A COMPREHENSIVC GENERALIZED MEAN VALUE BASED ON THE DIRICHLET DISTRIBUTION

Abstract

The Hardy, Littlewood, Polya class of power means (w_iz^a ₁ + …+w_n z^a _n)^1/a, including the usual harmonic, geometric, and arithmetic mean values, has been generalized by Bruno deFinetti and B. C. Carlson. These two generalizations are here simultaneously extended to a comprehensive generalized mean value involving an arbitrary continuous strictly monotonic function and a linear form in the data values with Dirichlet-distributed coefficients. Properties are given which relate the new mean naturally to its deFinetti and Carlson subclasses. Statistical interpretations and possible further extensions arc discussed.