Estimation and tests of the discrete probability law based on the empirical generating function, (two dimensional case).
Jamal-Dine Chergui · Hispana · 1994
A large portion of statistical literature pertains to the theory and application of nonparametric methods of inference.This work presents a new approach to some well-known statistical problems based on observation of stochastic process.The recent development of the theory of probability allows us to consider these observations as ones of random variables with values in an infinite dimensional vector space.This paper will be devoted to estimation and test by means of empirical generating function G^.We use a similar procedure to that of CRAMER-VON MISES for various hypotheses testing problems.m and let IP^ ""^ ^(x >;) ^^ ^^^ corresponding empirical measure.G^ its generating function.We denote by: *r = [o,i]x[o,i] *C = C{T) be the separable Banach space of all continuous functions endowed with a norm ||/| = sup|/(5',i)|.B{C) its Borel o -field.T *M = M{T) the space of all bounded measures defined on {T,B{T))*C and M are paraid spaces, by the pairing functional = 'v'dii(u,vy,(k,l)e'N\iieM . JT^m"In order to be more explicit and so that the subject can be accessible to anyone wishing to read it let us list some important concepts and results that are necessary to prove several results stated in laters parts.They are moreover presented in perspective with the historical development and strong interaction between Probability theory and Analysis.