freedom and scale matrix $$\Sigma$$ (e.g., Pham-Gia & Choulakian, 2014, p. 331).

What is the distribution of $$S$$ (and how can $$S$$ be calculated) if the vectors $$\boldsymbol x_{j}$$ contain missing values (under the assumption that the values are missing completely at random, MCAR)?

References:

Pham-Gia, T., & Choulakian, V. (2014). Distribution of the sample correlation matrix and applications. Open Journal of Statistics, 4, 330–344. http://www.dx.doi.org/10.4236/ojs.2014.45033