It is well known that $X/(X+Y)$ has the beta distribution when $X$ and $Y$ follow the Dirichlet distribution. Linear combinations of the form $\alpha X+\beta Y$ have also been studied in Provost and Cheong [S. B. Provost and Y.-H. Cheong: On the distribution of linear combinations of the components of a Dirichlet random vector. Canad. J. Statist. 28 (2000)]. In this paper, we derive the exact distribution of the product $P=XY$ (involving the Gauss hypergeometric function) and the corresponding moment properties. We also propose...

We give a stochastic expansion for estimates $\widehat{\theta }$ that minimise the arithmetic mean of (typically independent) random functions of a known parameterθ. Examples include least squares estimates, maximum likelihood estimates and more generally M-estimates. This is used to obtain leading cumulant coefficients of $\widehat{\theta }$ needed for the Edgeworth expansions for the distribution and densityn1/2θ0) to magnitude n−3/2 (or to n−2 for the symmetric case), where θ0 is the true parameter value and n is typically the...

We give a stochastic expansion for estimates that minimise the arithmetic mean of (typically independent) random functions of a known parameter θ. Examples include least squares estimates, maximum likelihood estimates and more generally M-estimates. This is used to obtain leading cumulant coefficients of needed for the Edgeworth expansions for the distribution and density n1/2 ( of − θ0) to magnitude n−3/2 (or to n−2 for the symmetric case), where θ0 is the true parameter value and n is typically...