**Abstract:** |
Let ,..., be a discrete probability distribution; all >0 and =1. Denote ,..., the corresponding frequencies in a sample of size n and consider the vector Yn of components of chi-square statistic , i= 1,...,m. As n 1 the vector has the limit distribution of 0-mean Gaussian vector Y= such that Y=X- is a vector of independent (0;1) random variables, , and (a,b) denotes inner product of vectors a and b. The distribution of Y depends on -it is only its sum of squares (Y,Y) = ; which is chi-square distributed and hence has distribution free from . It is for this reason that we do not have any other asymptotically distribution free goodness of fit test for the discrete distributions but the chi-square statistic. In this presentation we introduce a vector = as follows: let be the unit length "diagonal" vector with all coordinates 1/, and put =-. Then (i) statistic based on is asymptotically distribution free; (ii)asymptotically, the partial sums , k m; will behave as a discrete time analog of the standard Brownian bridge; (iii) the transformation from to is one-to-one. We will explain the nature of transformation of to , give different form of such transformations and then show that it produces new results also for continuous distributions in and for both simple and parametric hypothesis. Therefore, a unified approach to distribution free goodness of fit testing for continuous and discrete distirbutions in is now possible. This talk is based on Khmaladze, E.V., Note on distribution free testing for discrete distributions, Ann. Stat. 2013, 2979-2993. |