THE EXACT DISTRIBUTION OF INDEFINITE QUADRATIC FORMS IN NONCENTRAL NORMAL VECTORS

SERGE B. PROVOST AND EDMUND M. RUDIUK

Department of Statistical and Actuarial Sciences, The University of Western Ontario, London, Ontario, Canada N6A 5B7

(Received March 29, 1994; revised August 7, 1995)

Abstract. The exact density of the difference of two linear combinations of independent noncentral chi-square variables is obtained in terms of Whittaker’s function and expressed in closed forms. Two distinct representations are required in order to cover all the possible cases. The corresponding expressions for the exact distribution function are also given.

Key words and phrases: Exact distribution function, exact density function, indefinite quadratic forms, noncentral chi-square variables, singular normal vectors, Whittaker’s function.