On the asymptotic distribution of the 'natural' estimator of Cronbach's alpha with standardised variates under nonnormality, ellipticity and normality

Following van Zyl, Neudecker and Nel (2000) we consider asymptotic properties of the 'natural' estimator of Cronbach's alpha when the variates are standardised. This means that the population correlation matrix P is the population variance matrix E, because now all diagonal elements of E are equal to unity. The 'natural' estimator (alpha) over cap (s) = (p-1)(-1)p[1-p(1'R1)(-1)], where R is the sample correlation matrix and p is the number of items (responses). We find the asymptotic distribution of (alpha) over cap (s) under nonnormality, ellipticity and normality. Use is made of a (0,1) 'duplication' matrix (D) over cap. This enables us to switch easily between vecA and w(A), where A is a symmetric zero-axial matrix (Ad = 0) and w(A) contains the 'free' elements of A.